Chapter 11.2, Problem 23PPSE

Solution

To state: The theoretical probability of 3 students from one’s school that are selected as contestants out of 9 possible contestant spots if a group of 30 students from one’s school is a part of the audience for TV game show. The total number of people in the audience is 150.

The resultant probability is $P=0.17879$ .

Given information:

A group of 30 students from one’s school is a part of the audience for TV game show. The total number of people in the audience is 150.

Explanation:

The total audience includes 150 people out of that 30 are from his school.

Now choose 3 students from his school as contestants out of 9 possible contestants, then these 3 are to be selected from 30 of your school that can be expressed as: ${}_{30}{C}_3$

The remaining 6 contestants can be chosen from remaining 120 $\left(150-30=120\right)$ audiences that can be expressed as: ${}_{120}{C}_6$

Total ways of selecting 9 contestants from 150 audiences: ${}_{150}{C}_9$

Now the probability is:

  $\begin{array}{c}P=\frac{{}_{30}{C}_3{\times }_{120}{C}_6}{{}_{150}{C}_9}\\ P=\frac{\frac{30!}{3!\left(30-3\right)!}\times \frac{120!}{6!\left(120-6\right)!}}{\frac{150!}{9!\left(150-9\right)!}}\\ P=\frac{\frac{30!}{3!27!}\times \frac{120!}{6!114!}}{\frac{150!}{9!141!}}\\ P=0.17879\end{array}$

Therefore, the probability is $P=0.17879$ .