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Tricky probability puzzle

Posted By: Timothy Chow
Date: Tuesday, 31 October 2023, at 2:36 a.m.

In Response To: Tricky probability puzzle (Jason Lee)

I agree with you that there are many cases where the mathematically correct answer remains counterintuitive even after all the mathematical modeling issues are clarified. Peter Winkler gives a good discussion on page 113 of his book Mathematical Puzzles (not to be confused with his earlier book, Mathematical Puzzles: A Connoisseur's Collection), in the context of the puzzle "Boy Born on Tuesday." Winkler writes:

I have a friend whose wife became pregnant with fraternal twins, and an amniotic test revealed the presence of Y chromosomes in the placenta. … I inquired and was told that the test always finds Y chromosomes as long as one or more fetuses is male.

In this case, the probability of mixed twins really is 2/3 (or at least much closer to 2/3 than 1/2), and for most people (including me!), that seems counterintuitive even after working through the math (or seeing a computer simulation).

Nevertheless, I feel that the specific example of the ace versus the ace-of-spades is a slightly artificial one, because our intuition comes from real-world experience, and the way we learn the identity of a card in real life often doesn't line up with the way this problem is usually presented in textbooks. As the title of the puzzle suggests, "Boy Born on Tuesday" is the following puzzle:

Mrs. Chance has two children of different ages. At least one of them is a boy born on a Tuesday. What is the probability that both of them are boys?

If we very carefully stipulate that Mrs. Chance was randomly selected from the population of mothers of two (non-twin) children, and that we asked her if at least one was a boy born on a Tuesday, and she said yes, then we get the counterintuitive answer of 13/27. In the real world, though, we never learn that at least one of Mrs. Chance's two children is a boy born on a Tuesday in precisely this manner, so IMO the 13/27 answer is rather artificial.

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