Martin Gardner was amazingly accurate and reliable. That he made a mistake is simply testimonial to the difficulty of this particular problem, which appeared in 1959 and was republished in [3]:
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Mr. Jones has failed to stir any controversy, so we ignore him and his two children [5]. Instead, we concentrate on Mr. Smith. Here is the solution that Martin Gardner published with the problem:
If Smith has two children, at least one of which is a boy, we have three equally probable cases: boy-boy, boy-girl, girl-boy. In only one case are both children boys, so the probability that both are boys is ⅓.
The corrected solution
Later Martin Gardner wrote a column titled “Probability and Ambiguity,” which was also republished (in [4]). In this column Gardner corrects himself, writing “… the answer depends on the procedure by which the information “at least one is a boy” is obtained.”
He suggested two potential procedures.
(i) Pick all the families with two children, one of which is a boy. If Mr. Smith is chosen randomly from this list, then the answer is ⅓.
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