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

Let B_i and G_i stand for i^th child be a boy and girl respectively. Then the sample space can be expressed as

S = {B₁B₂, B₁G₂, G₁B₂, G₁G₂}

Consider the following events: 

A = Both the children are boys; B = One of the children is a boy:

C = The older child is a boy.

Then, A = B₁B₂, B = B₁G₂, B₁B₂, G₁B₂ and C = B₁B₂, B₁G₂

So, A ∩ B = {B₁B₂} and A ∩ C = {B₁B₂}