A family has two children. What is the probability that both the children are boys given that at least one of them is a boy?

$\frac{1}{3}$

The correct answer is Option (2) → $\frac{1}{3}$ ##

Let $b$ stand for boy and $g$ for girl. The sample space of the experiment is

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

Let $E$ and $F$ denote the following events:

$E$ : 'both the children are boys'

$F$ : 'at least one of the child is a boy'

Then $ E = \{(b, b)\}$ and $F = \{(b, b), (g, b), (b, g)\}$

Now $E \cap F = \{(b, b)\}$

Thus $P(F) = \frac{3}{4}$ and $P(E \cap F) = \frac{1}{4}$

Therefore $P(E|F) = \frac{P(E \cap F)}{P(F)} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}$