Probability: find the probability of event B given that event A occurs My Problem,
Suppose a family has 2 children,
if one children is randomly selected and it is a girl then what is the probability of the second child to be a girl ?
Please help. Thanks

Let $A$ be the event that the you randomly selected a girl. 
Let $B$ be the event that both children are girls.
Let $C$ be the event that one child is a boy and one is a girl. 

$$P(A) = 1/2$$
$$P(B) = 1/4$$
$$P(C) = 1/2$$

What is $P(A \,\mid\,B)$? The probability that you randomly select a girl, given that both children are girls. This is clearly $1$. 

We are looking for $P(B \,\mid\,A)$. We can use Bayes' rule:
$$P(B\,\mid\,A) = \frac{P(A\,\mid\,B)\cdot P(B)}{P(A)}$$
$$P(B\,\mid\,A) = \frac{1 \cdot \frac{1}{4}}{\frac{1}{2}}$$
$$P(B\,\mid\,A) = \boxed{\frac{1}{2}}$$
