Why 92 Heads in a Row Is Not Not Surprising

Estimated read time (minus contemplative pauses): 111 min.

1. Introduction

In his 2017 essay, “Why Throwing 92 Heads in a Row Is Not Surprising,”¹ philosopher Martin Smith argues that it is irrational—more on this qualifier shortly—to feel surprise at seeing 92 Heads in a row. He grounds the claim on what I’ll organize here into roughly three arguments.

The first argument he labels the conjunction principle, which states that if multiple independent events are individually unsurprising, then those events are collectively unsurprising. The second is a commonly cited argument that serves as a fundamental source of tension in other articles I’ve written on this topic (more about which below); I’ll call this the equiprobable principle: if two events have the same probability of occurring and one is unsurprising, then the other is also unsurprising. The third argument is broader and has to do with the relation between belief, surprise, and investigation; for convenience, I’ll term this the surprise function, more details about which soon.

I’ll address each of these arguments roughly in turn (i.e., with unavoidable overlap), beginning with the conjunction principle, which I find to be the least convincing of the three—though, to be clear, I will attempt to show that all three arguments fail.

1.1 An Introductory Note on Rationality

I only intend to discuss the term “rational” insomuch as I feel compelled to by the arguments at hand. It turns out that will be quite a lot, starting with these foundational comments.

I consider “rational” to roughly mean “consistent” in the sense of there being mutual alignment between one’s beliefs, attitudes, emotions, and so on. And I take the phrase “is rational” to mean something like “is warranted” or “is justified” for the person who is, or wishes to be, consistent in this way. “Irrational,” then, roughly means “inconsistent”.

This is in the vicinity of the usual, basic understanding of the term as it’s used by analytic philosophers, social psychologist, behavioral economists, decision theorists, and such—Smith included.

Precisely when and how to apply the terms may not always be clear, as we’ll soon see. Here’s a fairly clear example.

If you say you are not surprised on hearing that a coin tossed 100 times yielded precisely 20 Heads, but are surprised on hearing that the coin yielded precisely 80 Tails in those 100 tosses, then you are being inconsistent, as those two statements describe the exact same event. I claim that this is close to being irrational, or at least is not yet obviously there, though it is obviously inconsistent (to those who recognize the inconstancy).

For it to be obviously irrational, I think you must recognize that the two statements describe the same event. If, on recognition and reflection of that fact, you still say one of those outcomes is surprising and the other is not, then you are now being obviously irrational.

Now, it’s debatable whether you must be aware of the inconsistency for your misaligned responses to count as irrational. I claim you must be so aware, because it’s easy to concoct realistic examples where inconsistencies are difficult to spot; for example, when you find two mathematical properties obvious while not having yet noticed or convinced yourself that they jointly imply some other mathematical property you have rejected or at least find surprising.

I’ll address these and other nuances surrounding the application of the term “rational” as they arise.

Now, I’m all too familiar with, and not without sympathy for, the eye-rolls the term “rational” inspires in folks who seem to be allergic to it in contexts such as the present one. And explicitly qualifying the word as a term of art doesn’t help, and in fact might do more harm than good because such arguments are meant to describe (or, more formally, to model), and even prescribe (more about which later), the behavior and states of mind of people going about their daily, complicated lives.

When I wish to avoid such baggage, I’ll opt for less tainted near-synonyms like “sensible” and “reasonable” and “consistent.” These more colloquial characterizations also get me off the hook from the technocrats who’d like to tell me I’m using the word incorrectly. Really, I’d rather be irrational than conform to what some folks insist is rational. But this is part of what’s at issue here. Is rationality like democracy, wherein the genuineness of the thing is measured by the degree to which outcomes match my beliefs about the world? Or is this a more objectively grounded project than that? And then there is the awkward, and increasingly prevalent, notion that no human is truly rational. Perhaps, but I think most are rational enough to follow my above coin example, and others I’ll mention shortly. That shouldn’t be too much to expect.

It should be apparent, then, that my concerns are not with the meaning of “rational,” but rather with how we go about applying the term. To raise such concerns is of course nothing special—see, for example, Eliezer Yudkowsky’s article, “What Do We Mean By ‘Rationality’?,” which he prefaces with with two definitions of “rationality”:

Epistemic rationality: believing, and updating on evidence, so as to systematically improve the correspondence between your map and the territory. The art of obtaining beliefs that correspond to reality as closely as possible. This correspondence is commonly termed “truth” or “accuracy,” and we’re happy to call it that.

Instrumental rationality: achieving your values.  Not necessarily “your values” in the sense of being selfish values or unshared values: “your values” means anything you care about.  The art of choosing actions that steer the future toward outcomes ranked higher in your preferences.  On [Less Wrong] we sometimes refer to this as “winning.”

Not to mention his (nearly) 2,400-page book compiling, reorganizing, and to some degree cleaning up two years’ worth of blog posts on the subject: Rationality: From AI to Zombies (2015).

That is deeper than I intend to get into the subject here, though sensitivity to such nuances surrounding the subject are certainly working on me as I write this. Indeed, I’ve probably already given it too much attention, as Smith’s argument does not hinge on much more than an informal, ordinary understanding of the term—the title of Smith’s article, for example, loses nothing crucial by the word’s absence. In fact, he takes the word for granted, giving no ink at all to what he means by it, which is reasonable. But the term does feature into his argument to some degree, and it is to that extent that I will put its usage on the spot.

2. The Conjuction Principle

2.1 Defining the Conjuction Principle

Smith introduces the conjunction principle by way of Tom Stoppard’s 1966 play Rosencrantz and Guildenstern Are Dead. Guildenstern, bewildered at flipping 92 consecutive Heads with a fair coin, considers some farfetched explanations before landing on a mundane conclusion: because it was unsurprising that any given toss landed Heads, it is unsurprising that all of them did.

It’s not so clear that Guildenstern’s reasoning isn’t a distraught coping mechanism, and Stoppard is on record as saying that the event is meant to establish that, “throughout the play, the laws of probability aren’t working” (see this footnote for the full quote and citation.²). Of course, none of this makes Guildenstern wrong. I bring it up, however, to highlight the air of mystique that surrounds this sort of event. I’ll try to convince you that the mystique is warranted and that Stoppard chose the right sort of event to make his point.

Smith formalizes Guildenstern’s reasoning as the conjunction principle:

If it’s unsurprising for event e₁ to happen, and it’s unsurprising for event e₂ to happen, and these two events are independent of one another, then it’s unsurprising for e₁ and e₂ to both happen. (Smith 2)

Such events—i.e., events that are both independent and unsurprising—may be concatenated indefinitely under the principle:

If it’s unsurprising for the first two coins to land heads and it’s unsurprising for the third coin to land heads and these events are independent, then it’s unsurprising for the first three coins to land heads. And so on right up to 92—or even further if we wish. (Smith 2)

He cautiously describes this “fair reconstruction of Guildenstern’s reasoning” (Smith 2) as “a ‘proof’ that 92 heads in a row is not a surprising event” (Smith 3). He also characterizes it as a special case of Wolfgang Spohn’s law of conjunction (for negative ranks)³, which falls within the broader framework of ranking theory, a mathematical system applicable to surprise described by Spohn in the 1980s, and an improvement on attempts made in the 1950s and ‘60s by George Shackle ⁴ to develop a rigorous mathematical theory of surprise, Axiom 7 of which is similar to Smith’s conjunction principle.

Before challenging the conjunction principle, I’d like to draw attention to some important distinctions.

2.2 Possibility vs. Surprisingness (and Other Distinctions)

As noted above, I’ve written before on the subject of landing many consecutive Heads. Those articles can be found here:

“Anthropic Bias (Ch 2, Part 2): Fine-Tuning in Cosmology & 100 Heads in a Row” (17,351 words) Call this “Fine-Tuning” for short.

“Nassim Taleb’s Fat Tony Example / And: Is It Possible to Flip 100 Heads in a Row?” (4,683 words) Call this “Taleb’s Example” for short.

“Fine-Tuning” extends and adds to ideas I present in “Taleb’s Example.” I’ll borrow from and refer to these often here, especially “Fine-Tuning.”

In those essays, I challenge/defend my admittedly naive, but powerful, intuition that it is impossible for a fair coin to land 100 (much less a trillion) Heads in a row with/against the admittedly more theoretically, and maybe even rationally, sound arguments of the sort Smith presents here. However, while those writings are focused on possibility, the present one is focused on surprisingness.

For example, I consider something much like the conjunction principle, but with an eye towards possibility:

If you can get one heads, why not another? But if my intuition is right (and I’m not saying it is), this does not generalize. Because it’s not how a fair coin behaves. (Wallace, “Fine-Tuning,” Point 21)

The idea there, tempered by my naive intuition’s pushback, is that you can apply the rule to any two flips you like in a sequence of arbitrary length. (And, to be clear, this may involve a single coin or a different coin for every toss.) My naive intuition is also challenged with what I call here the equiprobable principle (addressed in Section 3, below): an n-Heads sequence has the same probability as any other n-results sequence.

I have yet to resolve these tensions within myself. And so my skepticism persists, as does my belief that 100 Heads in a row must be possible, at least in principle. If it weren’t, the implications would be unimaginable—e.g., we’d need to remove all Heads and all Tails and all other “patterned” outcomes from the sample spaces of anything more than, say, 29 flips. Absurd!

My intuitions about 100 consecutive Heads also apply to 92 Heads and maybe as few as 30, though I don’t know where to draw the line (I tend to use the number 29; but here again arises the tension: after 29 consecutive Heads, there’s no reason you can’t get another one… to really see this, imagine putting the coin in a drawer then retrieving it 50 years later to perform that 30th flip).

Smith goes further than to argue that 92 Heads in a row is possible. He claims, rather, that such an event is not—or at least should not be seen as—surprising. This stronger claim poses less of a challenge to either of my opposing selves. Indeed, I take for granted, intuitively and maybe even theoretically, that seeing even just 20 Heads in a row would be surprising, and I’m not the only one: Peter van Inwagen used that number of consecutive Heads to make the same point about surprisingness, while fiercely opining that the equiprobable principle “must be one of the most annoyingly obtuse arguments in the history of philosophy.”⁵

Something I just said demands qualification: “would be surprising.” Smith emphasizes that he is not “making a prediction about what [he], or anyone else, would feel” if confronted with 92 consecutive Heads (Smith 2). He is, rather, arguing that we should not feel surprise. Namely because it is not surprising (note the use of the word “is” in his title), though this is something of an oversimplification.

I’ll say more about this as I go along, as it raises difficult questions. Indeed, there turn out to be several interconnected parameters at play here—for example, between possibility, belief, surprise, and the broader project of making sense of surprise as a subjective state that sometimes seems inappropriate (i.e., irrational). In other words, there’s a case to be made that if people—even philosophers of probability and the like—are nearly universally surprised by seeing some event, then that event is surprising; or that at least a response of surprise is natural, reasonable, and maybe even rational. While I grant that we should not establish our rationality criterion simply as “whatever it happens to be that most people feel or do,” there must be something significant about how people actually think and behave if developing such criteria has any point at all to it.

I’m getting ahead of myself. With preliminary attention having been brought to this interconnectedness and to the distinctions between concepts such as possibility & surprise; causing or being responded to with surprise & being surprising; would & should be responded to with surprise; and so on—let’s return focus to the conjunction principle.

2.3 Challenging the Conjunction Principle

Smith claims that “the conjunction principle allows us to prove that throwing 92 heads in a row is not surprising, even though it’s extremely unlikely” (Smith 4).

My challenge to this claim is that counterexamples are easily built. Below, I present three in increasing order of intensity (I’ll present yet others in later sections and invite you to imagine as many others as you like). I readily admit that, by “intensity,” I refer to the relative intuitive potency of a given example. My task today is, in large part, to convince you that this approach is appropriate—that such intuitions matter.

Calling a Coin Flip: There’s nothing surprising about correctly guessing the outcome of a single coin flip. I claim, however, that there is something obviously surprising about doing so 92 times in a row.

Maybe you disagree with me and think such an event unsurprising. Remarkable, maybe. Extremely unlikely for sure. But not surprising.

Fair enough. But note well that we are now debating—perhaps in reasonable terms—what does, or should, count as surprising.

Well-Ordered Aleatoric Music: Suppose you randomly toss a hefty coin into the inside of an opened grand piano and the note C is sounded. There’s nothing surprising about this. Suppose you do it again and there sounds an E-flat. Again, not surprising. There is no note, given a single toss, that will be surprising (particularly if you don’t try to predict the note before tossing). Assume all tosses are independent and any string is as likely to be hit with any other.

But now suppose several people randomly toss a stream of coins at the inside of that piano and the result is a note-for-note rendition of Bach’s 13th Invention. Would this not be surprising? Would surprise be unwarranted? The conjunction principle tells us that, because each constitutive event is independent and unsurprising, the final result is unsurprising. I think you know where I stand here.

Spooky Cards: If such a thing as the aleatoric music example were to happen, I agree with Smith that, once any rigging and such have been ruled out, nothing supernatural has happened. I cannot stress enough that I am not a believer in the supernatural. The final explanation, then, must be: it just so happened that the coins landed that way and, boy was it fun and perplexing, but that’s all there is to it.

But that is a tremendous if. So tremendous, that I can’t trust my worldview would continue to have even the smallest corner in which to accommodate such cool and rational logic. Just as Smith seems to grant that he himself may, despite himself, feel surprised at seeing 92 consecutive Heads, I can’t, despite myself, be sure that no such event would lead me to revise my beliefs in God and ghosts before accepting (if not believing) an explanation of blind chance.

If the Aleatoric Music case wouldn’t affect my in that way, the Spooky Card one likely would.

Imagine a deck of 37 cards: 26 cards depict one each of the capital letters of the English alphabet; ten depict the digits 0–9; one is blank. (Feel free to extend the deck to include punctuation, foreign alphabets, symbols, and so on.)

Randomly pulling the “W” card from that deck would not be surprising. Follow that with an “8.” Still not surprising. No particular pull is surprising. All pulls are independent, due to being performed with replacement.

Now suppose you go on to randomly pull many thousands of cards that spell out elaborate and correct answers to a thousand questions to which only you know the answers, and many other questions whose answers you’d like to know (including a correct lottery number). Imagine this goes on for hours running into days and weeks as you cannot walk away due to the spellbinding grip these cards have on you—these cards, which are more eloquent and wiser and better at spelling than you’ve ever been. They even know languages you don’t know (you test it against Google translator to make sure).

One of the cards’ spookiest phrases is, “Stop tempting us—we are prescient, not loaded.”

But I beg the question by making you the bedazzled subject of this story. So I’ll ask. Would you, experiencing this at home alone with no cheaters in sight, be surprised? Would you be more surprised than had you selected cards that mostly spell out gibberish—maybe with a “FART” here and a “DOBALD TLUMP” (so close!) there? (This is an early invocation of the equiprobable principle, more about which very soon.)

Would you be spooked? Freaked? Psychologically jarred and jolted? Would you guard the deck with your life or burn it out of fear of it getting into the wrong hands? Would you think you’ve stumbled on some portal to a serendipitous glitch in the Matrix?

More to the point, would you think it reasonable, sensible, appropriate, or even rational for you to be so moved? To reject blind chance, even though you take to shuffling the cards to point of bleeding fingers before each pull?

I can only imagine that I would be those things and more. I can only imagine such a thing being life-changing, world-bending, reality-shattering. What I cannot imagine is it actually happening. I unwaveringly believe it would not ever happen, not if I spent every day of my life trying and not if there are countless me’s splintering off at every moment into countless worlds manifesting every possible world no matter how improbable*—any more than I believe that in one of those worlds my eyeballs will morph into spherical cubes and of which I will somehow have in my skull both 2 and 17 because in that world 2 equals 17.

(*I admit that this must be wrong: if—and that’s an EPIC if—there are branching worlds, there are, at every coin toss, worlds in which it comes up Heads, and worlds in which it comes up Tails. There is also at least one world, by the end of several flips, in which the coin has come up all Heads [perhaps a great many such worlds, as the coin might land Heads slightly to the left of where it lands Heads in other worlds]. There is also a world in which the alpha-numeric cards have come up Spooky. Maybe worlds of magic are those in which the incredibly improbable just so happens to happen often—like one’s body happening to phase neatly through a wall whenever one happens to want it to.)

