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} $.