# Solved (Free): A committee of three people is to be selected at random from a council consisting of five men and three women

#### ByDr. Raju Chaudhari

Apr 6, 2021

A committee of three people is to be selected at random from a council consisting of five men and three women. What is the expected number of women on the committee?

#### Solution

Council consists of 5 Men and 3 Women. Total 8 members in a council.

Let $X$ denote the number of women in a committee. Random variable $X$ take the values 0,1,2,3.

Number of ways of selecteding 3 members from a council of 8 is $\binom{8}{3} = 56$.

$P(X=0) = \frac{\binom{5}{3}\binom{3}{0}}{56}=\frac{10}{56}$

$P(X=1) = \frac{\binom{5}{2}\binom{3}{1}}{56}=\frac{30}{56}$

$P(X=2) = \frac{\binom{5}{1}\binom{3}{2}}{56}=\frac{15}{56}$

$P(X=3) = \frac{\binom{5}{3}\binom{3}{0}}{56}=\frac{1}{56}$

X P(X=x)
0 $\frac{10}{56}$
1 $\frac{30}{56}$
2 $\frac{15}{56}$
3 $\frac{1}{56}$

The expected number of women on committee is

 \begin{aligned} E(X) &= \sum_x x* P(X=x)\\ & = 0*\frac{10}{56} + 1*\frac{30}{56}+ 2*\frac{15}{56}+3*\frac{1}{56}\\ &= \frac{63}{56}\\ &= 1.125\\ &\approx 1 \end{aligned}