A committee of 3 is formed from a panel of 5 men and 3 women. Find the : (a) number of ways of forming the committee ; (b) probability that at least one wom...
Question 1 Report
A committee of 3 is formed from a panel of 5 men and 3 women. Find the :
(a) number of ways of forming the committee ;
(b) probability that at least one woman is on the committee.
Answer Details
The panel has \(5\) men and \(3\) women, i.e. \(8\) people, from which a committee of \(3\) is chosen. Order does not matter, so we use combinations.
(a) Number of ways of forming the committee
\[\binom{8}{3}=\frac{8\times 7\times 6}{3\times 2\times 1}=56.\]
There are \(56\) possible committees.
(b) Probability that at least one woman is on the committee
Use the complement "no woman", i.e. an all-men committee chosen from the \(5\) men:
\[\binom{5}{3}=\frac{5\times 4\times 3}{3\times 2\times 1}=10.\]
\[P(\text{no woman})=\frac{10}{56}=\frac{5}{28}.\]
Therefore
\[P(\text{at least one woman})=1-\frac{5}{28}=\frac{23}{28}.\]
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