91Ó°ÊÓ

A group has eight members. Where in January, three members are randomly selected on a committee. In February, four members are randomly and independently selected from another committee. In March, five members are selected randomly and independently of the previous two selections to serve on a third committee.

The total number of ways to pick the committee members i.e., the sample space is,

\(\left( {\begin{aligned}{{}{}}8\\3\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}8\\4\end{aligned}} \right)\left( {\begin{aligned}{{}{}}8\\5\end{aligned}} \right)\)

With all members at least once= The Total number of members – At least one member + At least two members -At least three members + At least four members-…

All members of at least one committee are,

\(\left( {\begin{aligned}{{}{}}8\\3\end{aligned}} \right)\left( {\begin{aligned}{{}{}}8\\4\end{aligned}} \right)\left( {\begin{aligned}{{}{}}8\\5\end{aligned}} \right) - 8\left( {\begin{aligned}{{}{}}7\\3\end{aligned}} \right)\left( {\begin{aligned}{{}{}}7\\4\end{aligned}} \right)\left( {\begin{aligned}{{}{}}7\\5\end{aligned}} \right) + \left( {\begin{aligned}{{}{}}8\\2\end{aligned}} \right) \times \left( {\begin{aligned}{{}{}}6\\3\end{aligned}} \right)\left( {\begin{aligned}{{}{}}6\\4\end{aligned}} \right)\left( {\begin{aligned}{{}{}}6\\5\end{aligned}} \right) - \left( {\begin{aligned}{{}{}}8\\3\end{aligned}} \right) \times \left( {\begin{aligned}{{}{}}5\\3\end{aligned}} \right)\left( {\begin{aligned}{{}{}}5\\4\end{aligned}} \right)\left( {\begin{aligned}{{}{}}5\\5\end{aligned}} \right) + 0...\)

\(\left( {\begin{aligned}{{}{}}8\\2\end{aligned}} \right) \times \left( {\begin{aligned}{{}{}}8\\3\end{aligned}} \right)\left( {\begin{aligned}{{}{}}8\\4\end{aligned}} \right)\left( {\begin{aligned}{{}{}}8\\5\end{aligned}} \right)\)

Comes from first choosing, where two members do not appear.

Thus, the probability of each eight members at least one of the three committees is,

\(\begin{aligned}{}\frac{{\left( {\begin{aligned}{{}}8\\3\end{aligned}} \right)\left( {\begin{aligned}{{}}8\\4\end{aligned}} \right)\left( {\begin{aligned}{{}}8\\5\end{aligned}} \right) - 8\left( {\begin{aligned}{{}}7\\3\end{aligned}} \right)\left( {\begin{aligned}{{}}7\\4\end{aligned}} \right)\left( {\begin{aligned}{{}}7\\5\end{aligned}} \right) + \left( {\begin{aligned}{{}}8\\2\end{aligned}} \right) \times \left( {\begin{aligned}{{}}6\\3\end{aligned}} \right)\left( {\begin{aligned}{{}}6\\4\end{aligned}} \right)\left( {\begin{aligned}{{}}6\\5\end{aligned}} \right) - \left( {\begin{aligned}{{}}8\\3\end{aligned}} \right) \times \left( {\begin{aligned}{{}}5\\3\end{aligned}} \right)\left( {\begin{aligned}{{}}5\\4\end{aligned}} \right)\left( {\begin{aligned}{{}}5\\5\end{aligned}} \right)}}{{\left( {\begin{aligned}{{}}8\\3\end{aligned}} \right)\left( {\begin{aligned}{{}}8\\4\end{aligned}} \right)\left( {\begin{aligned}{{}}8\\5\end{aligned}} \right)}}\\ = \frac{{613}}{{219520}}\\ = 0.279337\end{aligned}\)

The probability of each of the eight members serving on at least one of the three committees is 0.279337