“In formulations that say nothing about how one’s knowledge is gained, my intuition (these days) is with the ‘textbook’ answer of 1/3.”

This is a classic example of Bertrand’s Box Paradox. And I mean the paradox that shows why the textbook answer can’t be right when don’t know how the fact was learned, not (as is usual in modern parlance) the problem Joseph Bertrand applied it to. And BTW, if you add a fourth box with mixed coins to that problem, it becomes the Two Child Problem. To remove possible preconceptions, I’ll do exactly that while making a slight change.

Suppose I have four identical boxes. In one box, there are two Gold coins minted in different years.In another, there are two Bronze coins also minted in different years. In last two, there is one Gold coin and one Bronze coin. In one of these two, the Gold coin is older and in the other, the Bronze coin is older. (Does this sound at all familiar? :)

You pick a box at random. The probability that it has two different coins is 1/2. Suppose I look in the box, see both coins, and choose one of them. I show it to you, and it is Bronze. Since I did not tell you how I choose the coin, this exactly matches the quote above, about two children. According to that statement, the probability that the box had two different kinds of coin are now 2/3 (note that I reversed the question).

But if I had shown you a silver coin, the logic has to be the same. The probability that the box held mixed types also would change to 2/3. But now we have a paradox: if this probability changes to 2/3 no matter what kind of coin I show to you, then I don’t have to show it. The probability changes if I take a coin out and keep it hidden. Since that can’t be, the statement above must be false.

Note that this doesn’t directly prove that the answer 1/2 is correct; but it does prove that any answer except 1/2 is incorrect.

]]>But most solutions apply Bayes Rule incorrectly. In the Parade version, where the contestant chooses door #1, the information isn’t “Hall reveals a goat..” It is “Hall reveals a goat behind door #3.” The correct solution is:

Pr(Car=2|Open=3) = Pr(Open=3|Car=2)*Pr(Car=2)/Pr(Open=3).

where

Pr(Open=3) = Pr(Open=3|Car=1)*Pr(Car=1) + Pr(Open=3|Car=2)*Pr(Car=2).

Since Pr(Car=1)=Pr(Car=2)=Pr(Car=3)=1/3, and Pr(Open=3|Car=2=1, we can reduce this to:

Pr(Car=2|Open=3) = 1/[Pr(Open=3|Car=1) + 1].

If we don’t know how Monty Hall chooses a door when C=1, we can only assume Pr(Open=3|Car=1) = 1/2. This makes Pr(Car=2|Open=3)=2/3. But if we let Pr(Open=3|Car=1) vary between 0 and 1, the answer varies between 1 and 1/2. Essentially, those who answer that switching can;t matter are implicitly assuming that Monty Hall was required to open door #3 if he could.

]]>Suppose, instead of picking a coin at random and revealing it, say I pick a coin at random and take it out of the box without revealing it. What it the probability that the coin still in the box is the same kind?

If I were to reveal it, and it turns out to be gold, we have the same original problem. If I were reveal it, and it turns out to be silver, we have an identical problem that must have the same answer. Since those are the only possibilities, and they have the same answer, we don’t need to look at the coin to answer the question. The answer to my question is the same as the answer to the original problem.

If that answer is 1/2, we have a problem. The probability that a random box has two identical coins is 2/3, not 1/2. Taking a coin out, without looking at it, cannot change that. By reductio ad absurdum, the answer 1/2 can’t be right. Note that if I look at both coins, and tell you that I will intentionally take out a gold one if I can, then the probability does change to 1/2

This approach can be applied to other problems, invalidating what are sometimes accepted as correct answers. Mr. Jones has two children, and we are told that at least one is a boy. What is the probability that he has two boys? Many “experts” will say it is 1/3; Bertrand’s Paradox says that can’t be right. An identically-worded problem that uses “girl” in place of “boy” has to have the same answer. The chances that a two-child family has two children of the same gender is 1/2. Learning the gender of one can’t change that, unless somebody looked at both with the explicit intent to say “at least one is a boy” if possible.

]]>I don’t think it’s a typo, just bad writing. I edited it. Thanks.

]]>“Jones owes Smith’s family (or, lacking any family, society) suffering. “

]]>Hi Kyle! Thanks for the thought-provoking comment.

I certainly agree that the readouts on measuring instruments shouldn’t be taken as the final word for what’s real (in my post, hopefully I seem more sympathetic to David Albert than to Lawrence Krauss on this point). That said, a couple of points of clarification.

I’m defining “frequency” in the usual sense. I’d say if there’s no medium that can be structured so that it will oscillate at a given frequency, then the frequency doesn’t exist. It seems likely that such limitations do in fact obtain, given that the infinite divisibility of matter—or at least of any such medium—seems unlikely. But who knows.

I define “pitch” as a mental event (i.e., as experience, or what is often called a “percept”). I’d say if there’s a pitch experience that cannot be had by any potentially existing mind, then the pitch doesn’t exist. In broader terms, if something is impossible to observe, such that no entity—human, nonhuman, god, or machine—can observe it, then, even though that thing exists, the observation of it doesn’t.

In light of the above, it strikes me that, when someone says “there are infinitely many frequencies/notes/pitches between the A4 and Bb4 keys on this piano,” it’s an open question whether that claim is true. Though my intuition is that it’s false. My sense, however, is that people don’t usually mean the claim in such broad or well-defined terms. In which case, whether or not it’s clear that something false is being said will depend on what the person actually means.

Great point about the two-/three-dimensional vibrating objects.

I definitely take your points regarding “whether pitch itself exists,” at least, as I understand you: i.e., our tidy models of how we experience the sounds made by musical instruments (for example) are just that: tidy models. More precisely, a family of models that includes our concepts of those things, so that we can talk about them and notate them; as well as our actual mental representations/perceptions of them (perhaps independent of whether we have concepts, words, or symbols for them). I see this as similar to the question of whether triangles exist as anything other than theoretical ideals.

]]>