Let b and g represents the boy and the girl child respectively.

Now if a family has two children, the sample space will be

S = {(b, b), (b, g), (g, b), (g, g)}, n(S)=4

(i) Let A be the event that both children are males, then

A= {(b, b)}, n(A)=1

So the probability that both children are males is

Let B be the event that at least one of the children is male

Then B = {(b, b), (b, g), (g, b)}, n(B) = 3

And the corresponding probability becomes

The sample space for the at least male and both being male will become

(A∩B) = {(b, b)}, n(A∩B)=1

And the corresponding probability becomes

So the conditional probability that both are males given that the at least one is male is

(ii) Let P be the event that both children are females, then

P= {(g, g)}, n(P)=1

So the probability that both children are females is

Let Q be the event that the elder child is a female

Then Q = {(g, b), (g, g)}, n(Q) = 2

And the corresponding probability becomes

The sample space for the elder child being female and both child being female will become

(P∩Q) = {(g, g)}, n(P∩Q)=1

And the corresponding probability becomes

So the conditional probability that if the elder child is female then both children are females is