Even if I could convince myself that it could happen, at least in principle, I cannot convince myself to relinquish the belief that it would not happen were I to literally perform the experiment right now. And so if it did happen, I’d be surprised—and, more to the point, sensibly so. Despite having contemplated the conjunction principle.

I have presented this as an intuitive argument, but it may also have logical suction. In “Taleb’s Example” I consider different sorts of possibility; e.g., practical and logical. I won’t delve into that here. Rather, I will float the idea—which will bob up here from time to time—that it is arguably irrational to not believe that certain things won’t happen, and to thus be unsurprised when they don’t, or surprised when they do. This, too, will require some intuitive justification, but no more than is required for accepting the conjunction principle.

For now, I’ll observe that I would worry about the capacities, rational or otherwise, of the mind that was not surprised by the above examples. At least, I see no reason whatsoever to question whether a human adult is reasonable, sensible, sane, coherent, intelligent, emotionally stable or a reliable decision maker due to seeing that person surprised by a fair coin landing 92 Heads in a row (not to mention 92 thousand). Those characteristics safe, what does it matter even if the person fails some theoretically grounded test of rationality?

Still, were I inclined to argue about such such things, I might suggest that one desideratum for a description of rationality might be that perfectly reasonable-seeming folks shouldn’t look at it and say, “If that’s what you mean by ‘rational,’ then I’m out.”

So much for counterexamples and peering a bit more closely into the intersection of possibility and the cognitive dimensions of surprise. I’ll close Section 2 with an analogy whose impact here I’m not quite sure about; but perhaps it bears mentioning.

2.4 Analogy from Degenerate Circles

One way to interpret the conjunction principle’s application to the present context is that if we assign a value of 0 to the degree of surprise experienced by any given instance of a flipped coin landing Heads, then calculating the appropriate degree of surprise at seeing 92 Heads in a row is a simple matter of multiplying 0 by 92. Perhaps I’m slyly begging the question here by saying “the degree of surprise experienced.” Should I, rather, say simply the “degree of surprisingness,” so as not to presume that it is our experience of an event that determines the appropriate degree? I’m not so sure, as it seems to me fundamentally a matter of intuition that a single coin flip “is” unsurprising. For now, I’ll leave the question open, and will assume that the the above interpretation is on the right track as a charitable expression of the conjunction principle.

This interpretation also makes sense in a less formal, qualitative sense. As Smith puts it:

Guildenstern seems to reason like this: Since there is nothing surprising about any one particular coin landing heads, there’s no point in the sequence of 92 consecutive heads at which anything surprising actually happens. (Smith 2)

Recall also that the rule extends to as many independent, unsurprising events as you like.

This line of reasoning may be correct. I of course think it’s obviously incorrect, and that it may be at least a little analogous to the following sort of fallacious reasoning.

Any point on the Cartesian plane has an area of zero. If we line up many of these points—sometimes referred to as degenerate circles, or circles with a radius and diameter of zero—we are lining up many objects with an area of zero. And so the line itself has an area of zero. As does the circle we claim to plot, impossibly, with a radius of 9, given that it too is in actuality made up of infinitely many degenerate circles of area zero.

While philosophically thought-provoking, and perhaps even demanding of some mathematically valid solution or another, it would be clearly fallacious to apply such reasoning to the real world. (For discussions of this and similar paradoxes, see Section 2.3, titled “The Argument from Complete Divisibility,” of this Stanford Encyclopedia of Philosophy entry: Zeno’s Paradoxes. See also this enjoyable discussion on the In Our Time podcast, also titled “Zeno’s Pardoxes” (9/22/2016), in which there arises disagreement over the extent to which mathematics and philosophy have [or at least take themselves to have] dealt with these problems, and in which a guest points out near the end that, “These paradoxes are still very relevant today in teasing out the nature of reality and our intuition about it.”)

I’m by no means claiming the analogy to be perfect or even good. Maybe it’s mostly nonsense. Or maybe it at best serves as an example of how similar mathematical reasoning to that of the conjunction principle, which relies on a similar mathematical model, can go wrong when held up to the real world. Or maybe its very inappropriateness says something about the inappropriateness of treating the cognitive state of surprise with numeric formality (though there would be no problem with borrowing the concept in a mathematical context, as it has been in information theory; see this footnote for more on this.⁶ In other words, surprise of the sort conscious animals actually experience has no substance, no particles, cannot fill a measuring cup. It may go too far, then, to expect its relative intensity to correspond directly to some numeric reification, where actually experienced, “phenomenological intensity” is figuratively treated as “quantity.” Again, I would distinguish this lived experience from the sort of technical formalisms found in, say, information theory (just as I always recommend caution with confusing technical and colloquial uses of a given word more generally!).

Perhaps, then, this does make the analogy with points appropriate. If so, then I might suggest that, by drawing a border around several individually surprising events, resulting in a larger event (e.g., a 92-Heads sequence), we have done something similar to when we establish borders (themselves conceptual objects and thus of area 0, of course) around a range of points on the plane. Something special happens, some kind of intuitively comfortable shift in our thinking about the ontological status of the conceptual materials at hand. We are not adding together points so much as we are viewing the new figure as a singular object in itself.

Interestingly, while points in space are conceptual objects that aid in contemplating the properties of real-world figures, a coin flip is itself an arbitrarily conceptualized event made up of countless smaller events, though we don’t know how small the smallest of these are. We may, then, arbitrarily conceptualize any 92-flip sequence as the sum total of the exact same countless smaller events that make up each individual flip in that sequence.

This sort of idea is also discussed in the above-referenced In Our Time podcast, and is at least implied in the below-referenced Mindscape podcast, e.g., in reference to efforts to quantize gravity, or, in other words, to reconcile quantum mechanics and gravity (as understood under general relativity).

The latter discussion is especially interesting in the present context of agglomeration (see below) and conjunction problems and so on. That is, it may well be that we can, in principle, predict exactly what a collection of particles will do when conceived of as the object coin, though we cannot so predict, even in principle, what any one of those particles will do (though we can work out a, as it were, built-in probability for what it will do; more on this distinction in another writing, coming soon). A variation on this comes up in the study of things like crowd behavior (a group of humans exiting a crowded stadium, flocks of birds), and social behavior more generally.

3. The Equiprobable Principle

3.1 Defining the Equiprobable Principle

As far as I know, this term is my own. And it leaves something to be desired. Consider it a shorthand for something I won’t attempt to define formally, as that would require an attempt—perhaps a hopeless one—to develop conditions relating, among other things, to reference class consistency. Instead, I’ll say merely that it denotes a certain kind of argument that I’ll demonstrate with examples.

The equiprobable principle says, for example, that any given string of 92 coin flips has the same probability of occurring as any other 92-flip string. Therefore, no given string that occurs is any more or less possible or believable or surprising, etc., than any other. An example of Smith’s expression of the principle is:

If we throw 92 coins in a row, then a 1-in-5,000-trillion-trillion event is bound to happen—and, as such, we shouldn’t be surprised when a 1-in-5,000- trillion-trillion event does happen. (Smith 4)

This argument, which I cover extensively in the above-linked essays, especially “Fine-Tuning,” constitutes one of the sturdiest obstacles to my naive intuition’s insistence that 92 Heads in a row is impossible. On the other hand, even my most theoretically enthusiastic side loses its cool given farfetched examples such as the following.

3.2 Well-Ordered Aleatoric Music and the Equiprobable Principle

Given Smith’s conjunction principle, producing Bach’s 13th Invention by randomly throwing coins at a piano is unsurprising. But the equiprobable principle works here as well—better, in fact, as far as I can tell.

Each coin tossed at the piano is bound to land on some string or another, with no string more individually probable or surprising than any other. Assuming equal probability for which strings are struck and the (perceived) durations between each sound produced (one could easily construct such an experiment), whatever collection of sounds is produced has the same probability as any other (given the same number of coin tosses). And so we should not be surprised by hearing Bach’s 13 Invention.

Again, I usually use such examples to bring into question whether highly improbable events of a certain kind—e.g., well-ordered or meaningful-to-minds-like-ours⁷, despite an overwhelming bias to be anything but—are possible. In fact, I conceive of this bias as something like a force that, in the case of a far coin (and similar setups), I call fairness bias or a fairness boundary (where “fairness” really just means “equiprobability”); more generally, we might call this a propensity boundary. I take it this is also what Stoppard meant above when invoking the “laws of probability.” This is not a literal force, like some invisible hand that reaches in and ensures a coin won’t land Heads for the 30th time. Indeed, I agree with Smith when he writes:

Consider the claim ‘there will be between 40 and 50 heads’—the kind of thing that we’re meant to “expect”. Although we can calculate this claim to be approx. 73.8% likely, it’s not as though there is some special force compelling the coins to fall in this way. (Smith 5)

I refer, rather, to something like the following, from my “Fine-Tuning” article:

I’m clearly not positing an explicit restriction on any particular coin toss, but rather a behavioral tendency that emerges from the conditions composed of the coin’s physical attributes (including its body, its environment, and the physical forces that mediate the relations between those things). The result amounts to a restriction of the sort of behavior the coin may exhibit. The coin will not, mid-toss, morph into a dragonfly or start singing the blues or spin in place indefinitely or land heads 100 times in a row. (Wallace, Fine-Tuning, Point 23)

Once again, I apparently cannot convince myself that it’s even possible to land a great number of Heads in a row. Much less land Bach’s 13th Invention. And if I don’t believe it’s possible, then, as will become evident soon enough, I’m allowed on Smith’s account to be both rational and surprised when it does. Well, provided I can justify my characterization of that belief as rational. I’m working on it.

One of the difficulties in that task is that I do recognize that if you line up 92 coins in a row and were to run through all 2⁹² permutations, all Heads is obviously among them, and there is no natural law or principle in The Great Book of Received Mathematical Models that I can invoke to argue that it is impossible for that particular permutation to come up should you, say, have 92 people with blind folds pick up a coin, jostle around in their hand, and set it back down. Again, it would be absurd to remove that or any other permutation from the sample space.

But I gratefully am not arguing for possibility here, but surprisingness. In which case I need only show that it’s rational to believe, even if in a here and now sense that such things won’t happen. The shear greatness of that number of permutations might help out here. Just how big is 2⁹²? Massive, if not as big as the number of possible outcomes in the Aleatoric Music or Spooky Cards cases (whose numbers are too large to intelligibly fathom, and thus not worth the effort of calculating). Let’s look more closely at the equiprobable principle, then, relying only on coin flips.

3.3 Coin Flips, the Equiprobable Principle, and the Interpretation Question (a.k.a., A Conceptual Slight-of-Hand)

Suppose I successfully predict Heads then Tails then three Heads in a row then two Tails in a row and so on. I’m correct 92 times in a row, with a probability of (1/2)⁹², which is the same probability for getting any given sequence of 92 coin flips, including 92 Heads (more on this in just a second). There’s nothing surprising about me predicting any given throw. But my predicting all 92 would surprise, upset, intrigue many people.

I could do worse for respectable researchers to mention me in fancy academic journals, even if to prove a statistical point, and I could even convince some folks of mental powers. Were I to predict enough flips, I might start to believe I have mental powers, human as I am. According to the conjunction and equiprobable principles, however, it would be unsurprising for me to correctly guess every coin flip I encounter from here to my hopefully very far-off death bed.

But let’s stick with just 92 flips. Suppose that, instead of experiencing an existential crisis, I accept blind chance as the explanation. I, and any witnesses, would still be very surprised, I’m sure. Smith doesn’t doubt that, of course, and in fact seems to imply that those who calibrate their surprise against the relative rareness of an event would likely register too little surprise in the case of 92 consecutive Heads: “It is indeed very unlikely to throw 92 heads in a row—perhaps even more unlikely than you might guess at first” (Smith 3). Though, in a certain sense, as Smith will correctly point out, we constantly encounter events even rarer than this—perfectly mundane ones, in fact.

To begin to get an intuition for how this works, consider two scenarios. Which of the following would you find more surprising?:

(A) You flip a coin 92 times and get a run-of-the-mill, random-appearing sequence.

(B) Rewind the tape. This time, you correctly guess all 92 flip results of (A) ahead of time.

I’m happy to call (A) unsurprising. On the other hand, (B) would be intensely surprising. What’s remarkable here is that, in a certain sense, both (A) and (B) have precisely the same probability of occurring. Of course, in one sense, (A) has the higher probability of occurring: one is extremely likely to get some random-appearing sequence when flipping a coin 92 times, and one is extremely unlikely to correctly guess which of those sequences will actually occur. Let’s look at this more closely.

Any give sequence of 92 flips has a probability of (1/2)⁹² power of occurring. Now suppose you try to guess a 92-flip sequence, either one flip at a time or by writing out all 92 results ahead of time. The probability of correctly guessing all 92 flips is, again, (1/2)⁹². These two events have the exact same probability and involve the exact same flip results—yet one is boring and the other apparently miraculous.

The point I want to home in on is the vague air of cheating—or perhaps conceptual confusion—that may result from observations of the above sort.

(A) does not designate a specific sequence. I easily could have specified one, but if I did, I fear it would seem that you’d need to imagine flipping that exact sequence in order to satisfy (A), which then undermines the conceptual slight of hand I’m trying to get across. Here’s what I mean. I could have actually asked you to flip a coin 92 times and note the results. Then, for (B), I could have asked, “How surprising would it have been had you predicted that sequence?” But this is a kind of cheat, because you are guaranteed to get some 92-result sequence when you flip the coin; and in this sense, (A) and (B) literally couldn’t be further apart, as there is a probability of 1 for getting some 92-flip result. And the probability is extremely close to 1 for getting a random-appearing (i.e., “un-patterned”) result. I won’t try to work that out precisely, but will provide some precise numbers that intuitively support the claim.

Here’s another way to get across the same idea. Suppose (A) you flip a coin 92 times a row and note the results. Now suppose (B) you flip another 92 times and get the exact same results. The low probability event of note here is obviously (B). Yet another way to put this would be for you to do the following 92 times in a row: flip a coin and get either Heads or Tails, then flip again and get the same result. The probability of these matching results occurring is precisely the same as the probability of getting any particular 92-flip sequence. But by using the word “particular,” I am implying something like a prediction (i.e., that we should have in mind some particular sequence), when what I’m really referring to in (A) is no particular sequence (which we will see with 100% probability).

A similarly illustrative fact is that to correctly predict 92 flips in a row has exactly the same probability as failing to predict all 92 flips. For example, getting about half correct has the same probability of getting all or none correct. This is initially perplexing, but only because I’m being ambigous. These all have the same probability provided we designate precisely how how one goes about failing to predict all 92 flips, given that you can fail in many ways: there is one way to get all 92 flips wrong; 92 ways to get one flip wrong; 4,186 ways to get two flips wrong; 125,580 ways to get three wrong; 2,794,155 to get four wrong; and the numbers get bigger and bigger from there, until you hit the pivot of 46 (410,795,449,442,059,149,332,177,040 ways⁸), at which point the number of ways begin to reverse symmetrically (e.g., there is one way to get all 92 flips right).

The point is that it’s critical to be clear about what we mean by “failing” to predict.

Now, the equiprobable principle says (as does the conjunction principle) that it should be no more surprising to correctly guess 92 coin flips in a row than to succeed in guessing roughly half of a given 92-flip sequence. This is implied, for example, from the following sort of application of the equiprobable principle:

If I’m surprised by 92 heads in a row, on the grounds that it’s so unlikely, then I’d have to be surprised by any sequence that came up—surprised no matter how the 92 coins land. (Smith 4)

It seems to me, however, that this seems less plausible when we are precise with what we mean by “how the 92 coins land.” There is a 100% chance that they will land somehow. Clearly this is not what is meant, as it is called the equiprobable principle. So it must be meant in the precise sense. In which case it’s implied something like predicting how the coins will land, though this seems misaligned with the phrase “no matter how the 92 coins land.” In other words, there seems to be a kind of conceptual confusion (if not intentional slight of hand) here.

