Do mothers have fewer boys?


It turns out that the question is ambiguous. There are (at least) two possible answers.

If we randomly select a mother from the population of all mothers with exactly two children where at least one is a boy, then the answer is 1/3. This is because the four possibilities FF, MF, FM and MM are equally probable in the general population and our question narrows things down to the last three possibilities.

Consider another formulation of the same question. Select at random a woman who has two children, then select one of the children at random. If this child is a boy then what is the chance that the other child is also a boy? Now the answer is 1/2.

The moral of the story is that the method used to select the sample population of mothers determines what proportion of them will have two boys. This is one of the simplest versions of Bertrand's Paradox.

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