Chapter 8.6, Problem 65E

(a)

To determine

To find: the probability that both shoppers paid for their purchases only in cash.

Answer to Problem 65E

The probability that the both shoppers paid for their purchases only in cash is 0.1024.

Explanation of Solution

Given information:

The circle graph shows the methods used by shoppers to pay for merchandise. Two shoppers are chosen at random.

Given graph is:

  

Calculation:

By the given circle diagram shoppers paid for their purchases only in cash is

  $\begin{array}{c}p\left(A\right)=32\%\\ =\frac{32}{100}\end{array}$

The probability that the both shoppers paid for their purchases only in cash is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A\right)=\frac{32}{100}\cdot \frac{32}{100}\\ =0.1024\end{array}$

The probability that the both shoppers paid for their purchases only in cash is 0.1024.

(b)

To determine

To find: the probability that at least one shopper paid only in cash.

Answer to Problem 65E

The probability that the at least one shopper paid for their purchases only in cash is 0.32.

Explanation of Solution

Given information:

The circle graph shows the methods used by shoppers to pay for merchandise. Two shoppers are chosen at random.

Given graph is:

  

Calculation:

By the given circle diagram shoppers paid for their purchases only in cash is

  $\begin{array}{c}p\left(A\right)=32\%\\ =\frac{32}{100}\end{array}$

And

Shoppers will not paid for their purchases only in cash is

  $\begin{array}{c}p\left(A^{\prime }\right)=\left(100-32\right)\%\\ =\frac{68}{100}\end{array}$

The probability that at least one shopper paid for their purchases only in cash is

  $\begin{array}{c}p\left(A\right)\cdot p\left(A^{\prime }\right)+p\left(A\right)\cdot p\left(A\right)=\frac{32}{100}\cdot \frac{68}{100}+\frac{32}{100}\cdot \frac{32}{100}\\ =0.2176+0.1024\\ =0.32\end{array}$

The probability that at least one shopper paid for their purchases only in cash is 0.32.

(c)

To determine

To find: the probability that neither shopper paid only in cash.

Answer to Problem 65E

The probability that the neither shopper paid for their purchases only in cash is 0.4624.

Explanation of Solution

Given information:

The circle graph shows the methods used by shoppers to pay for merchandise. Two shoppers are chosen at random.

Given graph is:

  

Calculation:

By the given circle diagram shoppers paid for their purchases only in cash is

  $\begin{array}{c}p\left(A\right)=32\%\\ =\frac{32}{100}\end{array}$

And

Shoppers will not paid for their purchases only in cash is

  $\begin{array}{c}p\left(A^{\prime }\right)=\left(100-32\right)\%\\ =\frac{68}{100}\end{array}$

The probability that neither shopper paid for their purchases only in cash is

  $\begin{array}{c}p\left(A^{\prime }\right)\cdot p\left(A^{\prime }\right)=\frac{68}{100}\cdot \frac{68}{100}\\ =0.4624\end{array}$

The probability that neither paid for their purchases only in cash is 0.4624.