Such ambiguities seems to be roughly the general move that Smith makes in his appeals to the equiprobable principle. Of course, he’s well aware of these distinctions and in fact addresses them (we’ll look at that shortly). Nevertheless, they seem significant here. That is, if you flip a coin 92 times each day for the rest of your life, you very well might not see the same sequence twice. If you do this for 50 years, that’s about 18,250 instances of flipping 92 times, while there are 2⁹² = 4,951,760,157,141,521,099,596,496,896 possible sequences (including all Heads, all Tails, and any seemingly patterned outcome).

What you are very likely to see in those 50 years are 18,250 mostly run-of-the-mill, random-looking sequences. We could do a little more work here to figure out the longest runs you’d expect to see and such, but I’ll direct you to “Fine-Tuning” for that sort of thing. Let’s just say that maybe you’ll have some days where you see some neat things, like 11 Heads in a row, which is only barely remarkable. But five Heads in a row will become boring.

One thing I agree with Smith on, however, is that, should you get 92 Heads in a row, while I’d be surprised, I would not be—nor should I be—any more surprised to learn you got in on the first day than on the last. If 92 Heads in a row is a possible event, then it’s possible, period, and it can happen at any time. This fact only bolsters my skepticism! I respect the laws of independence. I’m just not convinced—not really and truly convinced—that those laws support things like the conjunction and equiprobable principles. Quite the contrary.

The takeaway here, though, is that, while (A) and (B) in one sense have the same, extremely low, probability of occurring (namely when we specify a given sequence for (A)), there’s a broader sense in which they do not (when there are many trillions of possibilities for (A), while (B) necessarily involves some specific series [i.e., the one you guessed]). This ambiguity, which I am not convinced I’ve done a good job of describing, must be avoided.

I’ll give it two more tries. Here are two uncharitable, but clear, illustrations of an exploitation of the ambiguity I’m aiming to describe.

The Murder’s Defense: Suppose a murderer’s defense attorney makes a show to the jury of chucking 20 quarters into the air and then says, “The result you’ve just witnessed has a probability of less than one in a million of occurring, which is far less than that of my client’s innocence. I can easily produce, before your eyes, sequences with probabilities of one in a billion or lower, and can do so over and over again. These events are mundane. Commonplace. Don’t be fooled by low probabilities!”

As we’ll see below, it may not make sense to inform belief about such different classes of event (coin flips, murders) only on the grounds of their respective probabilities. But we need not go so far here. A simpler argument against the defense attorney is obvious. “You’ve performed a conceptual slight of hand. You tossed 20 quarters into the air knowing that some sequence would occur with a probability of 100%. And then some sequence indeed occurred. What we want to see is for you to predict the sequence ahead of time. That’s the low-probability event in question. Try it with even just ten quarters.”

Raffle Confusion: There’s a high probability you’ll lose the 1,000-name raffle due to your name not being pulled. There’s a low probability of losing it due to your nemesis’s name being pulled.

Smith is obviously aware of such confusions, and addresses the worry in a discussion of statistical expectation, which really just means “average.” For example, that if you flip a coin 92 times, the “expected”—in its technical sense—number of Heads is 46. Which is to say that the average number of Heads among all the ways the coin can land is 46. To make vivid the theoretical nature of expectation in this sense, reflect on the fact that if you flip a coin three times, the expected number of Heads is 1.5, which can be verified by listing the eight possible outcomes and noting the average number of Heads.⁹

So, what we expect on average is obviously not what we should explicitly expect (in the ordinary sense of the word) in actuality, particularly from relatively short runs (which are the only sort we can see in actuality). You’re not too likely to get exactly 46 Heads in a given 92-flip sequence. The probability is about 8% for that, though this is still the mostly likely number of Heads to get. Compare this to the meager, roughly 0.03% probability of getting exactly 30 Heads in no particular order, for example.

Of much greater significance is that there are likely and unlikely ways to get 46 Heads. For example, you’re unlikely to get 46 consecutive Heads and then 46 consecutive Tails, or to get them alternating with Tails: THTHTHTH…

Random-appearing sequences containing 46 Heads are much more prevalent than the patterned-appearing ones. There are, again, over 410.8 septillion (or 410.8 million billion billion or approximately 4.108 × 10²⁶) 92-flip sequences containing 46 Heads. So, if I told you that a sequence landed 46 Heads, there’d be less than a three out of 410.8 septillion chance that it’s one of the three patterned examples I just mentioned. (At the very least, I would argue that the “fairness bias” I referred to earlier is simply whatever collection of facts it is that accounts for the differing frequencies among sequence types.)

Of course, while random-appearing results far outweigh patterned-appearing ones, it’s also true that any given sequence has the same low probability of any other: (1/2)⁹². And one of them is guaranteed to happen. So you shouldn’t be surprised when any one of them does. Thus the frustrating allure of the equiprobable principle, even when stated unambiguously!

Smith wholeheartedly embraces the mind-boggling-ness of all this, though of course seems (at least on paper) to do a better job than I do of resolving any resulting cognitive dissonance, which for me is the dissonance between confidently asserting both that there are 2⁹² permutations of 92 coin flips and that some permutations are impossible—or at least surprising. Part of this dissonance arises from the sheer difficulty of conceiving of the numbers involved.

Smith illustrates just how unimaginably small the probability is for getting any specific 92-flip sequence, i.e., (1/2)⁹²:

This is (much) less than the chance of two people being asked to randomly choose a single grain of sand from anywhere on the Earth and happening to choose exactly the same one. Surely if a 1-in-5,000-trillion-trillion event were to actually happen, then this would be near miraculous and certainly very surprising. If you have this reaction, then you’re in good company. (Smith 3)

This is a hypothetical experiment that cannot be literally run, even if there were a way to give two people equal access to all regions of space on Earth that contain sand, and as much time as they’d like in order to choose one. It would be easy to simulate a scaled down version, however. What it loses in its multidimensional grandiosity (imagine traveling the world in search of a grain of sand) is made up for with flat mathematical tidiness. These come down to differing rhetorical appeals to intuition. Pick whichever most impresses you.

There are estimated to be 7.5 × 10¹⁸ grains of sand on Earth¹⁰, a number much smaller than 2⁹². Tell a computer to randomly generate two integers from 1 to 7.5 × 10¹⁸. The numbers will match with a probability of 1 in 7.5 × 10¹⁸. That’s approximately the probability of getting 63 Heads in a row, by the way. There’s something starkly mundane about the example now, no? But it would have been odd and intuitively unimpressive indeed for Smith to say, “Getting 92 Heads in a row is even harder than getting 63 Heads in a row.”

Perhaps it doesn’t make sense to compare the numerical probabilities of two people choosing grains of sand and landing 63 (much less 92) Heads in a row. These are very different sorts of events, both qualitatively and in their complexity. Indeed, the unfeasibility of two people choosing among all the grains of sand on Earth—the real probability of which may depend on much more than how many grains the Earth carries—is perhaps part of what makes the example effective. That said, the equiprobable principle is about comparing probabilities, and all the better if they’re numeric. This is how Smith means to present it, too, given that part of his point is that the probability of choosing the same grains of sand is lower than that of landing 92 consecutive Heads.

In other words, the primary effectiveness of Smith’s example is meant to come from our sense that there is a whole lot of sand on Earth. The number of permutations of 92 coin flips, however, is well over 660-million times bigger than that.

The long and short of it is that it would be much harder to predict 92 coin flips in a row than it would be to guess which grain of sand someone else chose or, if you prefer, than guessing which integer the simulation generated between 1 and 7.5 × 10¹⁸. On Smith’s view, as I understand him, none of these events should surprise you any more than any other. Indeed, the equiprobable principle makes it so that nothing is ever any more surprising than anything else, at least on the mere grounds of probabilities. The probability of any event, no matter how rare, will be less than the probability of landing some number of Heads in a row, making it so that we shouldn’t be surprised, say, by a 120-sided die cycling the numbers 1 through 120 indefinitely, as many times as you’d like to throw it (it must land some sequence of numbers; why not that one?).

But if you are surprised, notes Smith, you are in “good company.” The “good company” he refers to are esteemed mathematicians with whom he promptly goes on to disagree: Jean le Rond d’Alembert, the “1760s the polymath… [who] questioned whether it was even possible to observe a long run of a single outcome when two equally likely outcomes could result on each trial”; 1840s mathematics of probability theory pioneer Antoine Augustin Cournot, who “claimed that it was a ‘practical certainty’ that an event with a very low probability won’t happen”; and Émile Borel, a “major figure of probability theory” who, in a 1943 book (see below) wrote of a “law of chance,” sometimes known as “Borel’s Law”: “Events with a sufficiently small probability never occur.” (Smith 3)

Smith most explicitly disagrees with Borel. “I’ve come to think,” writes Smith, “that Borel’s law is not even close to being right—in fact it’s almost the exact opposite of the truth” (Smith 4).

And here Smith seems to make something like the sort of move about which I express suspicion above—i.e., that which interprets (A) as having the same low probability as (B).

Smith rightly points out that whatever sequence you happen to get when flipping a coin 92 times has exactly the same probability of occurring as 92 Heads in a row. And so, he argues, if you’re surprised by 92 heads in a row, you should be surprised by any sequence that comes up, even a run-of-the-mill, random-appearing one. Alternatively, you should be unsurprised by both. In short, you should have the same reaction in both cases.

To bolster this idea, he points out that we constantly go unsurprised by far less likely events. I survey a variety of such events in “Fine-Tuning.” For example, the “mundanely unsurprising” (as I term it) but highly improbable event of stepping on a pebble—improbable given precisely those particles coming together in precisely that way at precisely that time, but unsurprising because some such collection of events was bound to occur. Another example is the less mundane, but still unsurprising, event that someone’s name is pulled in a trillion-name raffle; well, it’s surprising to the winner, for whom winning carries an extremely low probability, but it’s unsurprising that someone is the winner. I claim that there are real differences between these and other sorts of improbable events—i.e., of the sort that we may rightly call “surprising,” of which I think the Spooky Cards case would be an undeniable instance.

Smith traces a similar line of thinking, but of course follows it to a different conclusion

It’s not only when we repeatedly flip coins that something unlikely is bound to happen—something unlikely is bound to happen with every intake of breath, every heartbeat, every step. If I’m surprised by throwing 92 consecutive heads, based just on its low probability, then I should be in a state of constant amazement. … in a sense, everything that happens is an unlikely thing, once it is seen in sufficiently high definition. (Smith 4)

It should be clear by now that I agree with Smith when he says that, in a certain sense, everything that happens is unlikely. Any given event event amounts to the unique convergence of (perhaps literally) countless sub-events. So any event is identical only to itself, however one arbitrarily defines an event, from the tiniest possible flicker of a particle to the sum total of all reality at a given time slice, or forget time slices: we may view all of existence as a single event. And so any given coin flip is only identical to itself: the precise motion and location of the coin, all happening at precisely the times they do, given the precise history of the coin itself and the person flipping it and so on. This set of events is not only rare; it’s literally the only event of its precise kind in the whole of existence. But as a token of the event type “coin flip,” nearly every coin flip is mundane, as are all of its constitutive sub-events.

And we haven’t even so far mentioned that events with a probability of zero happen all the time. For example, the probability is precisely zero that a continuously distributed random variable takes on a precise value. So, the probability of randomly choosing the number 2 out of all real numbers between 1 and 3 is zero, though the probability that some number will be chosen is 1. One way to understand this is by observing that the area above 2 on a graph of this uniform distribution is zero, while the area on that graph between the numbers, say, 1.9999 and 2.0001, is 0.0001—which is the probability of randomly choosing a number in that interval (though the technically expected value actually is 2, as it is the midpoint between those numbers; also, echoes here of the above example of adding up area-less points). This can play out in surprising ways: the probability of an accumulation of a day’s snowfall hitting precisely 4.5 inches is zero, as is that of a randomly chosen human’s height being precisely 6 feet (which is a different question than, say, what the person’s driver license reports). Even if obtaining precise measures or really being able to choose from all real numbers is hopeless, the point is only that, of the theoretically available numbers in these ranges, any one of them has a precise probability of zero of arising.

Even if it turns out that, in the case of human height for example, there is some physical granularity that restricts what actual heights are possible—which raises the question: as you grow from 5′ to 5’1”, do you measure infinitely many heights in between? I think a more interesting related question is whether there really are infinitely many notes between any two keys of a piano, as so many music teachers are prone to say. My short response: Between any two frequencies, say 440 and 466.16 (that’s A4 and A#4), there are infinitely many numbers that can be characterized as frequencies, but this does not prove that there are infinitely many frequencies that can actually be taken on by physical materials—in other words, that a tuning fork could be made to resonate at each frequency—and, even this were true, there certainly is no reason to think that there are conscious entities in the world that could perceive them all distinctly. My point in belaboring these examples is that what we are dealing with hear are models of the world—whether mathematical or phenomenological.

Rarity per se, then, particularly as strictly described by some probability model, is not, nor should it be, intuitively impressive. What matters is the relation between the sorts of models we can construct of the world, and the world as it is—which we can only know through models. But which sorts of models to choose? The coin has to do something when you flip it—something in accordance with the laws of physics, whatever those may be. Experience, even of the strictest and most rigorously scientifically minded sort, says it will do something roughly normal. Landing 92 Heads in a row, however, is not normal. Fair coins don’t behave that way. Not usually. And there’s no evidence that they ever land 92 million Heads in a row. I claim in fact that by definition that is simply not a fair coin no matter how symmetrical it seems. Period. In fact, there may be an implicit contradiction in the very notion that it even could seem symmetrical on close examination. (Unless a deep inspection reveals the confounding influence to be, say, tiny cheating nanobots; this raises a question of how much investigation is required in order to justify belief, more about which below). And so there may be some logical mistake in the equiprobable principle even when just restricted to the domain of coins. (See “Fine-Tuning” and “Taleb’s Example” for more on logical impossibility.)

My point here is that there is some important difference between events that tend to surprise on probabilistic grounds and those that don’t, which I call mundane—no matter how impossible it would see for some prognosticator at the beginning of time, or even a mere 2,000 years ago, predicting the precise event of you stepping on that pebble, I simply cannot be amazed by stepping on a pebble, just as Smith recommends.

Nor can I take his next recommended step, however, and convince myself that I therefore should not be amazed by the more likely event, viewed a certain way, of a fair, symmetrical, two-sided coin landing 92 Heads in a row, much less 92 nonillion.

It seems to me that there is some justification for these differences in response even just on the grounds of practical benefit. Appeals to evolutionary psychology may even be in order for bolstering the idea that reliably feeling surprise says something important about the nature of certain kinds of events, or at least our relation to them. As we’ll see in a moment, when I turn to Smith’s discussion of the point of surprise, such observations may in fact be well in line with his account.

For now, I’ll note that I agree with Smith that rareness simpliciter does not warrant surprise. Though rareness of a certain kind does. I won’t formally define this, but will say that it seems to come down to events that seem—yes, seem—practically impossible, even if they seem theoretically possible. This points to a milder claim on Smith’s part, which is that, at the very least, low likelihoods are not enough to ground belief that something won’t happen. In other words, if you believe that all permutations of a coin are possible, then you should not believe that the coin you’re about to flip will not land that way. And if you don’t believe it won’t land that way, then you shouldn’t be surprised when it does. I trust that Smith would agree that this line of reasoning may be drawn from his account.

It’s a fair line of reasoning, though it becomes harder to accept as events attain a certain kind of extreme rareness—see again the Aleatoric Music and Spooky Cards examples. Surely most people would have a difficult time not believing those things won’t happen, even if they believe them theoretically possible “on paper.” As for myself, I need not go as far those outlandish examples. I believe that the coin I’m about to flip will not land 92 Heads in a row. I justify this belief with appeals to something like the fairness bias. A harder belief to justify is that I can’t guess the outcome of 92-flip sequence, though I still think it can be justified from experience, if not something like the fairness bias (though that might apply here as well).

