Question

A committee of three persons is to be randomly selected from a group of three men and two women and the chair person will be randomly selected from the committee.The probability that the committee will have exactly two women and one man, and that the chair person will be a women, is / are

• A. $\frac{1}{5}$
• B. $\frac{8}{15}$
• C. $\frac{2}{3}$
• D. $\frac{3}{10}$

Answer 1

The correct option is A $\frac{1}{5}$

No of ways of selecting 3 persons from the group of 3 men and two woman = ${}^5{C}_3$

No. of ways of selecting exactly 2 women and 1 man = ${}^2{C}_2{\times }^3{C}_1=3$

$\therefore$ Probability of selecting exactly 2 women and 1 man = $\frac{3}{{}^5{C}_3}$

$=\frac{3}{\frac{5!}{3!2!}}=\frac{3}{10}$

probability that the chair person selected from the committee is a woman = $\frac{2}{3}$

$\therefore$ Probability of selecting exactly 2 women and 1 man for the committee and chairman selected i.e a woman = $\frac{3}{10}\times \frac{2}{3}$

$=\frac{1}{5}$