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
$$ \begin{aligned} E(X) &= \sum_x x* P(X=x)\\ & = 0*\frac{10}{56} + 1*\frac{30}{56}+ 2*\frac{15}{56}+3*\frac{1}{56}\\ &= \frac{63}{56}\\ &= 1.125\\ &\approx 1 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