Here arises an interesting distinction. That is, it seems to me that it’s possible to believe, without a worrisome degree of cognitive dissonance, that such a thing could happen at some point in the history of the multiverse, but that it won’t happen here and now, even in full light of the lessons of the Gambler’s Fallacy and the Law of Large Numbers and so on.

To some degree, our beliefs in these situations are matters of intuition. And they very well may at times be obviously irrational in just the same way I’ve demonstrated about surprise: it would be irrational to believe it’s possible to land 92 Heads face-up in a row, but not 92 Tails face-down in a row. A less obvious case might be to believe it’s possible to guess which number the Grain of Sand simulator generates, but not to land 63 Heads in a row, which has roughly the same probability of occurring, though I don’t think so. I admit that my degree of belief is higher, if only slightly so, in the former rather than latter case, and that this comes from my contemplation of the aforementioned fairness bias, which seems to apply in the latter case (e.g., the sorts of things a symmetrical coin can actually do when vigorously flipped in your living room), but not the former (where extreme [mis]fortune is not restricted within finite sets—though big enough numbers might whittle away at what little belief I have in this idea).

This difference in degrees of belief may be warranted on Smith’s view, given that they are not strictly formed by probabilities.

At any rate, it becomes quickly apparent that it is difficult to reasonably inform one’s belief on purely probabilistic grounds. The most important aspects of engaging Smith’s discussion may be that it encourages one to think deeply about this question. Smith and I seem to be on the same page in this regard: probabilistic reasoning is fraught and, most importantly, this fact matters.

Real-world concerns are commonly informed by probabilistic models. Sometimes detrimentally so, counterbalancing (and then some!) perhaps the aforementioned practical benefits of our probabilistic intuitions. Such problems sometimes arise due to a lack of nuance about probability—the mother, for example, who, after two of her children die of SIDS, gets wrongly imprisoned due in part to a failure by an expert testifier to properly condition, that is, to treat the deaths as dependent events (e.g., see the infamous case of Sally Clark).

But sometimes rare events happen, sometimes they happen often. People get rare cancers and get struck by lightening and win lotteries. Even the rarest of these can’t touch the rarity of 92 consecutive Heads. According to this National Weather Service probability that you get struck by lightening in your lifetime is about 1 in 14,600, which is a tad higher than the probability of flipping a coin right now and getting 14 consecutive Heads (or any particular 14-flip sequence you’d like to predict). This assumes an estimated lifespan of about 80 years and of course doesn’t factor in your love of swimming outdoors under thunder storms. The probability that you’ll get struck by lightening in a given year is around 1 in 1,171,000, which is a tad lower than the probability of landing 20 Heads in a row.

I believe 20 Heads in a row is a live possibility, though I’d be surprised to see it. Should I be surprised to see it? Should I view it any differently than landing some higgedly-piggeldy 20-flip sequence?

The Lucky Team: Sometimes coin flips play an important role in professional sports. Suppose some team were to win 20 such flips in a row. I hope this would be cause for investigation. And for obvious reasons. For one thing, it has a probability of 1 in 1,048,576 of occurring. But wait: So does any combination of wins and losses. Why, then, should it raise suspicion? Because mixed, random-appearing results are far more common than patterned ones. To be clear, we are not necessarily talking about 20 Heads in a row here. Rather, we can suppose that the 20-flip sequence is itself random appearing. What we also expect to be random appearing is the sequences of wins and losses of a given team.

Given a particular 20-flip sequence, there is only one way to win all 20 flips, while there are 125,970 ways to win eight flips and 167,960 ways to win nine flips. Which is to say that more-or-less symmetrical coins very rarely, if ever, are seen yielding 20 wins in a row to anyone. They reliably do not, though such a fluke may well be possible. Thus our reliance on them as fair deciders, on average.

I note something similar in “Fine-Tuning” when I point out that there’s no mystery about the statistics professor’s ability to recognize when the student assigned to document 100 coin flip results has faked the data. I also remark there that I’d bet anything that the winning guess (provided there is one) when trying to predict 100 flip results will be of the sort a statistics professor is at an advantage to produce: one that follows the most common behavior of flipped coins, one that doesn’t just have mixed Heads-Tails, but has a certain number of Heads and Tails in a row, and so on (e.g., alternating HTHTHTHT… would be a bad bet). For much the same reason, the host of this Numberphile video fails to fool the mathematician: “Randomness Is Random.”

The finer details about the technical expectations involved with 92-flip sequences (e.g., What is the longest expected string of Heads?), is more than we need to get into here. The point is that there are certain kinds of results that command notice, that may rightly raise red flags. Indeed, I would bet that never in the history of human civilization has anyone guessed as many as 92 Heads in a row, and quite possibly far fewer even, though far less likely events have indeed occurred, when interpreted a certain way—see again the mundane, but highly improbable, examples above, or imagine someone winning a raffle of as many names as you like: someone will win, even if the probability of a given person winning is far less than that of landing 92 million Heads in a row.

Winning a mere 20 coin tosses in a row should be enough to raise suspicion, making that event exceptional on what do seem to come down to probabilistic grounds—making it grounds for investigation. Even though there is no surprise at anyone winning a coin toss, and even though every sequence of wins and losses has the same probability as any other, someone winning 20 in a row warrants investigation.

Before finally getting to investigation, I’d like to clarify and address Smith’s anticipation of the arguments I’ve just made.

To be clear, none of the foregoing is meant to suggest that Smith claims no events are surprising. His claim is that low likelihood alone isn’t enough and that, if you are surprised by any event purely due to its low probability of occurring, you should find yourself surprised by all sorts of mundane events. Though he also grants that some statistical outliers may indeed be surprising: “Undoubtedly, there are cases in which it’s surprising to observe an extreme divergence from an average” (Smith 5). Like what? He doesn’t give examples. I take it the examples of surprising events he does give, like a car going missing, do not count as an extreme divergence (more about such examples below). Would a 4-foot tall person being able to dunk a basketball or someone who grows to 10-feet tall count?

At any rate, Smith’s point about the “divergence from the average” in raised in his discussion of statistical expectation. His anticipation of objections citing the conceptual ambiguity I’ve described here—e.g., the competing interpretations of (A)—is captured in the following excerpts, which I’ve mildly edited:

…the probability of getting between 30 and 60 heads [in 92 flips] is around 99.9% … Obviously, the outcome in which we get 92 heads is located right in the extreme tail of the curve (over 9 standard deviations above the mean, if we want to put it in these terms). Does this mean that we should regard 92 Heads in a row as a surprising result? …

Although we can calculate [getting between 40 and 50 Heads] to be approx. 73.8% likely, it’s not as though there is some special force compelling the coins to fall in this way. … the only reason it is likely to be true is that there are so many different ways in which it could be true (each of which is extremely unlikely). In fact, there are about 3,700 trillion trillion different sequences of 92 coin throws that feature between 40 and 50 heads in some combination. This is a large set, but there’s nothing special about the sequences that make it up—no reason to prefer them over the 1,300 trillion trillion or so remaining sequences. As we’ve seen, all of the sequences are equally likely, and any one could come about just as easily as any other. In fact, we could pick any set of 3,700 trillion trillion sequences, on whatever basis we like, and it will be approx. 73.8% probable that the actual sequence will be one of those in the set. But this doesn’t mean that we should suddenly regard the sequences outside the set as surprising. (Smith 5)

These are fair points, though I worry that the next-to-last sentence is misleading. That is, we cannot pick roughly 3.7 octillion sequences on whatever basis we like. There are only two sequences, for example, that feature one repeating outcome: all Heads or all Tails. More broadly, I view any sort of patterned-appearing result on the same footing as getting all Heads or all Tails and for the same reason: fair coins just don’t behave that way. And I am confident that there are not enough of those to make a dent in whatever 3.7-octillion sized set you’d like to construct. I haven’t attempted to calculate the size of that dent, but have noted elsewhere (i.e., “Fine-Tuning”) the sorts of restrictions we might consider for getting started with that project.

Consider it explicitly acknowledged, then, that our cultural mystique around a lone permutation—all Heads—is in some sense arbitrary. But I of course claim this isn’t as arbitrary as the math seems to indicate. Namely, I could exchange that permutation with one of all Tails or of alternating Heads-Tails, or with any permutation that defies the fairness bias. Of course, were I to have chosen some specific random-appearing sequence, that, too, would be very hard to come by. But it would lack the same force, and not just because the human brain is designed for pattern recognition, but because, were we to put up every possible permutation as a candidate for mystique-ness, it is most likely to be a random-appearing one that actually arises in practice, a possibility that disqualifies it from the running. Well, at least when it comes to the question of possibility. Given that the focus here is on surprisingness, I suppose any given sequence would do, so long as it is viewed with a precise, rather than general, interpretation.

Even aside from these perhaps admittedly idiosyncratic speculations, and in just the most mathematically flat terms, it strikes me as odd to suggest that we can organize, with no overly clever gerrymandering, the expected results of 92 flips in a way that categorizes 92 Heads in a row as anything but the most extreme sort of outlier.

On the other hand, just because something is a statistical outlier doesn’t mean it should be seen as surprising. And I concede that there really is an important and explicit sense in which we shouldn’t expect a given sequence over any other given sequence, even though we should view certain kinds of sequences as far more likely than others. Thus my own internal tension on these questions.

The notion of a fairness bias—a term I introduced in Section 3.2 along with an excerpt from Smith that reappears in the passage cited just above—is one way I cope with my cognitive dissonance, though I’m not fully convinced it’s correct. A more convincing defense may simply be one of statistical justification for the belief that the coin I’m about to flip right now will not land 92 Heads in a row; namely: such a particular thing is so rare that I’ve never seen anything remotely like it occur, and mathematical models are not enough to override that fact as far as my beliefs about the world are concerned.

The sorts of examples Smith gives of justified surprise, on the other hand, involve event types that I have observed that are not all that improbable. Though, again, their probabilities are not what makes them surprising for Smith. I’ll wonder what it is, then, that makes them surprising in a moment. The thing to highlight first is the connection made between surprise and justified (or rational) belief:

When I park my car on the street, it’s rational to believe that it will still be there an hour later. If it isn’t, then it would be rational to be surprised and to look for an explanation. If my work colleague tells me that she will be at the meeting at 3:00pm, it’s rational to believe that she will. If she isn’t, then it would be rational to be surprised and to look for an explanation. It’s rational to believe that the lights will come on when I flick the switch. It’s rational to feel surprised, and seek an explanation, if I flick the switch and the room remains dark. (Smith 7)

I see no problem with the idea that it’s rational to believe one’s car will still be there, and thus rational to be surprised when it isn’t. I agree with this even for the person who believes the plain fact that cars go missing from time to time. But does this not point to a tension in Smith’s account? Given that it is a simple and unremarkable fact that cars sometimes go missing, should a rational mind not resist the belief that it won’t, and thus the surprise that follows when it does? I trust this isn’t too nit-picky a question. Though it probably would be uncharitably nit-picky to commit Smith, given the nature of his disagreement with Borel’s law, to the fact that something unlikely was going to happen to the car—a fly could have landed on it in three precise spots at three precise moments in time, or any one of a nonillion, or even infinitely many, other things could have happened, depending on how we slice up and establish references classes for the events surrounding the car. So why be surprised when something unlikely does happen? Even more generically and far sillier: some event was going to happen, so why be surprised when one does?

I’ll happily pull back on these latter lines of reasoning and will leave them for you to contemplate, or not. I stick to the question, though, of whether Smith’s account allows for rational surprise in the above cases. As we’ll see in the next section, he will go on to justify his beliefs about the above events, at least to some significant degree, as not only “natural” but “rational” on the grounds that “there would have to be some explanation” if the car goes missing (Smith 6).

I’ll conclude this section by reiterating certain key points I would like to have at the forefront when moving to the next section.

I can’t view 92 Heads in a row as an extremely rare, singular event, particularly when comparing its unique features (all the same result) to the features of a the vast majority of 92-flip run (higgedly-piggeldy results). Patterned events such as 92 Heads in a row are so rare, in fact, that I’ve never seen a coin give even a hint of exhibiting such behavior, and so I simply cannot get myself to believe that, at the very least, the coin I’m about to flip right now will fail to exhibit such behavior. If that belief is justified, then it is justified for me to surprised should it happen. And given just the rareness of the thing, I am also justified in investigating its cause when the results matter (e.g., when a team wins 20 flips in a row). This is mostly in line with Smith’s account, though it’s not clear that his account allows for the belief that 92 Heads won’t happen right here and now is justified. Though I have not reconciled why it would not be with his claim that it is rational to believe the car will still be there after an hour, and thus to be surprised when it isn’t.

The discussion so far seems to demonstrate that it may be possible to talk ourselves out of (and maybe even into) surprise about anything. Sometimes such reversals will be appropriate. But when is it appropriate? The potential arbitrariness involved in answering this question seems to fall under the broader difficulty of attempting to design any sort of nuanced, quantified systemization for the purposes of prescribing the correct way for humans to interact—behaviorally and cognitively—with the world.

Ultimately, it seems to me that arguments over how to effect such systemization will always boil down to appeals to intuition about whether or not it’s rational to believe that, say, this series of 92 flips I’m about to executive will land all Heads. In particular, what rates higher: consistency with the best (or at least most fashionable) mathematical models one has learned of and understands, or consistency with the personally (if largely unconsciously) constructed models derived from one’s actual and extensive experience of the world?

Finally, it’s worth remarking that just because something has not happened doesn’t mean it won’t. One day, the sun won’t rise. One day, you might get into a car accident while driving. There are reasons to believe these things will or might one day happen. Volumes could be written on the complex of ideas attending these and similar comments. Every event is only identical to itself, but some events share enough in common to establish a reference class, to count as types, some more reasonably so than others: there’s driving without an accident, driving tired, driving drunk, driving fast, driving ill or in the rain, while wearing green underwear, on a day starting with the letter T. This is a discussion whose full scope I don’t have space for here but that is working in the background, falling under the general question of how probability—how statistical models of various sorts—should, or do, inform belief (not to mention insurance rates).

That in mind, let’s attend more closely to the aspects of Smith’s account having to do with the connections between possibility, cognition (especially, belief and surprise), and investigation. I’ve already mostly covered possibility. The rest, I loosely file under the heading of the Surprise Function.

4.0 The Surprise Function: Belief, Surprise, and Investigation

Smith considers surprise to play an important role in our lives, particularly, it seems, in connection with belief and investigation. It also seems to me that the strong points he raises in this regards work against his fundamental claim that 92 Heads in a row is unsurprising. I’ll begin by looking yet more closely at belief, then will turn to investigation.

4.1 Surprise and Belief

I’ve already well noted that, on Smith’s account, if you believe it possible for a fair coin to land 92 Heads in a row, while you would be justified in believing that event highly unlikely, you would not be justified in believing that it won’t happen, and thus shouldn’t be surprised when it does.

This also works in reverse. Surprise, for Smith, functions as a signal for rational belief, much in the way, I presume, that fear seems to do for danger. A more complicated analog might be the claim from many contemporary moral philosophers that emotions of certain kinds serve to direct our attention to the morally salient features of a situation. In other words, properly tuned surprisingness detectors may help us arrive at more reliable beliefs about the real world:

While questions about what is surprising do have some interest in their own right, what makes them really significant is precisely the way in which they seem to be bound up with questions about what we should believe … Generally speaking, surprise is what we experience when the world doesn’t match our beliefs … Surprise is a guide to belief. …

…if we have reason to believe that something isn’t going to happen, then we have reason to be surprised if it does happen. Or, to put it another way, if we have no reason to be surprised if a certain event happens, then we have no reason to believe that it won’t happen—we should keep an open mind about it. Rational surprise is a guide to rational belief. If it’s right that we have no reason to be surprised by throwing 92 heads in a row, it follows that we shouldn’t believe in advance that this won’t happen. (Smith 5–6)

In particular, I agree that if we rationally believe something is not going to happen, then surprise is warranted when it does. From this, it seems fair to say that a feeling of surprise may signal either a need to investigate (e.g., should my car go missing) or a need to revise my belief (e.g., about 92 consecutive Heads being possible). There are, however, some points worth quibbling with here. If my car goes missing, I don’t need surprise to tell me to investigate (more on this in the next sub-section). And I don’t need surprise to tell me I’m wrong about 92 consecutive Heads being possible. If I see it happen, then I’ll know it’s possible.

