Chapter 9.7, Problem 42E

a.

To determine

Find the probability of each event.

Answer to Problem 42E

  $0.375$

Explanation of Solution

Calculation:

Total paycheck $=5$

Total envelopes $=5$

Let event $E=$ Exactly $1$ paycheck inserted correctly.

  $n\left(S\right)=5!=120$

Number of ways in which $1$ letter is put in correct envelope $=5$

Number of ways of putting other letters in correct envelope $=4$

Total ways $=5+4$

  $=9$

  $\begin{array}{l}n\left(E\right)=5\times 9\\ =45\\ P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ P\left(E\right)=\frac{45}{120}\\ P\left(E\right)=\boxed{0.375}\end{array}$

Hence, the probability of getting exactly one pay check is inserted in the correct envelope is $0.375$ .

b.

To determine

Find the probability of each event.

Answer to Problem 42E

  $0.6333$

Explanation of Solution

Calculation:

Total paycheck $=5$

Total envelopes $=5$

Let event $E=$ At least $1$ paycheck inserted correctly.

  $\begin{array}{l}n\left(S\right)=5!=120\\ n\left(E\right)=5!-5![1-\frac{1}{1!}-\frac{1}{2!}-\frac{1}{3!}-\frac{1}{4!}-\frac{1}{5!}]\\ =76\\ P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ P\left(E\right)=\frac{76}{120}\\ P\left(E\right)=0.6333\end{array}$

Hence, the probability of getting at least $1$ pay check is inserted in the correct envelope is $0.6333$ .