The Monty Hall reference just seems confused to me. The author wrote: > In the Monty Hall problem, the probability suddenly switches from 1/3 to 2/3 when the

The Monty Hall reference just seems confused to me. The author wrote: > In the Monty Hall problem, the probability suddenly switches from 1/3 to 2/3 when the player (“observer”?) chooses to switch boxes. [..] For example, assuming x = 1 means a win and x = 0 a loss, the expectation value would change from 1/3 to 2/3... Usually I think of the Monty Hall problem as a discrepancy between 2/3 and 1/2: the probability that switching boxes will lead to a win is 2/3, but most people's intuition strongly expects it to be 1/2, because the boxes were originally equally likely to contain a prize. However, I can think of one way to treat it as a change from 1/3 to 2/3. For any given box (call it 'our' box), the Bayesian probability (from the player's perspective) that it contained a prize was originally 1/3. Assuming the player chooses a different box, then if and when the host rules out a third box, the Bayesian probability that 'our' box contains a prize rises from 1/3 to 2/3. (On the other hand, if the host rules out 'our' box, then the probability falls from 1/3 to 0.) Importantly, though, that change occurs at the time the host rules out a box: it gets to the heart of the 'paradox' because it reflects the change in expectations due to new information provided (in a counterintuitive way) by the host. But that's not what the paper is referencing. It talks about a change occurring later, once the player chooses to switch boxes. It's true that the probability of winning changes from 1/3 to 2/3 at that time - but that has nothing to do with what makes the Monty Hall problem special! The same thing would happen even for a much simpler scenario where the underlying probabilities are fixed. Like if for some reason you're given the choice between opening box #1, only, or both box #2 and #3 - and then the host (always) gives you the opportunity to change your mind, without revealing any additional information. Very boring game. Maybe you get a bonus reward if you choose box #1 and win, or else there would be no reason to ever choose it, but that doesn't matter. The point is, you get to switch from a 1/3 bet to a 2/3 bet, and if you do, the expectation value of winning changes from 1/3 to 2/3. Nothing paradoxical there at all. The author does correctly describe the Monty Hall paradox in the appendix, but that doesn't explain the relevance to quantum mechanics. The other invocation of a classical paradox, the boy or girl paradox, is just wrong: > The probability of “both children being boys if one is a boy” is 2/3 if the children are completely indistinguishable except by gender or 1/2 if the children are somehow distinguishable. For normal life, such a distinction is silly, which is one of the reasons this problem was so confusing, but in quantum mechanics indistinguishable particles are a real possibility, … If a family is selected randomly from all two-child families with at least one boy, the chance that both children are boys is 2/3. Or in other words, the conditional probability of "both children being boys" given that "one is a boy" is 2/3. If you know a family has two children, and you see a boy in their yard, the Bayesian probability that both children are boys is 1/2. This is true regardless of whether you have any way of "distinguishing" them. It makes no difference if, say, all boys in the world look exactly the same as each other and the same for girls. For the purpose of mapping out the probabilities, it's fine to 'distinguish' between "the child I saw" and "the child I didn't see". If you walk past their yard frequently, but at such a distance that you can only tell if you see a boy or a girl, without being able to tell whether you saw the same child on different occasions… and on any given occasion, you only ever see either one boy or nobody… and you know that both children spend time in the yard… then the Bayesian probability that both children are boys approaches 1 the more observations you make. At least here it matters whether you can distinguish the children; if you could, then the probability would go to 1 or 0 as soon as you knew you'd observed both of them, and remain 0.5 until then (assuming that 'which child is outside at this time' is uncorrelated with gender). But given that you can't distinguish them, the probability doesn't depend on whether the children are distinguishable in principle (e.g. by getting closer to the yard and seeing their faces), or whether they're not, because e.g. all children of a given gender look identical. I'm not saying the boy or girl paradox isn't a paradox (i.e. a nonintuitive result, not a literal contradiction), or that it doesn't generally have to do with some ontological concept of "distinguishability". It may well be a good analogy to some aspect of quantum mechanics, similar to the other ways the author explains quantum mechanics in terms of analogies with classical probabilities (while being careful to say they're analogies, not exact interpretations). But I think it's fundamentally incorrect to say the paradox is confusing because people aren't indistinguishable in real life, when that really doesn't affect things much.