Tom Ellis, author of the puzzle that is even harder than the hardest logic puzzle ever, writes with another problem:
Alice and Bob are at a party and Alice says to Bob “Mrs Smith over there has two children. At least one is a boy.” What is the probability that both her children are boys?
The first trick is that the naive puzzler will answer “one half”, whereas his mathematical challenger will explain that the correct answer is one third.
There are three ways that Mrs Smith can have at least one boy: her elder a boy and her younger a boy; her elder a boy and her younger a girl; her elder a girl and her younger a boy. These are all equally likely, so the probability that both are boys is one third.
The naive puzzler will smile at the correct answer, pleased that he now understands the application of conditional probability theory.
But there’s a second trick. The second trick is that the answer is not “one third”. The answer is “it depends”; what it depends upon is how Alice reached her decision to tell Bob about Mrs Smith’s children.
Suppose that Alice has met exactly one of Mrs Smith’s children, a boy. Then she can quite truthfully say “At least one is a boy.” But now there are only two possibilities, equally likely: the child Alice hasn’t met is a boy; and the child Alice hasn’t met is a girl. Thus the probability that both are boys is one half.
It’s actually *harder* to construct a realistic motivation for Alice in which the probability of two boys is one third. Perhaps whilst checking her e-mail on Mrs Smith’s computer she saw an icon for the GI Joe website.
Political correctness aside, she might take that as a cast iron indicator that one of the following three possibilities holds: Mrs Smith’s eldest is a boy; her youngest is a boy; both her children are boys. In this case the chance that both are boys *is* one third.
Mathematics is an indispensable means of understanding the world, but if someone says to you “At least one of my children is a boy” then the reason that they said it and the precise meaning of what they said are far more important than the probabilistic content. The mathmatical form of reasoning should not trump the psychological and the linguistic.
Deep waters. For a discussion of the even-harder “Tuesday Boy” problem, check this out.
