Epictetus comments on My Bayesian Enlightenment - Less Wrong
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This is why I sometimes hate probability. The probabilities here strongly depend on how the family and boys are chosen.
If you took a list of families with exactly two children and threw out the ones that had no boys, then you'd find that of the remaining families, 1/3 have two boys.
If you took a list of boys who have exactly one sibling and asked how many of them had a brother, you'd get the answer 1/2.
The difference is whether the child is chosen at random. Even a minor change in the phrasing of the question can change the correct answer. Always be cautious with probability.