But perhaps this isn’t quite what Smith means, given that I shouldn’t have been surprised about the 92 consecutive Heads in the first place, given rational beliefs, which is itself is brought into question by the fact that 92 consecutive Heads is supposed to be unsurprising, a claim that is further borne out by there being no need for explanation. I find myself confused in trying to parse all this out, so please bear with me; hopefully my blurry thoughts will have come into focus by the time we’ve discussed the role of investigation (i.e., the need for explanation) in all this.

As I understand Smith’s account so far, among his foundational claims is that if there’s no reason to be surprised by an event’s occurring, or if surprise would be irrational, then one should not believe the event will not occur. This reasoning concerns me. The a priori arguments for the possibility of 92 consecutive Heads simply don’t convince me. And no amount of counterfactual what-if?-ing or mathematical-model-building aimed at ruling out the rationality of surprise is enough to replace, for me, the actual seeing of the thing—or, more pointedly, the lack thereof. I’d be willing to bet a lot that no one has seen 92 Heads in a row.

In other words, if my degree of surprise at such a thing is meant to guide my belief about it, this is a sort of hypothetical surprise abstracted from a mathematical model of the event. The substantive claim, then, isn’t that I should revise my belief just on seeing the thing happen (though, to be clear, even this isn’t trivial, because I then must convince myself that I really did see it happen!), but is rather that I should align my beliefs with a priori reasons about whether I should be surprised were I to see it. This also means that, on the mathematical model’s authority, I should not be too slow in convincing myself that I really saw the event (more on this, too, when we discuss investigation.)

As it stands, I remain, yet again and for better or worse, involuntarily unconvinced by such models to revise my belief that 92 consecutive Heads absolutely would not happen were I to flip this coin right now. And this “local” (let’s call it) belief is in part what leads me to reject a centrally prescriptive surprise model (more below on what I mean by “prescriptive”).

Again, there my be something of an uncomfortable circularity here, where the irrationality of surprise is justified by the irrationality of belief that is exposed as irrational due to the irrationality of surprise; or something like that. But let’s suppose it’s not a fatal flaw and that there are intuitively satisfying interpretations of these aspects of Smith’s account—in particular, investigation’s part in all this may take care of such worries altogether. In fact, I will happily make use of that aspect of Smith’s account myself in order to justify surprise and thus belief.

But for now I’ll rely on the separate claim that the local belief that 92 consecutive Heads won’t happen. In fact, I claim that this is true of any particular event when the probability whose probability of being predicted in advance is nearly guaranteed to fail—i.e., is so close to zero that, as far as any human’s capacity for grading belief in degrees goes, is practically is zero. Correctly predicting what this coin will do right now should I flip it 92 times has a probability of about 0.0000000000000000000000000002 of occurring. Not zero, but close enough as far as my belief is concerned. Epistemologists like to challenge such claims by asking stuff like, “Would you bet on it so that if you’re right you win a penny and if you’re wrong you lose your soul to Satan?” Who knows. Insert a few more trillions after the decimal and I think my belief would survive such tests—though I’d have to factor in the fact that I now apparently believe in Satan, and whatever changes that means for the laws of nature.

Assuming less fantastical circumstances, were I to flip this coin right now, it would be safe and rational—practically a sure thing!—to bet against any specific 92-flip sequence appearing, including a run-of-the-mill, random appearing one. Smith agrees that this bet is “perfectly rational,” but maintains that it’s irrational to believe a specific sequence won’t come up and thus, “If the coins did land THTTHHTH… or land TTHTHTHH…, then it would not be rational to be surprised” (Smith 6).

I’ve already claimed that I take such a belief to be rational. If not for 92 flips than certainly for 92 thousand. I would bet the galaxy that I couldn’t manage to correctly guess that many flips in a row, and even that such a thing has never happened in the history of coins in this or any other galaxy. It strikes me that, at some point, and some point well before 92 thousand flips, rational willingness to bet converges to rational belief.

But I don’t have a knock-down argument for this other than that such a thing is unheard of, in a certain sense, and that this fact is intuitively indomitable.

I also believe that this intuition is reasonably unmoved by what epistemologists refer to as an agglomeration problem. That is, I seem to have committed myself to the following awkward scenario. If you gave me a list of all 2⁹² permutations of 92 coin flips and ask me to check the boxes of each permutation I believe won’t occur right now, I would click Select All. And yet, I do believe one of them will occur. I think I can avoid paradox here by making my beliefs clear: I believe that, were I to select one permutation at random, I will fail to select the one that actually arises in

There is a temptation here to make some Bayesian claim—namely, that I believe to a degree of 1–(2^-92) ≈ 0.999999999999999999999999999798 that it won’t happen, because that’s approximately the probability that it won’t, and thus my belief that it will happen is indeed non-zero; but, as I’ve already alluded to above, it’s a fantasy that belief could ever be so microscopically granular.

There are perhaps more thorough and convincing ways to deal with agglomeration¹¹, but I’m satisfied for present purposes to maintain that I rationally believe the coin I’m about to flip right now won’t land any specific sequence you’d like to suggest. And so if it did, I’d be rationally surprised.

At the very least, I’d be moved to investigate in search of an explanation.

4.2 Surprise and Investigation (and Much Else Related)

4.2.1 The Point of Surprise

Smith reasonably observes that

…part of the purpose of surprise is to spur us into action. If an event surprises us, then that prompts us to investigate why and how it happened—to try and explain it. (Smith 6)

He extracts from this observation a principled, biconditional (i.e., if-and-only-if) view of the relation between surprise and investigation, to which he seamlessly adapts the conjunction principle:

Roughly speaking, it’s rational to be surprised by an event if and only if that event requires investigation and explanation. … If there’s no need to explain the fact that e₁ occurred, and there’s no need to explain the fact that e₂ occurred, and the events have nothing whatsoever to do with one another, then there’s no need to explain the fact that e₁ and e₂ both occurred. (Smith 7)

Unsurprisingly, my concerns about the conjunction principle apply here, too. Explanation is required when a team wins not one, but 20 consecutive coin tosses. Granted, if cheating turns out to be behind it, anything surprising about it would not reside in the coin tosses themselves. So maybe suspicion is at least as likely to be aroused as is surprise, and appropriately so given that you’d have to believe the 20 consecutive tosses were won fairly, and it’s unlikely that they weren’t. But if an investigation shows that the team won fairly, then surprise may dominate suspicion, even if only because the winning streak turned out to be due to blind chance rather than cheating.

Though I’d bet many fans of opposing teams would never stop being suspicious. Come to think of it, fans of the toss-winning team may well decide that the world’s (super)natural forces recognized that their team deserved to win, no cheating necessary, which aligns with the general observation that the typical fan’s rationality-regulating machinery, at least when it comes to their sports fandom, has its own physics, the specific laws of which change from play-to-play, and so, then, does what counts as “rational”—but there is a simple law that guides all others: their team is to have the most points on the board by the end-of-game buzzer. Or maybe they’d just rest firmly on the thought that lightening does sometimes strike and why not here.

It seems to me that these folks are within the reasonably subjective range of suspicion and surprise, that to feel or not feel those things is not a question of rationality but personality, or at least personal circumstances—which inclines me to say that rejecting the event as cause for investigation would be irrational insomuch as that rejects the feeling of surprise or suspicion on the grounds of their being irrational.

These are nit-picky distractions from the central point: it seems reasonable to say that 20 wins in a row warrants investigation precisely due to how unlikely that event is, even on reflection of the conjunction and equiprobable principles; and this is so even if surprise is or isn’t rational, pointing again to the reasonable debatability of the question, to the subjective dimensions of the question, at least in the case of 20 flips, particularly once the explanation of blind chance has been rooted out.

Significantly, I assume here that blind chance counts as an explanation; or, more precisely, the features that compose it—symmetry, vigorous tosses in a stochastically interesting environment—do. I also assume that the more intense the (potential) surprise, and the higher the stakes, the deeper the investigation must go and the more features of the event must be investigated. Suppose every team from a given city has won every coin toss for several years, amounting to hundreds of consecutive wins. This might, just might, be enough to even make home-city fans suspicious (if not willing to endorse an investigation).

And I of course insist that the Aleatoric Music and Spooky Cards examples merit yet deeper investigation—to the point of maniacal, soul-searching obsession—if only for the stakes they raise about a certain picture of reality. Should they occur. (They won’t.)

That these assumptions seem reasonably aligned with Smith’s account, however, suggests a tension in the account. Namely, he maintains that low likelihood alone is, in general, insufficient for warranting surprise, and thus is not enough to prompt investigation—i.e., once chance is understood to be at the root of things, there is nothing more to explain. But at the same time, his account seems to recommend investigation in such cases; or, at least, it doesn’t seem to provide a clear, principled reason to rule it out.

To begin teasing out the tension with appropriate delicateness so as not to overstate it, consider again the following biconditional statement, which must be kept in mind from here on out:

Roughly speaking, it’s rational to be surprised by an event if and only if that event requires investigation and explanation. (Smith 7)

The “if and only if” implies that events require investigation and explanation if and only if it’s rational to be surprised by the event. I’m happy to accept this strong principle, but on two (I think reasonable) conditions.

The first condition is that it deals in surprise of a certain kind, picked out intuitively, I presume—one can imagine an assortment of situations in which surprise does not point to a need for explanation, and others in which surprise would be a strange response though there is a need for explanation.There are also other reasonable (evolutionary psychological, etc.) accounts of the point of surprise—e.g., to alert one to a change in environment (which may or may not require action). Smith and I do seem to have in mind the same sort of surprise; namely, that which he claims to experience fairly easily, such as when he “flick[s] the light switch and the room remains dark” (Smith 2), and which I presume I’d experience when landing 30 Heads in a row. (Actually, as I’ve already noted, Smith seems to imply that he would be surprised here as well, if despite himself and irrationally.)

The second condition is that the task at hand should be seen to some degree as descriptive rather than as a purely prescriptive—e.g., in which we arbitrarily define surprise as something that should only be felt according to some arbitrary standard of usefulness, and then enjoin folks to thereby abide. I would like to discuss this point further before taking on the above-noted tension.

4.2.2 Descriptivist vs. Prescriptivist Accounts of Surprise, Rationality, and Intuition

To make my meaning clear:

Descriptivist: What are the sorts of things people in general are or tend to be surprised by?

Prescriptivist: What are the sorts of things people should be surprised by?

Smith’s project leans prescriptivist (or what we might alternatively call “normative”), in the sense of correcting judgments that inform surprise by aligning them with basic standards of rationality. Recall my earlier example in which I claimed that it would be irrational to rate as unsurprising 20 Heads out of 100 flips, but as surprising 80 Tails out of 100 flips; this is irrational because those are exactly the same event. It’d be like saying the water in the lake is wet, but the H₂0 (or agua, the Spanish word, if you prefer) in the lake is not—assuming, of course, the speaker knows the meaning of all the relevant terms. Facts about the world sometimes demand corrections in judgment, including that which informs surprise. (I’m reminded here of an exchange in Dostoyevski’s Crime and Punishment (1866), in which Lebezyatnikov argues that…¹²)

On the other hand, there is also some range in which surprise is reasonably subjective. It is conceivable that, just as two people with similar cultural and educational backgrounds may look at the same evidence, even within the “hard sciences,” and reasonably form different beliefs or degrees of belief, they may also reasonably experience differing degrees of surprise.¹³

The boundaries of that range are fuzzy. But there are obvious examples outside of them. It would be inappropriate for me to experience sustained surprise by waking up tomorrow morning in the same room and bed in which I went to sleep and have done for hundreds of nights in a row (I have no reason at all to believe I won’t wake up there, though I acknowledge that such things do happen unexpectedly—a vivid dream may even be enough to do it, for at least a moment). And if a math teacher sincerely told me she found it surprising that 2 and 3 make 5, but was not similarly surprised by any other arithmetic statements (including that 3 and 2 make 5), I would question her rational capacities.

Exactly what makes surprise inappropriate in these obvious examples seems to differ, but it may be difficult to say how.

The math example seems to be a question of logical or internal consistency. The inconsistency here is obvious, but there may be similar statements that would surprise someone with even a fairly proficient command of basic math. For example, some folks are surprised when shown that .9-repeating equals precisely 1. Are they irrational to be surprised, particularly given that they have no problem with accepting that 1/3 = .3-repeating, nor with adding those figures together three times to clearly yield 3/3 = 1 in fractional form, and .9-repeating in decimal form? Some folks go further than surprise here and flat-out reject the identity. Are they being irrational? Defense of the rejection is usually framed mathematically—“it really just rounds to 1” or “it never actually reaches 1”—and the skeptic is unmoved when reminded that this should also mean that .3-repeating never “reaches” 1/3 and when shown how inconsistent their view is with their existing understanding of the real number line, etc.

“Rational” here becomes again a question of whether they are to be internally consistent with what they believe about math, which is really all we’re asking for when claiming that .9-repeating is 1, or are to be externally (let’s say) consistent with their experience of the world as a discretely, finitely experienced set of events. Which brings us to the waking-at-home example, where the standard of consistency is based on my experience of the world. In short, the world as I understood it at the time of going to bed was such that I had no reason to be surprised by waking up in the same bed and room I fell asleep in.

So, on the one hand we have a kind the internal logical consistency of mathematics, and on the other we have an external logical consistency of how one understand’s the world to be from experience. Interestingly, rationality here seems to be regulated by the same basic notion of logical consistency—when I believe the water in the lake is wet, but the agua is not, there is a severe logical inconsistency at play. Even more interestingly, while the intuitive designation of internal and external logical consistency may actually be arbitrary, there is a point to separating out distinct sets of facts or propositions as mutually involved in constituting a harmonious or consonant or coherent or consistent etc., local system.

In other words, the mistake the .9-repeating deniers are making is, again, that they are being inconsistent with their basic understanding of math. Accepting the internal (or local, if you prefer) consistency of .9-repeating being 1 need not have anything at all to do with the “external” or “real” world—what we might broadly call “nature.” Any more than does conceiving of the world in Euclidean terms when faced with a 9th-grade geometry test.

The question before us today is what sorts of standards of consistency should be assumed given basic probabilistic models that are uncomfortably situated at the intersection of math and nature (there are some that are comfortably situated, but those aren’t at issue here). A probabilistic model of 92 coin flips involves a sample space of 2⁹² equiprobable outcomes. I have no problem holding up this model when aiming to be mathematically consistent. I do have a problem with as my guide when I examine nature—it seems starkly inconsistent with the world as I understand it from experience. I have a harder time reconciling these because I’m supposed to view the model as a correct description of nature (according, at least, to folks like Smith), unlike with, say, Euclidean geometry (though it was for a long time assumed to be a correct description of nature).

Here again, though, I find myself more angled than needed towards the question of possibility. What counts as surprising can’t just be that which materially contradicts what one explicitly disbelieves on the grounds of being impossible (Smith must agree, given his examples of genuinely surprising events). I’ll zero in, then, on the observation that, however the lines are laid in demarcating distinct systems of facts or propositions, working out what is or should be surprising isn’t clear even when the rules are pretty well understood. I mentioned before that it seems possible to justifiedly believe to two mutually incompatible propositions while justifiedly rejecting something they jointly imply. Perhaps no one has ever had cause to notice the contradiction. Stumbling on it would likely be surprising, then. It would be interesting to construct such an example, but instead I’ll present a less severe example, one of my own surprise—of an exquisitely delightful sort—at a mathematical discovery.

Last summer, in working out the probability of getting two Heads in a row within a certain number of coin tosses, I found that the number of ways to get each result is described by the Fibonacci numbers. I stumbled on a similarly wonderful surprise later when I noticed that the Tribonacci numbers achieve this for three Heads in a row. I share the math in “Fine-Tuning.” Perhaps someone else, for some reason or another, would not be surprised by this observation. This example is of course not about the relation of surprise and low likelihoods, but rather about rational consistency in one’s views of the world. At least, it is about the observation that the threshold at which surprise will ensue may reasonable vary from person to person, even those who share many fundamental facts about the system in question.

