A couple has $2$ children. Find the probability that both are boys, if it is known that (i) one of them is a boy (ii) the older child is a boy.

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Solution

Let 'A' is the event that Both are boys (BB)
'B' is the event that one of children is boy (BB),(BG),(GB)
'C' is the event that older child is boy (BG)(BB)
i) An $P (A / B) = \frac{P (A \cap B)}{P (B)} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}$
ii) $P (A / c) = \frac{P (A \cap C)}{P (C)} = \frac{\frac{1}{4}}{\frac{2}{4}} = \frac{1}{2}$

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