The elder is a boy

Question

A family has two children, each independently a boy or a girl with probability 	frac{1}{2}. You learn that the elder child is a boy. What is the probability that both children are boys?

Accepted Answer

This time the information points at a specific child, the elder one, and pins its sex. List the two children in age order (elder first). The four equally likely families are BB, BG, GB, GG. "The elder is a boy" keeps only the families whose first entry is B: that is BB and BG, each still equally likely. Among those two, both boys is BB, so P(	ext{both boys} \mid 	ext{elder is a boy}) = \frac{1}{2}. Equivalently, once the elder is fixed as a boy, the younger child is still an independent coin flip, boy or girl with probability 	frac{1}{2}. Contrast this with the weaker statement "at least o…

The elder is a boy

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