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Let a family have two children. It is known that one of the children is a boy. What is the probability that both the children are boys.

So for this we build the sample space $S=\{(b,b)(b,g)(g,g)\}$

Let our event E be the case where both children are boys

$E=\{ (b,b) \}$

Let the conditional be F

$F=\{(b,g),(b,b)\}$

Hence $P(E|F)=\frac{P(E\cap F)}{P(F)}=\frac{1/3}{2/3}=\frac{1}{2}$

But the answer in my book is given as $\frac{1}{3}$ and I can't seem to understand why.

Answers