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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.
Solution
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`
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