Smith has two children. At least one of them is a boy. What is the probability that the other child is a boy? Jones has two children. The older one is a girl. What is the probability that the other child is a girl?
If Smith has two children, at least one of which is a boy, we have three equally probable cases: BB, BG, GB. In only one case are both children boys, so the probability that Smith’s other child is a boy is 1⁄3. Jones’s situation is different. We are told that his older child is a girl, so there are only two equally probable cases: GG, GB. Therefore the probability that the other child is a girl is 1⁄2.
If this were not so, we would have a simple method of guessing, with better than even odds, whether a flipped coin, covered by someone’s hand, was heads or tails. We would simply flip our own coin. If it came up heads, we could reason: at least one of the two coins is heads, therefore the probability that the concealed coin is tails is 2⁄3. This reasoning is incorrect because we know which coin is heads. On the other hand, if someone flips two coins and informs you that at least one came up heads (picking this coin at random and not on a prespecified basis of naming the shiniest coin, the coin on the left, and so on), you can bet that the other coin is tails and in the long run expect to win two out of three times.
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A version of this puzzle originally appeared in the May 1959 issue of Scientific American.