Chapter 4, Problem 4.2.12RE

Purchasing Sweaters During a sale at a men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters, and 7 yellow sweaters were purchased. If a customer is selected at random, find the probability that he bought

a. A blue sweater

b. A yellow or a white sweater

c. A red, a blue, or a yellow sweater

d. A sweater that was not white

To determine

To find: The probability that the randomly selected customer bought a blue sweater.

Answer to Problem 4.2.12RE

The probability that the randomly selected customer bought a blue sweater is 0.257.

Explanation of Solution

Given info:

In the sale of men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters and 7 yellow sweaters were purchased.

Calculation:

Let event A denote the customer bought a blue sweater.

The number of blue sweaters are 9.

The total number of sweaters is obtained as follows,

$\begin{array}{c}\text{Total number of sweaters =}\left[\begin{array}{l}\text{Number of white sweater + Numbeer of red sweaters + }\\ \text{Number of blue sweaters + Number of yellow sweaters}\end{array}\right]\\ \text{=}16+3+9+7\\ =35\end{array}$

The probability that the randomly selected customer bought a blue sweater is represented by $P\left(A\right)$.

The formula for obtaining the probability that the randomly selected customer bought a blue sweater is,

$P\left(A\right)=\frac{\text{Number of blue sweaters}}{\text{Total number of sweaters}}$

Substitute 9 for ‘Number of blue sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(A\right)=\frac{9}{35}\\ =0.257\end{array}$

Thus, the probability that the randomly selected customer bought a blue sweater is 0.257.

To determine

To find: The probability that the randomly selected customer bought a yellow or a white sweater.

Answer to Problem 4.2.12RE

The probability that the randomly selected customer bought a yellow or a white sweater is 0.657.

Explanation of Solution

Calculation:

Let event A denote the customer bought a yellow sweater and the event B denote the customer bought a white sweater.

The number of yellow sweaters are 7, number of white sweaters are 16 and total number of sweaters is 35.

The probability that the randomly selected customer bought a yellow or a white sweater is represented by $P\left(A\text{ or B}\right)$.

The formula to obtain the probability that the randomly selected customer bought a yellow or a white sweater is,

$P\left(A\text{ or }B\right)=P\left(A\right)+P\left(B\right)$

The formula to obtain $P\left(A\right)$ is,

$P\left(A\right)=\frac{\text{Number of yellow sweaters}}{\text{Total number of sweaters}}$

Substitute 7 for ‘Number of yellow sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(A\right)=\frac{\text{7}}{\text{35}}\\ =0.2\end{array}$

The formula to obtain $P\left(B\right)$ is,

$P\left(B\right)=\frac{\text{Number of white sweaters}}{\text{Total number of sweaters}}$

Substitute 16 for ‘Number of yellow sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(B\right)=\frac{\text{16}}{\text{35}}\\ =0.457\end{array}$

Substitute 0.2 for $P\left(A\right)$ and 0.457 for $P\left(B\right)$ in $P\left(A\text{ or }B\right)$

$\begin{array}{c}P\left(A\text{ or }B\right)=0.2+0.457\\ =0.657\end{array}$

Thus, the probability that the randomly selected customer bought a yellow or a white sweateris 0.657.

To determine

To find: The probability that the randomly selected customer bought a red, a blue or a yellow sweater.

Answer to Problem 4.2.12RE

The probability that the randomly selected customer bought a red, a blue or a yellow sweater is 0.543.

Explanation of Solution

Calculation:

Let event A denote the customer bought a red sweater, the event B denote the customer bought a blue sweater and the event C denote the customer bought a yellow sweater.

The number of red sweaters are 3, number of blue sweaters are 9, number of yellow sweaters are 7 and total number of sweaters are35.

The probability that the randomly selected customer bought a red, a blue or a yellow sweater is represented by $P\left(A\text{ or }B\text{ or }C\right)$.

The formula to obtaining the probability that the randomly selected customer bought a red, a blue or a yellow sweater is,

$P\left(A\text{ or }B\text{ or }C\right)=P\left(A\right)+P\left(B\right)+P\left(C\right)$

The formula to obtain $P\left(A\right)$ is,

$P\left(A\right)=\frac{\text{Number of red sweaters}}{\text{Total number of sweaters}}$

Substitute 3 for ‘Number of red sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(A\right)=\frac{\text{3}}{\text{35}}\\ =0.086\end{array}$

The formula to obtain $P\left(B\right)$ is,

$P\left(B\right)=\frac{\text{Number of blue sweaters}}{\text{Total number of sweaters}}$

Substitute 9 for ‘Number of blue sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(B\right)=\frac{\text{9}}{\text{35}}\\ =0.257\end{array}$

The formula to obtain $P\left(C\right)$ is,

$P\left(C\right)=\frac{\text{Number of yellow sweaters}}{\text{Total number of sweaters}}$

Substitute 7 for ‘Number of blue sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(C\right)=\frac{\text{7}}{\text{35}}\\ =0.2\end{array}$

Substitute 0.086for $P\left(A\right)$, 0.257 for $P\left(B\right)$ and 0.2 for $P\left(C\right)$ in $P\left(A\text{ or }B\text{ or }C\right)$

$\begin{array}{c}P\left(A\text{ or }B\text{ or }C\right)=0.086+0.257+0.2\\ =0.543\end{array}$

Thus, the probability that the randomly selected customer bought a red, a blue or a yellow sweater is 0.543.

To determine

To find: The probability that the randomly selected customer bought a sweater that is not white.

Answer to Problem 4.2.12RE

The probability that the randomly selected customer bought a sweater that is not white is 0.543.

Explanation of Solution

Calculation:

Let event A denote the customer bought a sweater that is white and the event $\overline{A}$ denote the customer bought a sweater that is not white.

The number of white sweaters is 16 and total number of sweaters is 35.

The probability that the randomly selected customer bought a sweater that is not white is represented by $P\left(\overline{A}\right)$.

The formula to obtain the probability that the randomly selected customer bought a sweater that is not white is,

$P\left(\overline{A}\right)=1-P\left(A\right)$

The formula to obtain $P\left(A\right)$ is,

$P\left(A\right)=\frac{\text{Number of white sweaters}}{\text{Total number of sweaters}}$

Substitute 16 for ‘Number of white sweaters’ and 35 for ‘Total number of sweaters’

$\begin{array}{c}P\left(A\right)=\frac{\text{16}}{\text{35}}\\ =0.457\end{array}$

Substitute 0.457 for $P\left(\overline{A}\right)$

$\begin{array}{c}P\left(\overline{A}\right)=1-0.457\\ =0.543\end{array}$

Thus, the probability that the randomly selected customer bought a sweater that is not white is 0.543.