Question

The Board of Directors at Smallville National Bank consists of 12 members - 9 male and 3 female. A revised mortgage policy manual is planned and a committee of 3 Board members will prepare the manual and be selected randomly.

                  a.              What is the probability that all committee members will be male

                  b.              What is the probability that at least one member will be female?

Answer 1

The selection of 3 board member follows binomial distribution with n = 3 and probability success (if male selected) = 9/12 = 0.75.

a)

$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$

For three male selected,

$P(X=3)=\binom{3}{3}0.75^3(1-0.75)^{3-3}=0.421875$

b)

Atleast one female member in 3 selected member, (or say atmost 3 male member selected such that k= 0 to 2)

$P(X \leq k)=\sum_{k=0}^{2} \binom{n}{k}p^k(1-p)^{n-k}$

For atleast one female selected,

$P(X\leq k)= \binom{3}{0}0.75^0(1-0.75)^{3-0}+\binom{3}{1}0.75^1(1-0.75)^{3-1}+\binom{3}{2}0.75^2(1-0.75)^{3-2}$$P(X\leq k)= 0.578125$