The same may even be true for the wake-at-home example, even just in terms of its probabilistic dimensions. Revisions to my belief about the world could lead me to assign a lower probability to waking up at home. It would be strange to insist that there is some absolute threshold at which surprise becomes irrational.

The long and short of it is that there are examples of clearly inappropriate instances of surprise, but there is also some fuzzily bordered range in which surprise is reasonably subjective. Those borders are matters of intuition and cannot be directly controlled. Again, Smith acknowledges this. His claim is not that we who’ve thought hard about the question would not feel surprise at seeing 92 consecutive Heads, but that we should not.

Which brings me to an important question for Smith. Why do some folks who’ve put their intuitions to the grindstone on the question of 92 consecutive Heads come away convinced not only that surprise would rational, but that it would be downright strange not to be surprised? See again, for example, van Inwagen’s damning characterization of the equiprobable principle.

[EDIT: See also the following conversation between physicist Sean Carroll and philosopher David Albert on the difficulties of making sense of probability under a many-worlds interpretation of quantum mechanics, one difficulty therein being that it seems to threaten the surprisingness of seeing the same result over many trials. Something Albert goes so far as to refer to as astonishing at one point, and which he seems to think signals deeper theoretical challenges to the many-worlds interpretation. For example, getting the same result over and over again—Albert’s specific example is finding a million x-spin up particles z-spin up, when there’s a 1/2 probability for the z-spin being up or down—would seem to count as evidence against quantum mechanics, but the many-worlds interpretation predicts with 100% certainty that there will be such a world in which this happens. They agree that such a thing would “amount to a gigantic surprise,” and, moreover, they seem to agree that surprise is justified. How to justify that surprise seems to be their point of disagreement.

Carroll seems to think the surprise is accounted for by reflecting back on the rarity of one’s situation, while Albert seems to find this unintelligible, along the lines of asking a person about to be perfectly cloned if they expect—or would be surprised—to end up on the left or on the right post-cloning. There is more nuance here than I’m doing just to, so I recommend watching the video to see for yourself (at YouTube; starts at around 40 minutes and 42 seconds): Science Saturday: Problems in Quantum Mechanics | Sean Carroll & David Albert (recorded July 2008). I’m particularly fascinated by Albert’s comment that attempting to make sense of probability in many-worlds scenarios is pushing us to think more deeply about what probability is. I’ve begun collecting papers to get a better handle on what he means.]

Additionally, why should we not view this distinction in intuitions as significant for our understanding of rationality? Question: How do we know that it’s appropriate—i.e., rational—to sit unsurprised by a single coin toss landing Heads? Answer: We agree that it’s intuitively obvious. It’s not so intuitively obvious, however, how to appropriately respond to the event of 92 Heads in a row. And so we find ourselves in disagreement.

It trikes me that 92 consecutive Heads may exist in a kind of middle ground for many folks. But it’s easy to imagine events that, should they occur, would astonish anyone who notices—such as the Aleatoric Music and Spooky Card events, if we need to go as far as those, which are far more farfetched—i.e., improbable—than the Grains of Sand case. It would strike me as odd to declare a rule from which it follows that surprise is irrational in such cases, and thus that nearly all humans are irrational, even those who are irrational despite their commitment to the conjunction and equiprobable principles.

But this points to yet another source of difficulty. It is pretty well understood that humans are not rational in any strict sense, though this is less true if we maintain that one must be aware of the parameters of the inconsistency to really be irrational, as people tend to be ready to revise their logical inconsistencies—including perhaps how they inform beliefs and emotional responses—when they are obvious (for the go-to bible of such experiments, see Daniel Kahneman’s 2011 book Thinking, Fast and Slow). But more broadly, it’s a good bet that humans are, strictly speaking, failures at rationality. Nor is it evident that we want, or even should want, to be otherwise.

It seems the sort of rationality Smith is—or would like to be—dealing in falls under the narrow view, wherein appeals to rationality can cure us of ill-informed emotions, and so on. That is, he characterizes rationality as having a point, as being desirable as being useful, as helping us form actual beliefs that inform how we engage and navigate the actual world, and so on. We are meant to strive to be rational. In its best-function mode, it cannot even be helped once the appropriate evidence has been displayed. Indeed, this isn’t just a correction of what we claim is rational in principle, but what we intuit as rational: so, this is also a correction of intuition.

It should by now be clear that I endorse such prescriptivist efforts. Not to mention that most of the articles I’ve written on probability are geared towards making counterintuitive aspects of probability intuitive. But I have also noted that such projects shouldn’t be taken too far. I’m shaky is on where to draw that line. I’m particularly suspicious of establishing definitions that set impossible goals for minds like ours—even rationality-striving ones—on the grounds of logical consistency. There are theoretically powerful arguments to be made that, for all you know, all creatures but you are philosophical zombies—i.e., are without consciousness—and, were you rational, you’d align your beliefs accordingly. Even more powerful is the so called skeptical argument, which persuades many a two-handed undergrad philosophy student, if only for a moment, that they don’t really know they have two hands—though it’s ok to believe they do, just not with certainty, perhaps (if they’d like to be philosophically and logically consistent, that is; i.e. rational). The skeptical argument exploits the limitations of evidence-from-experience, thus overriding that mode of rationality in favor of a logical, deductive one.¹⁴

That said, even if it is in some cases true in some strict or technical sense that no human is fully rational (this is definitely true), it may not be useful to set an impossible threshold even in that strict domain. Particularly when that domain is appealed to in practical application—as Smith, along with most theorists about rationality, seems to do. That is, the discussion here isn’t just one of how in some meta-cognitive attitudes should be a function of a sample space (or perhaps the sample space because a function that maps real-world events to some appropriately graded attitude)—but rather is a discussion about how we really should react when confronted with events of a certain kind of probabilistic status. That is, events that are viewed as exceedingly rare—even to the point of being unheard of—under certain interpretations (e.g., any specific 92-flip sequence is extremely unlikely, while some 92-flip sequence is guaranteed).

I’ll try (but will almost certainly fail) to say this in plainer English. I take surprise to be a psychological state, an emotion. Like fear or elation. So, when Smith aims to persuade the reader that being surprised by 92 Heads in a row is just like being surprised by getting Heads in a single flip, it’s designed to do the same work as bringing to the attention of the person who’s scared of heights the fact that, when they are afraid on the second latter rung, their fear must be misfiring given that they are now no higher off the ground than when walking up a given step of the stairs they fearlessly ascend every day. Just as the fear then melts away, so does the surprise.

But what if the fear doesn’t melt away? Is the person being irrational, in the sense of being logically inconsistent? Or is there some sort of internal consistency here? For example, the fear on the ladder is consistent with going vertically ever higher while stairs do something else—and why shouldn’t one be afraid of going vertically high? This sort of irrationality seems to come down to a heightened sensitivity to stimuli—more precisely a certain kind of autonomic response following a process of assigning meaning to a situation, a series of events for which “sensitivity to stimuli” is shorthand—and as such is only “rational” to the extent to which the business of meaning assignation tempers the autonomic response. There may be little connection in this regard at all, making it a kind of category mistake to refer to it as (ir)rational in a logical sense, but here we do have a somewhat technical background to refer to.

A typical characterization of phobia is as an irrational fear. So I’m giving an incomplete picture here by invoking internal logical consistency. The question surrounds a higher-order rationale connecting the cognitive state to some model of the world, not the world itself. The typical picture holds the measure of rationality to be the measure of the extent to which that model matches the world itself. The guy who insists on frantically checking every page of every book on his Kindle for signs of bedbugs is, in this sense, irrational, particularly if he’d rather be doing other things with his time while knowing very well that that next page he swipes to has no chance of bearing traces of bedbug poop. The higher-order rationale rates irrational given not just its local disconnect from reality, but to the extent that it is relied upon for the phobic’s well being within reality, by way of the phobic’s complicated, meaning-processing mind’s model of that reality, etc. (and, to be clear, we may suppose it’s not about the bedbugs; such a compulsion could be a way of warding off, or of satisfying the desire to ward off, some other unwanted event—some greater tragedy—entirely). This may not always be so debilitating (e.g., the typically superstitious sports fan, of the sort that maintains every call benefiting the home-team to be a correct call), but if you’re missing work to search your Kindle for bedbugs, there’s a problem.

Here again the lines between a technical, or we might say clinical, picture of rationality, which surely has its uses, and what we mean in a colloquial sense blurs. There is of course a tendency from certain factions of thinkers to reject any notion of rationality outright. But this is due to baggage we need not get into here (e.g., certain aspects of Enlightenment ideals about “the true and universal nature of man” and so on), which may be avoided by holding onto whatever technical—and genuinely useful—definition one may be using the word “rational” for in a given context (e.g., dealing with phobias), while simply using a different word. That said, we are presently concerned here with a picture of rationality that involves an unremarkable, run-of-the-mill model of reality. One in which the typical, say, human adult New Yorker would be surprised to see 92 Heads in a row but, according to Smith, should not be—at least not if that New Yorker is, or aims to be, rational. I’m trying to get a handle on just what sort of corrective or curative is involved here. That is, if the aim really is to adjust the model of reality held by most of us, to better align their respective geometries; or if, rather, it aims to simply set up a kind of ideal or rationality, in some technical or at least quasi-technical sense, to which those who esteem contemporary notions (e.g., decision-theoretic) of rationality should assent. I take to be principally the former.

A better example, then, may be something like the following. You think you see a snake on the floor of your bathroom. But it turns out to be a shadow of your own arm. It would be irrational to fear the shadow as you might a snake. Or how about this, replacing fear with jealousy and anger. You think you see your significant other making out with someone in the park. You’re shocked, as you’ve never suspected such a thing. The person turns around, and is clearly not your partner. It would be irrational to then go on a jealous rampage.

As often happens here, these topics admit of far more discussion than I can hope to get into. I bring them up to raise questions about what sort of cognitive state surprise is meant to be. Are misfirings of fear and surprise responses analogous? Are consecutive Heads meant to be like standing continually on the second rung of the ladder, no matter how many you achieve? Like seeing that your arm is a shadow? Like seeing that the person in the park is not your lover? And here’s a worse example.

I’ve been told that my loathing of cars is irrational, that car accidents are relatively rare. I don’t have a phobia, but folks who do are told they’re being irrational. Good capitalists have cars. Cars are necessary for the economy. And so on. I’ve had multiple friends and even more acquaintances and classmates and coworkers killed in car accidents. My mother’s mother was killed in a car accident and my mother was herself hit by a car making an illegal left turn (she healed, but it wasn’t fun). My father was hit by a truck (he also healed, with some disability). I recently learned that the dean of my old college was hit by a car and put in critical condition. There are about 40,000 people killed in the U.S. in car accidents each year. It’s not so rare. I personally think it’s irrational not to be at least somewhat afraid of cars—and not just of “other drivers.” My favorite expression of our collective irrationality surrounding cars is the irony that it is taboo these days to leave one’s kids in a car to pop into a store, perhaps could even get your kids taken from you by the authorities, while transporting children in a moving car is perfectly acceptable. The latter is obviously far, far more dangerous.¹⁵

But maybe your attitudes about cars is more positive than mine. Maybe your experience of cars is less horrific than mine or the data just don’t move you. Which of us is right? There’s no mathematical model to tell us, though we may well agree, on intuitive grounds, that it would be irrational for you to feel safest when driving a car at 100mph blind folded in busy traffic, and it’d be irrational for me fear moving a car 20 feet at a snail’s pace in a wide-open lot. In between there somewhere, the fear becomes personal, so that to feel it is, if not exactly rational, is at least not irrational.

This goes, too, for the ladder. Some human adults would be afraid to stand—let’s assume they are able to stand, etc.—on the 12th but not the 11th ladder rung. While others feel no fear at the top—or even thrive on a kind of thrilling fear there, which must be qualitatively different than a debilitating fear and which, at some point, may rightly give us cause to worry about the correspondence between that person’s judgment, or model of the world, and the world itself: i.e., about that person’s rational capacities. Somewhere between the first ladder rung and icy-handrail handstands a thousand feet up, fear is a question of personality and not rationality.

By this point, it is apparent that systematically connecting rationality and surprise threatens to become a tangle of different projects falling under cognitive psychology, evolutionary psychology, behavioral economics, and decision theory, to name only three. Teasing apart how these and other projects relate and don’t relate, overlap and don’t overlap, are purely technical and at-least-somewhat colloquial… is beyond daunting.

Rather than attempt even the beginnings of such a thing, I’ll simply reemphasize the distinction I’m principally concerned with here, which is that between descriptive and prescriptive motives. That alone is difficult to parse. I have been treating surprise as a kind of cognitive state similar to that of fear or envy. Phenomenological lines are difficult to draw—that which lies between heartbreak and physical pain, if there even is such a line as far as the nervous system or person are concerned.

It’s worth briefly reflecting on this aspect of surprise as well, which constitutes not only a phenomenological dimension, but a behavioral dimension amounting to an external report—voluntary and involuntary—of that experience; e.g., the loudness of one’s exclamations, the roundness of one’s eyes, the stamina of one’s investigative drive, the rate of one’s heartbeat. I take it the focus here is largely phenomenological—”you shouldn’t experience surprise,” but also to some degree behavioral, in the sense that, even if we knew we would feel surprise, we should see ourselves as committed to the idea that the surprise is irrational.

Compare this to, say when a patient is asked to rate her pain on a scale of 0 to 10. In general, it would seem strange to tell her that, given the nature of the stimuli in question, she should report some specific number. On the other hand, there may be times when a sincere report can be brought into question or scrutinized, such as by reminding the patient of past ratings she’s given in seemingly nearly identical conditions.

Meaning may play an important role here, too, as the pain felt during, say, receiving a massage or working out or getting a tattoo or having a tooth repaired registers in psychologically distinct ways in comparison to how we may process materially identical stimuli in different contexts. The same stimulus, in strict terms of object-meets-nervous-system, may thus warrant different ratings, due to the degree of worry or arbitrariness accompanying the meaning of those stimuli.

I won’t belabor this comparison, but will just point to the interesting interplay here between phenomenology, meaning, behavior, and so on, and to the interesting possibility that pain itself, in its most stripped-down physical forms, may be in some cases irrational, if only in terms of its degree or as a matter of interpretation (e.g., when pain may be properly interpreted as “suffering”).

This is obviously not simple stuff. I’m not so sure surprise is any simpler, or that it even constitutes a class of its own as far as responses to stimuli go. Particularly if one’s aim is to systematically match surprise responses to event types, especially in some prescriptivist sense (and super especially when this implies the existence of some inherent quality of surprisingness in events; see the comments below on “The Objectivity of Surprisingness,” Section 5.2.4, for more on this).

Having sufficiently confused myself on these matters, I’ll go forward assuming that all involved in the present discussion recognize the need for, and elusiveness of, a proper balance of prescriptivist and descriptivist motives. Elusive indeed: it seems to me that much of our apparent disagreement very well may come down to differing opinions about where the dividing borders lie between those motives, and which features therein outlined merit greater emphasis, and so on.

4.2.3 Comments on the Objectivity of Surprisingness

These comments may be needless digression, though I think the difficulties surrounding the notion of anything like objective or absolute surprisingness at least merits acknowledgement.

To be clear, I don’t mean to imply that there are cases in which something is ever surprising in some objective, observer-independent sense. Just as sugar is sweet to an observer and fire is hot to an observer, the 4-foot dunker is surprising to an observer.

Beyond this basic observation, however, I’m not sure of how to parse out the notion of objective surprisingness. For example, it may be that things can be objectively unsurprising in a vacuous sense: if there are no observers in the world, then everything is unsurprising due to no one being there to be surprised. This is debatable, as we may think it more appropriate to view surprise counterfactually. If, in an uninhabited world, some object like a coin were jostled around 92 times and came up on the same side every time, we might say it would be surprising were a “mind like ours” there to see it. In that sense, the event just is surprising. I suppose.

