Question

From a group of 8 boys and 5 girls, a committee of five is to be formed. Find the probability that the committee contains at least 3 boys.

Answer 1

The group consists of 8 boys and 5 girls
i.e., 8 + 5 = 13 persons.
A committee of 5 is to be formed from this group.
∴ 5 persons from 13 persons can be selected in ¹³C₅ ways
∴ n(S) = ¹³C₅ 
Let B be the event that the committee contains at least 3 boys (i.e., 3 boys and 2 girls or 4 boys and 1 girl or 5 boys and no girl)

∴ n(B) = ⁸C₃ × ⁵C₂ + ⁸C₄ × ⁵C₁ + ⁸C₅ × ⁵C₀

∴ P(B) = `("n"("B"))/("n"("S")`

= `(""^8"C"_3  xx ""^5"C"_2 + ""^8"C"_4 xx ""^5"C"_1 + ""^8"C"_5xx""^5"C"_0)/(""^13"C"_5)`