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') = ${\displaystyle \frac{{}^8{\text{C}}_1}{{}^{12}{\text{C}}_4}}$

= ${\displaystyle \frac{70}{495}}$

= ${\displaystyle \frac{14}{99}}$

⇒ P(A) = ${\displaystyle 1-\frac{14}{99}=\frac{85}{99}}$

Now P(A ∩ B) = P(2 boys and 2 girls)

= ${\displaystyle \frac{{}^8{\text{C}}_4\cdot {}^4{\text{C}}_2}{{}^{12}{\text{C}}_4}}$

= ${\displaystyle (6\times 28)/495}$

= ${\displaystyle \frac{56}{165}}$

Thus P(B|A) = ${\displaystyle \frac{\text{P}(\text{A}\cap \text{B})}{\text{P}\left(\text{A}\right)}}$

= ${\displaystyle \frac{56}{165}\times \frac{99}{85}}$

= ${\displaystyle \frac{168}{425}}$