Boy or girl paradox

Uncommon card · No. 113652

The boy or girl paradox surrounds a set of questions in probability theory, which are also known as the two children problem, Mr. Smith's children and the Mrs. Smith problem. The initial formulation of the question dates back to at least 1959, when Martin Gardner featured it in his May 1959 "Mathematical Games column" in Scientific American. He phrased the paradox as follows: Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls? Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys? Gardner initially gave the answers ⁠1/2⁠ and ⁠1/3⁠, respectively, but later acknowledged that the second question was ambiguous. Its answer could be ⁠1/2⁠, depending on the procedure by which the information "at least one of them is a boy" was obtained. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Maya Bar-Hillel and Ruma Falk, and Raymond S. Nickerson...

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