A son born in December

Question

A parent has two children. Each child is independently a boy or a girl with probability $\tfrac{1}{2}$ , and each child's birth month is uniform over the 12 months, independent of everything else. You learn that at least one of the children is a boy born in December.

What is the probability that both children are boys?

Accepted Answer

Attach a tag to each child: "a boy born in December." For a single child that tag has probability 	frac{1}{2}\cdot	frac{1}{12}=	frac{1}{24}, so the child lacks the tag with probability 	frac{23}{24}. Probability that at least one of the two children carries the tag: P(	ext{at least one December boy}) = 1-\left(	frac{23}{24}ight)^2 = \frac{576-529}{576} = \frac{47}{576}. Probability that both are boys and at least one was born in December. Both boys has probability 	frac14, and given both are boys, the chance at least one of the two boys is a December birth is 1-\left(	frac{11}{12}…

A son born in December

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