Question

A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. Given that there is at least one girl on the committee, calculate the probability that there are exactly 2 girls on the committee.

Answer 1

Let A denote the event that at least one girl will be chosen, and B the event that exactly 2 girls will be chosen.

We require P(B | A).

Since A denotes the event that at least one girl will be chosen

A′ denotes that no girl is chosen

i.e., 4 boys are chosen. 

Then P(A') = `(""^8"C"_1)/(""^12"C"_4)`

= `70/495`

= `14/99`

⇒ P(A) = `1 - 14/99 = 85/99`

Now P(A ∩ B) = P(2 boys and 2 girls)

= `(""^8"C"_4 * ""^4"C"_2)/(""^12"C"_4)`

= `(6 xx 28)//495`

= `56/165`

Thus P(B|A) = `("P"("A" ∩ "B"))/("P"("A"))`

= `56/165 xx 99/85`

= `168/425`