This threatens, however, to pin the ontology of surprisingness down to its potential to surprise at least one mind—or more vexing still, one “rational” mind, even if no such minds exist or could exist. Even if many minds exist that don’t find the event surprising. Surprisingness thus threatens to become something we don’t have access to, which seems absurd, as it is a word we give to certain cognitive states that we have meta-cognitively observed ourselves taking on, rather than it being a phenomenon that pre-exists cognition. Though the potential for that phenomenon clearly pre-exists cognition.

I take this to be more interesting than the, in some ways analogous, question, “If a tree falls in the woods and no one is there to hear it, does it make a sound?” No, it doesn’t “make” a sound. When the lonely tree falls, nearby air molecules vibrate so that, were there present some audition-endowed creature wired up with eardrums and auditory neurons and so forth, the air molecules would stimulate that delicate machinery so that the creature’s brain would produce an experience of the sort we call “sound.” Vibrating air molecules alone won’t trigger such an event—the event is phenomenological and as such requires a hearer. (So how do ghost’s and angel’s hear?)

But there does exist here the potential for such an event. In this way, perhaps the tree and surprise questions are interestingly analogous. The idea is that a tree makes a sound for any sort of creature that has something like what we’d call audition. There may be other sensory responses to the air molecules’ oscillations, of course. Depending on ones phenomenological machinery, they could produce a visual experience (and not just in synesthetes: it’s often noted these days that blind folks may in some literal sense see in this way). But the point is that there must be an observer present. Surprise is more complicated than being a phenomenological representation of vibrating molecules—it requires meaning, for example. Which gets me to my point. I have been conceiving of a potential for minds like ours. Which is to say minds at least as complex as ours. We conceive of much more complex—god-like, let’s say—minds as having our capacities and then some. Such minds don’t just contain a representation of—don’t just “hear”—the falling tree. They know the trajectory of every single one of those air molecules, they hear every leaf, every drop of moisture in the air, every hair on tails of the squirrels scurrying out the falling tree’s way, and on and on.

If there are supreme beings watching over us, then everything makes a sound. Or if we can imagine such beings counterfactually existing, then everything potentially makes a sound. But is anything surprising? Why would that mind be surprised by anything? If it would be, then I suppose we would say that the surprisingness of an event could be inherent in the event itself, purely qua event—i.e., as a phenomenological “object”—if not in any physical substance. In other words, surprise is about the event constituted by the observer, which is arbitrarily connected to the phenomenological capacities of the observer.

This brings us back to the fact that we are thinking here of minds like ours. I take it the laws of rationality apply as much to our minds as to the god-like minds. Or is this capacity-dependent? For example, would the unfathomably complex and diaphanously fine machinery of god-like brains represent pain, real pain like we humans feel? Or would they simply perceive—with one of their trillions of senses that we lack—that there is a stimulus that would be painful for minds like ours? But this casts the god-like mind as lacking some capacity that our human minds have. (Is there a god-like Thomas Nagel asking, “What Is It like to Be a Human?”)

This question of what it could possibly mean for a phenomenal quality to be inherent in a physical thing is as vexing as the more fundamental mind-body problem itself. To posit that it could be admits of the possibility that things can be surprising even when no such observers are surprised. But maybe this sounds like the nonsense of saying it may turn out that sugar is in actuality bitter, we all just taste it wrong; aside from the fact that I have no idea what “sweet” means to you experience-wise (let’s assume that our experience of sweetness is sufficiently similar), and the fact that a planet of sufficiently constituted brains would experience sugar in some other way.

At any rate, I imagine the sweetness of sugar is more phenomenologically robust, in its greater simplicity, than the surprisingness of an event, as this is a vaguer psychological state—an emotion—that involves meaning. As such, surprise, as a concept attached to an experienced bodily state, is more assuredly roughly similar between persons than, say, sweetness. That is, sweetness is learned by association with paradigmatic stimuli, such as sugar and honey—however you experience those things, it is what you will call “sweet” (provided you have a sense of taste etc., and maybe even if you don’t, behaviorally speaking); and I will too. If all things now sweet were to become bitter for all, the word “sweetness” could persist, for example if all things currently bitter became sweet. Degrees and subtleties of sweetness may differ. Perhaps in the same ways experience of colors differ. But there are baseline, paradigmatic anchors for establishing the relations between object-experience-concept-word. Is surprise like this? Are there paradigmatic events? A surprise birthday party? (Does that inspire investigation? I don’t see why it would need to.) Let’s go earlier: the re-materialized face in a game of peekaboo. (Maybe this is the first mystery to be explained for any human child with a present caregiver.)

And here again I find myself in the tangle of projects. Add phenomenology, cognitive neuroscience, and developmental psychology to the list. And did I mention math and logic?

All of this is simply to say that it strikes me as doomed project to attempt to argue what is or is not (see again Smith’s article’s title) or should or should not be responded to with surprise, particularly once we are within the fuzzily-bordered range of subjectively. These questions are famously difficult even for basic sense experiences, such as of color; all the more so for the meaning-laden experiences of psychological states such as surprise, fear, envy, moral repugnance, disgust, and so on, which seem something of a bridge between basic stimuli experience (world-meets-body) and meaningful experience (pain).

4.2.4 A Tension in Smith’s Account?

I’ll now (finally!) address what strikes me as a tension resulting from the purpose Smith assigns to surprise on the one hand, and his account of why 92 Heads in a row is not surprising on the other.

Smith claims that surprise would be

…an inappropriate reaction to a sequence of 92 coin throws, like THTTHHTH…. As unlikely as this sequence might be, it can just so happen that this is the sequence that came up, and there’s nothing more for an investigation to reveal or unearth. Surprise simply isn’t the right response to an event like this. (Smith 6–7)

I thoroughly agree that surprise wouldn’t be the right response to such a run-of-the-mill event, assuming no surprise-making context—e.g., predicting it ahead of time. Smith makes this argument in order to support the claim that, however, that if surprise isn’t appropriate there, it’s not appropriate given all Heads, either, as “the situation is really no different when it comes to a ‘patterned’ outcome like Rosencrantz and Guildenstern’s run of 92 heads” (Smith 7).

A puzzling claim given what he says next:

When faced with [92 heads in a row], of course it is sensible to check (as Guildenstern does) whether the coins are double-headed or weighted or anything of that kind. Having observed a run of 92 heads in a row, one should regard it as very likely that the coins are double-headed or weighted. But, once these realistic possibilities have been ruled out, and we know they don’t obtain, any remaining urge to find some explanation (no matter how farfetched) becomes self-defeating. As difficult as it may be to accept, there doesn’t have to be an explanation for this—and it’s not rational to relentlessly search for one. (Smith 7)

In other words, Smith notes that if you see 92 Heads in a row, it would be reasonable to make sure there’s no funny business going on. In other words, to investigate and look for some explanation. Is the investigation into the coin’s physical properties and so on not the first steps towards landing on some explanation? An explanation that could turn out to be that the coin is fixed or there is a cheat or, finally, that it’s a fair coin and that’s just how things went?

To be clear, it would not be in the most charitable spirit to invoke the sports team example here (though that and far stranger examples are fair game for the conjunction and equiprobable principles). What we need to deal with is the most direct case: You yourself vigorously flip a typical, symmetrical coin, and it lands 92 or 920 or 9,200 Heads in a row. In this case, I agree that the final explanation will be chance, but I disagree that surprise is irrational and, by extension, that it is not rational to continue a search for explanation. For example, were such a coin to land 9,200 Heads in a row, my picture of the world would be revised—my conception of “randomness” and “symmetry” and “fairness” and so on would be shattered. My beliefs—my truest, most sincere and real beliefs—about the world would be at least challenged, and eventually, as with Spooky Cards, demolished. And if this were, again, the case in which I correctly guess 9,200 random-appearing flip results in a row, and I truly accepted blind chance as the explanation, my beliefs about myself would likely be transformed. But how much work would it take to get me to accept blind chance in this case, much less given Spooky Cards? Would it really be rational to accept that explanation in that case?

Now, the reason I said the sports team case is uncharitable is that it is difficult to be sure that the investigation is complete—unlike when you flip the coin yourself. It seems to me that the following example exposes the tension I’m trying to highlight, while staying within the realm of charitableness.

The Repetitive Random-Number Generator: Suppose you create a program that generates random numbers. Suppose it doesn’t rely on pseudo-randomness, but some genuinely random, finely grained process in the physical world. Suppose the generator produces the number 2,136.003 over and over again. You investigate the code and see no issues, yet it still continues generating 2,316.003, over and over and over again. Would this be surprising? Frustrating? Would you continue to investigate? Would you scrap the program?

I’m reminded here of a question asked in Leonard Mlodinow’s 2008 book The Drunkard’s Walk: How Randomness Rules Our Lives: “And how would a scientist react upon flipping open a newly purchased book of random digits only to find that all the digits are zeros?” (Mlodinow pp 174–175).

I maintain that such events may be rationally experienced as surprising, for reasons already noted and, frankly, that strike me as consistent with the connection Smith draws between surprise and investigation. If a typical-appearing quarter landed Heads day in and day out for years on end, I’d need a new explanation of everything—of the world in total. And suppose after the first 63 Heads I switch to a new coin, and it continues the pattern of all Heads for another 29 flips. And then I begin to switch to a new coin with each flip, and every one of those land Heads. This wouldn’t just be surprising. It would be world-shattering.

This would be so even if the explanation came down to nothing more than blind chance. And, what I’m struggling to get across, is that this explanation would not be sufficient in itself, but rather would be the start of a deeper sort of investigation into the logical meta-space in which the various systems of coherence one travels between exist.

In perhaps simpler terms, I aim to convince you—or at least myself—that Guildenstern’s result merits being moved to investigate the properties of the coin (and the reliability of his eyes, and so on) because, as Smith notes, it’s “very likely” that the coin is double-sided or some such. It seems to me that the reasons that justifies his being moved to investigate are precisely the reasons that would merit surprise, and are precisely the reasons he would not be moved to investigate, and should not be surprised, given a random-appearing sequence; or, at least, these qualities are intimately intertwined.

If not rational, surprise here is at least reasonable, sensible, understandable, and entirely non-worrisome as far as the surprised person’s presumed rational capacities go.

Finally, it seems to me that if it’s rational to investigate a sequence, it’s rational to feel surprised by that sequence even if the answer turns out to be nothing but chance. Smith’s account not only allows for this conclusion, but recommends it. That is, surprise alerts one to investigate a coin that is “very likely” fixed, and thus reveals that it is not fixed. I do not see how the coin turning out to not be fixed somehow makes invalid the surprise that inspired the investigation.

Smith may reply here that what rational surprise would warrant is some sort of deeper investigation. A fair point, which I’ll address under a sub-header named after what we might call Smith’s “Just So Happens” principle.

4.2.5 “Just So Happens” and Surprise After Investigation

Smith characterizes unlikely events whose explanations come down to blind chance as events that “just so happen” to come about, and that thus merit no surprise. It’s an intuitively appealing claim, though one I find unconvincing on reflection. Smith makes the just-so-happens quality vivid by contrasting it with a genuinely surprising event:

Earlier on I gave some examples of things that I thought would be genuinely surprising—like this: If I park my car on the street and then return an hour later to find that the car is no longer where I parked it, then that’s surprising. While this may well be an unlikely event, what seems more significant is that it’s an event that demands explanation of some kind. Perhaps someone broke into the car and stole it. Perhaps I parked illegally and the car was then towed. Perhaps I didn’t properly apply the handbrake and the car rolled away…. Whatever the truth, it can’t “just so happen” that the car is now gone and there’s nothing more to the story. If I park my car on the street, it would be natural to believe, in an hour’s time, that it’s still there. I think it can be rational to believe this too, because there would have to be some explanation if it turned out to be false. (Smith 6) …

When I park my car on the street, it’s rational to believe that it will still be there an hour later. If it isn’t, then it would be rational to be surprised and to look for an explanation. If my work colleague tells me that she will be at the meeting at 3:00pm, it’s rational to believe that she will. If she isn’t, then it would be rational to be surprised and to look for an explanation. It’s rational to believe that the lights will come on when I flick the switch. It’s rational to feel surprised, and seek an explanation, if I flick the switch and the room remains dark. There are many things that we can rationally believe—but the claim that we won’t throw 92 heads in a row is not one of them. I can rationally regard it as extremely likely that I won’t throw 92 heads in a row, but I can’t rationally believe it. (Smith 7)

I wonder, would it be irrational on Smith’s account not to be surprised at his examples? I assume it would be, but won’t push the point. At any rate, surprise seems a reasonable response to finding one’s car gone so quickly (barring special circumstances etc.). Also, I do find interesting the contrast Smith draws between the above sort of example and 92 Heads in a row. Cars don’t just so happen to disappear. But if a fair coin lands 92 Heads in a row, then it would seem there is no further explanation needed than that it just so happened to land that way.

That’s a huge if. And I’m not convinced that it really is correct that no further explanation is needed than “it just so happened.” I’m not convinced that I wouldn’t conclude that I live in the Matrix or a computer simulation in which pseudo randomness is in effect. There are plenty of fancily credentialed philosophers and physicists and mathematicians you can talk to these days who give pretty strong support as it is to propositions along these lines, ¹⁶ so it’s not positing anything supernatural or all that farfetched. Given, that is, a sufficiently improbable event—like the clouds over every town and city one day spelling out “your world is a computer simulation.”

Or, from a Bayesian perspective and starting with even a very small prior probability for a coin being non-fair, by the thousandth consecutive Heads, one’s rational belief should be very nearly 1 that the coin is non-fair, even if you can’t figure out how. Even if once convinced that the coin is fixed, there may be (on my view) rational surprise about it landing Heads indefinitely, precisely because the coin has been analyzed with the best equipment in the world, and seems as symmetrical as any other coin, yet continues to land Heads.

At any rate, I’ll momentarily set all that aside and, for the sake of argument, will agree that if a coin lands Heads indefinitely despite the best equipment in the world showing it to be fair, that the explanation must be blind chance. In other words, I’ll agree that if this were to actually happen due to blind chance, then I would have to agree that it actually happened due to blind chance.

This tautological expression of Smith’s argument may be an unfair one. Or maybe it is fair, or maybe it at least highlights an aspect of Smith’s argument that makes me uneasy. That is, his claim seems to boil down to the argument that, if it occurred with no further explanation than blind chance, then no further explanation than blind chance is needed. In other words, once you’ve found the explanation you seek—that your surprise urged you towards—you are finished with your investigation. My unease here is with the claim that there is any meaningful difference between the coin and car cases in this regard.

Smith points out that his rational surprise at finding the car gone is not about the low probability of that event occurring. It would be bizarre for him to claim otherwise, of course, given that cars are stolen near-constantly when compared to the rareness with which we’d encounter even relatively small strings of Heads, even when flipping a coin once per second for years on end. That’s why I wouldn’t leave the winning Powerball ticket in my car no matter where I parked it. Would you? But if surprise is not about the rarity of disappearing cars, what is it about? What justifies the belief that it won’t disappear?

This question is, in my experience, a common sort of example encountered in epistemology classes, an examination of which would lead us into an area of thought we don’t have space to get into here. So let’s accept that, in typical car-parking contexts, it is rational to believe (even while rationally saying you don’t know) your car will still be where you parked it an hour later. On Smith’s account, the fact that the belief was thwarted is cause for surprise. And the point of this is surprise is that it spurs a search for explanation.

Moreover, the belief that the car would be there is justified by the need for explanation when the car goes missing. One is alerted to the need for explanation by the surprise felt on seeing that the car has gone missing, and, again, that surprise is justified by the belief that it wouldn’t go missing which is justified by the need for explanation when it does go missing which, again, justifies the surprise, and on and on.

I’ve already pointed out my uneasiness about this circular dynamic between belief and surprise, in which each justifies the other (though I take it the potential for surprise is not enough, for Smith, to justify believing a coin won’t land 92 Heads in a row). And now we add need for explanation into the mix. Perhaps there is some clean way to parse all this out that I’m not seeing, and perhaps could see with more effort; but it so far strikes me that there is a better account to be given here.

