Conditional Probability

Problem:

Suppose a family has 2 children, one of which is a boy. What is the probability that both children are boys?

Solution:

The reaction is $\frac{1}{2}$, but that is wrong.

Let the event $A$ be the event that we have one boy, that event should be ${ BB, BG, GB }$, this event has probability $\frac{3}{4}$.

Let the event $B$ be the event that we have two boy, that event should be ${ BB }$, this event has probability $\frac{1}{4}$.

By the definition of conditional probability, we have:

$P(B|A) = \frac{P(AB)}{P(A)} =  \frac{P(B)}{P(A)} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}$.