Question

video lesson 5.5 example 3 – finding a probability using permutations

1. drea and bryan are auditioning for a part in the school play. there are 15 people auditioning, and the order of their auditions is chosen at random. find the probability that drea auditions last and bryan auditions second to last.

Explanation:

Step1: Find total permutations

The total number of ways to arrange 15 people is \(_{15}P_{15}=\frac{15!}{(15 - 15)!}=15!\) (since permutation formula is \(_{n}P_{r}=\frac{n!}{(n - r)!}\), here \(n = 15\), \(r=15\)).

Step2: Find favorable permutations

We fix Drea at the last position and Bryan at the second - to - last position. Then we need to arrange the remaining \(15-2 = 13\) people in the first \(13\) positions. The number of ways to arrange these 13 people is \(_{13}P_{13}=\frac{13!}{(13 - 13)!}=13!\)

Step3: Calculate probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}\). So \(P=\frac{13!}{15!}\). Since \(15!=15\times14\times13!\), then \(P = \frac{13!}{15\times14\times13!}=\frac{1}{15\times14}=\frac{1}{210}\)

Answer:

\(\frac{1}{210}\)