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A committee of 4 students is selected at random from a group consisting of 7 boys and 4 girls. Find the probability that there are exactly 2 boys in the committee, given that at least one girl must be there in the committee.

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#### Solution

Let A: exactly 2 boys are in the committee,

B: at least one girl must be in the committee.

So,

P(B) = `(""^4"C"_1 xx ""^7"C"_3 + ""^4"C"_2 xx ""^7"C"_2 + ""^4"C"_3 xx ""^7"C"_1 + ""^4"C"_4)/(""^11"C"_4)`

= `(59)/(66)` &

`"P" ("A" ∩ "B") = (""^4"C"_2 xx ""^7"C"_2)/(""^11"C"_4) = (21)/(55)`.

= Now

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

= `(21/55)/(59/66)`

= `(126)/(295)`.

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