Namely, I take it that my belief that the car won’t go missing is justified by my experience of the world. When parking my car in a context like this one, it has never gone missing before. Such things are rare, particularly within an hour of parking. This is wrapped up with the fact that, in such contexts, it is unlikely, though of course not impossible, that a car goes missing. When it does go missing, I am surprised because my belief was wrong. And, for understandable reasons, I aim to seek an explanation. I don’t need to be surprised to seek an explanation, but, fair enough, perhaps my surprise is a kind of signal for my investigatory mind that there is an important mystery to be solved.

There are, as always, deeper questions to ask here, such as whether justification for belief must amount to some sort of real evidence (e.g., my experience with parking my car) or whether it can be something I’ve never even considered (e.g., I doubt many people would cite “because investigation-urging surprise is a guide to rational belief” as their basis for believing their car won’t go missing). But I’ll leave those details unconsidered here.

I assume this “cleaner” story conflicts with Smith’s account due to its reliance on experience and, more specifically, on the low likelihood of the car going missing as a justification for belief. In particular, this seems to allow for the possibility of being rationally surprised at seeing 92 Heads in a row.

However, I’ll set this too aside and will assume that the relations between surprise, belief, and need for explanation can be cleanly parsed. Instead, I will focus on the question of what counts as too much investigation may depend on context. This is actually amounts to a variation on the question of justification for belief, but here we are asking what counts as a sufficient uncovering and consideration of evidence for justifying the belief in some explanation. When is one justified in believing the store is closed, the clock on the wall is correct (and not broken), the car won’t disappear in an hour, the human-shaped being before one is actually human, the coin fair won’t land 92 Heads in a row this time, and so on?

The answers to some, maybe all, of these questions is the same. It depends on context. It may depend on the stakes, on one’s experience, one’s physically available tools, one’s technical expertise, one’s informational resources, what the alternative explanations may be and the extent to which those have been investigated, and so on. This may even depend on the epistemological expectations of one’s era, one’s culture. One person’s Gettier problem¹⁷ is another’s mere lack of justification (depending, for example, on the stock one’s present moment puts into the explanatory power and scope of science, logic, divination, theology, etc.).

This is not the place to elaborate on these thoughts, but it does bear mentioning that they make important contributions to the intricacies of the present discussion’s conceptual framework. Namely, we do generally seem justified in having at least some degree of belief in the proposition that our car will be where we parked it after an hour. And consideration of this point confuses me, given Smith’s arguments, appeals to need for explanation notwithstanding.

At any rate, the possibility of rationally believing the car will be there in an hour and that it might not be points to the value of Bayesian epistemology, in which we might say, for example, that the belief in the car’s being there is much stronger (i.e., “to a higher degree”) than that of it not being there. The conjunction of those beliefs, properly qualified, is justified. In this way, you might say, for example, that you believe the fair coin will land Heads to the same degree you believe it will land Tails. Until you’ve seen it land Heads 92 times in a row. Then it would be rational (again, even on technical, Bayesian grounds) to have a higher belief in its landing Heads on the 93rd flip. Unless you are utterly convinced it’s a fair coin, in which case I suppose the Gambler’s Fallacy would be applicable—but I maintain you must believe the coin is fair for the Gambler’s Fallacy to apply. (I would call Gambler’s Fallacy on expecting Heads [or Tails] to a degree of 1 after five Heads in a row, but would call sucker if you expected Heads to a degree of 0.5 after seeing the coin land Heads 92 times; see “Taleb’s Example” for more on this.)

But let’s suppose we haven’t yet flipped this here fair coin. I’m convinced that I can justify my belief that the coin won’t land 92 consecutive Heads, were I to flip it right now, on the grounds that there are 2⁹²–1 other possibilities. Those are TREMENDOUSLY favorable odds—practically guaranteed even if I do believe 92 consecutive Heads could happen. There are no such (practical) guarantees for a car not being stolen.

Furthermore, I’ve never seen anything remotely like 92 Heads in a row occur, and have no good reason to believe I will any time soon, certainly not right now.

Were I to see it happen, this would demand investigation not entirely like that of the disappearing car, even if we finally land on the hard-to-believe answer of blind chance. Namely, once we discover why the car disappeared, it would be irrational to continue investigating its disappearance. But would it necessarily be irrational to continue being surprised? I suppose it would depend on the scenario. Many come to mind, with varying degrees of likelihood—it sank into a sink hole that, by chance, was quickly repaired by a passing construction crew; its particles suddenly jumped in such a way that the car phased through the asphalt, though this would amount to getting no explanation as nobody guesses this, even if the car is mysteriously found underground. Suppose it’s a run-of-the-mill theft (you know, it wasn’t stolen by aliens). In other words, chances are that, if you get an explanation, your surprise should cease. If you get a vague explanation—“Hey, cars disappear without a trace sometimes. It just happens.”—your surprise may persist, as may your drive to investigate, depending on the stakes. And some explanations—“It turned out to be a Transformer and got called to duty; bystanders caught it on video”—may astound indefinitely.

Interestingly, the least likely scenarios here seem to be ones that least align with one’s experience, understanding, and expectations of the world, and the greater the misalignment, the greater the potential for surprise. In this way, I might well be more surprised to see that the coin landed 92 consecutive Heads due to blind chance rather than due to being two-Headed. More to the point, I believe the surprise would be appropriate, even on the grounds laid out by Smith. I (arguably) justifiedly believe the fair coin won’t so land, I investigate to the extent recommended by the degree or intensity of that belief belief (e.g., higher for 92 nonillion consecutive Heads than for 30), which correlates positively with the degree of surprise. Depending on the depth of investigation recommended, I arrive at a satisfying explanation or I don’t. The explanation may or may not be a surprising one.

4.2.6 A Final Example: Spooky Pixelation and Justified Belief

I would like to mike one last push for the claim that relative rarity matters for belief, despite the conjunction and equiprobable principles. And the more extreme the rarity, the more it matters. A patterned sequence of 10,000 coin flips is an extremely rare class of event, given that all but a minute handful of sequences of that length are mixed results. I, for one, expect (in the colloquial sense of the word) mixed results there just as I expect mixed results when shaking up a can in which I’ve poured red and yellow paint, no matter how much I shake it up.

And I expect mixed results, as it were, when flinging paint at a canvas, rather than seeing a perfect reproduction of the Mona Lisa.¹⁸

Spooky Pixelation: Here’s a more controlled version of the paint-flinging experiment. Tell a computer to randomly assign colors to pixels within a reasonably large rectangle. Of course, were a computer to shuffle through every possible configuration—a task that would take longer than there are years left in our universe—you would end up creating the Library of Babel with the addition of a visual wing (and, come to think of it, a sheet music wing).

Every recognizable variation of the Mona Lisa would be there. But is it not rational to believe that a reproduction will not emerge, no matter how many times you hit run on this program? Despite all pixel configurations being equally likely? And thus, if it did emerge, to be surprised? To be astonished? Freaked out? And should it begin to cycle through perfect portraits of all your family members, would anything in the world convince you that it was random? Would the equiprobable principle? Would a thorough investigation of the program, hardware, or anything else? What if, interspersed between each portrait, were the words, “You are a computer a simulation”?

I think you know where I stand on these questions by now. I tried an approximation of this experiment with a random bitmap generator, and got something like this every time:

I’m about to hit the GENERATE! button one more time. I claim I’m justified in believing that I won’t get anything like the Mona Lisa.

Smith’s arguments about 92 Heads in a row apply here, of course.

5. Concluding Remarks

In my brief time so far on Earth, I’ve never seen anything remotely close to 92 Heads in a row from a fair coin. And such a thing has, at least in our culture, enough mystique around it to inspire articles like mine and Smith’s and to by chosen by Tom Stoppard to show that “the laws of probability aren’t working”¹⁹ Maybe the mystique is warranted. I for one would bet a lot—even stronger, I believe—that 92 Heads in a row has never happened on this planet and never will. But in response to Smith’s belief conditions, I need only support the claim that I’m justified in believing it won’t happen right now, even if I believed it could happen right now. Given this belief, why shouldn’t I be surprised to see it happen? How could I not be? How could Spock not be? Furthermore, I believe that the intuitive strength of the rightness of feeling surprised says something important about its rational status; namely, that it is rational. I think that the various examples I’ve presented here bear this out, even when considered in the context of certain key aspects of Smith’s account, particularly those that deal with the relation between belief, surprise, and investigation.

But if considering 92 consecutive Heads as unsurprising (while assuming that surprise would indeed ensue) does turn out to be irrational, or, perhaps even more importantly, if rejection of the principles that justify its classification as irrational turns out to be irrational*, then I suppose I’m not rational. And who is?

(*This is an important distinction. That is, on the one hand, one may wholeheartedly endorse Smith’s arguments while failing to avoid feeling surprised by, say, Spooky Pixels. On the other hand, one may actually reject Smith’s argument that surprise is irrational in that case, as I’m doing. If I’m wrong to do so, then this is something like rejecting that it’s irrational to be surprised by a coin landing Heads-up 92 times in a row, while not being surprised by the coin’s landing Tails-down 92 times in a row.)

But these are expressions of an intuition that leans a certain way, and apparently not the way of Smith’s, to whom I’ll leave the final word. He ends his article on a humble note and, in fact, seems to feel some of the same tension I do, despite leaning counter-wise. I view this conversation not as a debate, but as a collaboration—one geared, perhaps, towards aiding one another’s self-interrogation (for me, at least, this is increasingly how I see “doing philosophy” in general).

I recommend his article. It’s brief and I probably haven’t done it justice, though I have tried to:

These ideas about rational belief are, of course, very sketchy, and I won’t try to pursue them further here. Maybe they aren’t even on the right track at all. But what I hope I have shown here, at the very least, is that there is a different way of looking at surprise and belief, and that a “Guildensternian” theory of surprise can be defended. I mentioned at the outset that it’s common for absurdist plays to feature fantastical events that are left unexplained. Another very common trope in absurdist drama is for characters to reason in nonsensical ways and to jump to bizarre conclusions. Guildenstern’s first three hypotheses about the coin-throwing episode are indeed bizarre. And so too, I suppose, is his fourth hypothesis—but it also, just maybe, happens to be true. (Smith 7)

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PS: I briefly outline an additional objection in this September 2019 post: “Another Argument That 100 Heads in a Row Is Impossible.”

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Further Reading

Further ReadingSelect CategoryDenotation Switch  (3)Epistemology  (13)Ethics  (16)Film  (8)Free Will  (14)Math  (21)Mind | Brain  (30)Music  (26)Philosophy  (78)Political  (14)Probability  (39)Science  (18)Social Group Ontology  (17)Technology  (14)Uncategorized  (7)Words  (17)

1. Martin Smith, Philosopher’s Imprint, Volume 17, No. 21, Oct 2017, pp. 1–8. The essay is the first-ever winner of the Sanders Prize for Public Philosophy.
2. “…Rosencrantz is not at all surprised by it, but Guildenstern is quite worried. It sets the stage, the suspension of the natural, right at the very beginning. Just like in Hamlet, where the ghost appears right away, the coin flipping is doing the same thing. It’s showing that throughout play the laws of probability aren’t working, free will isn’t working. After all, all their behavior and actions are scripted in play Hamlet.” From this October 2009 interview at Physics Central: “Theater for Physics Fans, and Physics for the Rest of Us.”
3. Spohn, W. (2012) The Laws of Belief: Ranking Theory And Its Philosophical Applications (Oxford University Press). See Chapter 5, Section 5.16, in particular.
4. Shackle, G. (1969) Decision Order and Time in Human Affairs, 2nd ed. (Cambridge University Press). See Chapter 10 in particular.
5. Peter van Inwagen, Metaphysics (2015, 2018; 4th Edition), p 204.
6. It’s interesting to compare uses of the word “surprise” (and variants) in probability textbooks. In one of my favorites, Introduction to Probability by Blitzstein & Hwang (2014), there are over 30 instances. Compare two, one informal and one formal.

   On page 140 the word is used informally when it’s noted that one may be surprised to learn that linearity of expectation is true of independent events—a case in which (reasonable) surprisingness is replaced by intuition as one pays closer attention to the implications of more basic properties with which one, by this point as a student of probability, should already be quite familiar. This certainly was my own experience.

   On page 426, “surprise” is used formally in discussion of an information-theoretic application of Jensen’s inequality: “The surprise of learning that an event with probability p happened is defined as log₂(1/p), measured in a unit called bits.”

   I earlier wrote extensive comments on these uses, but decided to forego discussing them here as, though thought-provoking, I’m on the fence about whether they’re as impactful on the present discussion as I originally thought they would be. So I leave it to you to investigate. I recommend the entire book for anyone looking for a solid probability text packed with intuition-building exercises (not to mention R code and accompanying online lectures). Or view the above-cited pages in Amazon’s “Look inside” function: Introduction to Probability.

7. But only meaning of a certain kind. See “Fine-Tuning” for discussion of the potential significance—in both psychological and “nature”-oriented contexts—of terms such as “meaningful.”
8. That’s 410 septillion 795 sextillion 449 quintillion 442 quadrillion 59 trillion 149 billion 332 million 177 thousand 40 ways.
9. In other words, it’s always just the number of flips times the probability of success, which here is defined as landing Heads. Technically, this is the expected value of any random variable that follows a binomial distribution. But we need not get into such technicalities here.
10. See this NPR article: “Which Is Greater, the Number of Sand Grains on Earth or Stars in the Sky?”
11. The problem usually comes up in the context of lottery and preface paradoxes. Read more at the Stanford Encyclopedia of Philosophy: Epistemic Paradoxes, Sections 3 and 4.
12. “…if you convince a person logically that they have nothing to cry about, they’ll stop crying. That’s clear. Is it your belief that they won’t stop?”

    “Life would be too easy were it so,” answered Raskolnikov. …

    … “…but do you know that in Paris they have been conducting serious experiments as to the possibility of curing the insane, simply by logical argument? One professor there, a scientific man of standing, lately dead, believed in the possibility of such treatment. His idea was that there’s nothing really wrong with the physical organism of the insane, and that insanity is, so to say, a logical mistake, an error of judgment, an incorrect view of things. He gradually showed the madman his error and, would believe it, they say he achieved results! But since he was also using cold showers while doing this, the results of this treatment can naturally be subject to question. … Or so it seems.”

    Raskolnikov had long ceased to listen.

    (Translation borrowed mostly from Constance Garnett [p 333] and Michael Katz [pp 466–476].)

    Sounds like CBT for Vulcans.

13. We are drowning in examples, so I’ll just reach for the closest at hand. Currently paused on my phone right now is Episode 2 (7/10/18) of Sean Carroll’s Mindscape podcast, in which physicist Carlo Rovelli defends quantum loop theory as a better bet than string theory for reconciling quantum mechanics and general relativity: Carlo Rovelli on Quantum Mechanics, Spacetime, and Reality.
14. I discuss the skeptical argument in “In Defense of Sensitivity: Nozick, Kripke, and Predicate Exclusivity.” It goes something like: You don’t know you’re not a brain in a vat, as your evidence would be the same whether you were one or not. And you have to know you’re not a brain in a vat to know you have two hands. So you don’t know you have two hands.
15. Folks used to share a healthier fear of cars. Early car manufacturers had a vested interest in shifting those attitudes. For a fascinating mini-history of this shift, see Episode 76 (4/4/13) of the 99% Invisible podcast: The Modern Moloch.
16. E.g., see Nick Bostrom’s 2003 paper, “Are You Living in a Computer Simulation?,” which I wrote about here: “You Are (Probably Not) a Computer Simulation.”
17. Broadly, when justified true belief (intuitively) fails to count as knowledge. See the Internet Encyclopedia of Philosophy for more: “Gettier Problems.”
18. I’m of course not the first to think of this. The day after writing this, I encountered the following passage from Cicero in James Franklin’s thoroughly recommendable 2001 book, The Science of Conjecture: Evidence and Probability before Pascal. If I understand him correctly (I’ve only read this excerpt), Cicero goes on to state that this observation is not cause for abandoning the application of probability models to the real world, something on which I agree: “It is possible for paints flung on a surface to form the outlines of a face; but surely you do not think a chance throwing could produce the beauty of the Venus of Cos?” (Franklin 164).
19. See again the October 2009 interview at Physics Central: “Theater for Physics Fans, and Physics for the Rest of Us.