10/143
Probability is the likelihood or chance that an event will occur.
If an executive committee consists of 13 members: 7 men and 6 women and 3 members are selected at random, the total outcome will be expressed according to the combination rule:
[tex]\begin{gathered} 13C_3=\frac{13!}{(13-3)!3!} \\ 13C_3=\frac{13!}{10!3!} \\ 13C_3=\frac{13\times12\times11\times10!}{10!\times6} \\ 13C_3=13\times22=286 \end{gathered}[/tex]If all the selected executive are women, then the expected outcome will be:
[tex]\begin{gathered} 7C_0\times6C_3=\frac{7!}{(7-0)0!}\times\frac{6!}{(6-3)!3!} \\ 7C_0\times6C_3=\frac{7!}{7!0!}\times\frac{6!}{3!3!} \\ 7C_0\times6C_3=\frac{1}{0!}\times\frac{6\times5\times4\times6}{6\times6} \\ 7C_0\times6C_3=20 \end{gathered}[/tex]The probability that all 3 selected are women will be:
[tex]\begin{gathered} Pr(all\text{ selected are women})=\frac{7C_0\times6C_3}{8C_3} \\ Pr(all\text{ selected are women})=\frac{20}{286} \\ Pr(all\text{ selected are women})=\frac{10}{143} \end{gathered}[/tex]Therefore the probability that all 3 selected are women is 10/143