There are three boys and two girls. A committee of two is to be formed. Find the probability of the following events:

Event A: The committee contains at least one girl

Event B: The committee contains one boy and one girl

#### Solution

Here, there are three boys B_{1}, B_{2}, B_{3} and two girls G_{1}, G_{2}.

A committee of two is to be formed.

Thus, the sample space (S) is given by

S = {B_{1}B_{2}, B_{1}B_{3}, B_{2}B_{3}, B_{1}G_{1}, B_{1}G_{2}, B_{2}G_{1}, B_{2}G_{2}, B_{3}G_{1}, B_{3}G_{2}, G_{1}G_{2}}

∴n(S) = 10

A is the event that the committee contains at least one girl.

Then, A = {B_{1}G_{1}, B_{1}G_{2}, B_{2}G_{1}, B_{2}G_{2}, B_{3}G_{1}, B_{3}G_{2}, G_{1}G_{2}}

∴n(A) = 7

`therefore P(A)=(n(A))/(n(S))`

`therefore P(A)=7/10`

B is the event that the committee contains one boy and one girl.

Then, B = {B_{1}G_{1}, B_{1}G_{2}, B_{2}G_{1}, B_{2}G_{2}, B_{3}G_{1}, B_{3}G_{2}}

`therefore P(B)=(n(B))/(n(S))`

`therefore P(B)=6/10`

`therefore P(B)=3/5`