The eldest is a girl, are all three?

Question

A family has three children. Assume each child is independently a girl or a boy with probability $\tfrac12$ each. You learn that the eldest child is a girl.

Given that, what is the probability all three children are girls?

Accepted Answer

The eldest being a girl is already settled, it uses up that piece of information and tells you nothing about the younger two, because the children's sexes are independent. So "all three girls" now just requires the two younger children to both be girls: P(	ext{all girls}\mid 	ext{eldest is a girl}) = \frac{1}{2}	imes\frac{1}{2} = \frac{1}{4}. Note this differs from conditioning on the vaguer statement "at least one is a girl," which would not pin down a specific child and would give a different answer. Identifying which child is a girl matters. Takeaway. Conditioning on a specific, indepen…

The eldest is a girl, are all three?

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