Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Give that

(i) the youngest is a girl.

(ii) at least one is a girl.

#### Solution 1

Let *B*_{i} and *G*_{i}, representing the *i*th child, be a boy and a girl, respectively.

Then, the sample space can be expressed as follows:

S = {*B*_{1}*B*_{2}, *B*_{1}*G*_{2}, *G*_{1}*B*_{2}, *G*_{1}*G*_{2}}

Consider the following events:

A = both the children are girls

B = one of the child is a girl

C = the youngest child is a girl

Then, we have:

A = {*G*_{1}*G*_{2}}*⇒ *P(A)* = 1/4*

B = {*B*_{1}*G*_{2}, *G*_{1}*B*_{2}, *G*_{1}*G*_{2}}*⇒ *P(B)* = 3/4*

Also, C = {*B*_{1}*G*_{2}, *G*_{1}*G*_{2}}*⇒ *P(C)* = 2/4=1/4*

`So, A∩B = {G1G2} ⇒P(A∩B)=1/4and A∩C = {G1G2} ⇒P(A∩C)=1/4`

* *

*Now, we have:(i) Probability that both the children are girls given that the youngest child is a girl: *

` P(A/C) = P(A∩C)P(C)=(1/4)/(1/2)=1/2`

*(ii) Probability that both are girls given that at least one is a girl:*

` P(A/B) = P(A∩B)P(B)=(1/4)/(3/4)=1/3`

* *

#### Solution 2

Let *b* and *g* represent the boy and the girl child respectively. If a family has two children, the sample space will be

S = {(*b*, *b*), (*b*, *g*), (*g*, *b*), (*g,* g)}

Let A be the event that both children are girls.

:. A = {(g,g)}

(i) Let B be the event that the youngest child is a girl.

The conditional probability that both are girls, given that the youngest child is a girl, is given by P (A|B).

(ii) Let C be the event that at least one child is a girl.