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.

]]>There’s also a sampling limitation. 110hz means 110 compression cycles within the space of a second. Unless your measuring device (and I’m not sure how computing and/or human apperception work here) can detect a fraction of a sine wave, you’re limited to whole numbers unless you can sample/hear more than a second of a note. And I doubt that human hearing is precise enough to continually evaluate pitch after a note has been sustained for more than a second.

I’ve run into this problem before when tuning instruments using the “graph spectrum” function on Audacity. The function only displays whole numbers, which means I have to settle for “close enough” in the low register, where there might be a difference of only 10hz between two pitches in 12TET.

If you want to go a bit deeper, you could interrogate whether pitch itself exists. The harmonic series is a theoretical ideal that is based on a one-dimensional shape, like a string or an air column. But once you consider things like the mass or stiffness of the string or the dimensions of the air column the upper partials might not be exactly perfect ratios to the fundamental. And two- or three-dimensional vibrating objects follow their own irregular harmonic series entirely; sometimes it is close enough to the expected harmonic series that we hear it as such, but sometimes it completely loses the appearance of pitch.

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