Chapter 10.5, Problem 75E

a.

To determine

To calculate:

Probability that you are a last performer before the intermission.

Answer to Problem 75E

Probability that you are a last performer before the intermission is $0.01$ .

Explanation of Solution

Given information:

From $10$ students you are one of them to perform in school talent show.

First $5$ performer’s goon stag before intermission.

Calculation:

Number of possible outcomes as the number of permutations of $10$ items taken $2$ at a time, $P{}_2$ because the order is important.

Need to apply permutation,

The expression for combination for $n$ objects taken $r$ at a time is as given by,

  $P{}_r=\frac{n!}{\left(n-r\right)!}$

The required number of combinations is as below,

  $\begin{array}{l}P{}_r=\frac{10!}{\left(10-2\right)!}\\ P{}_r=\frac{\mathrm{10.9.8}!}{8!}\\ P{}_r=90\end{array}$

To find the favorable outcomes, that is one combination in which person concerned will perform last and rival will perform before her.

To calculated the probability,

  $P$ (Required order),

  $\begin{array}{l}=\frac{1}{90}\\ =0.01\end{array}$

Thus, answer is $0.01$ .

b.

To determine

To calculate:

Probability that you are not a first performer.

Answer to Problem 75E

Probability that you are not a first performer is $0.9$ .

Explanation of Solution

Given information:

From $10$ students you are one of them to perform in school talent show.

First $5$ performer’s goon stag before intermission.

Calculation:

Number of possible outcomes as the number of permutations of $10$ items taken $2$ at a time, $P{}_2$ because the order is important.

Need to apply permutation,

The expression for combination for $n$ objects taken $r$ at a time is as given by,

  $P{}_r=\frac{n!}{\left(n-r\right)!}$

The required number of combinations is as below,

  $\begin{array}{l}P{}_r=\frac{10!}{\left(10-1\right)!}\\ P{}_r=\frac{10.9!}{9!}\\ P{}_r=10\end{array}$

To find the favorable outcomes, that is one combination in which person concerned will perform first.

To calculated the probability,

  $P$ (concerned person does not perform first),

  $\begin{array}{l}=1-P\left(P_{1^{st}}\right)\\ =1-\frac{1}{10}\\ =\frac{9}{10}\\ =0.9\end{array}$

Where,

  $P_{1^{st}}=$ person concerned perform first

Thus, answer is $0.9$ .