Chapter 4.2, Problem 29P

a.

To determine

Find the values of $P\left(S\right)$, $P\left(S/A\right)$, and $P\left(S/Pa\right)$.

Answer to Problem 29P

The value of $P\left(S\right)$ is 0.5914.

The value of $P\left(S/A\right)$ is 0.465.

The value of $P\left(S/Pa\right)$ is 0.717.

Explanation of Solution

It is given that the events A, Pa, S, and N are defined as follows:

$A=\text{Aggressive approach}$, $Pa=\text{Passive approach}$

$S=\text{Sale}$, $N=\text{no sale}$

There are 686 customers in the event sale and 1160 are participating customers.

The probability of sale is as follows:

$\begin{array}{c}P\left(S\right)=\frac{\text{Favorable cases}}{\text{Total number of cases}}\\ =\frac{686}{1,160}\\ =0.5914\end{array}$

Thus, the probability of sale is 0.5914.

There are 270 customers in sale and aggressive approach.

The probability of sale given that aggressive approach is used as follows:

$\begin{array}{c}P\left(S/A\right)=\frac{P\left(S\text{ and }A\right)}{P\left(A\right)}\\ =\frac{270}{580}\\ =0.465\end{array}$

Thus, the probability of sale given that aggressive approach is 0.465.

The probability of sale given that passive approach is used is calculated as follows:

$\begin{array}{c}P\left(S/Pa\right)=\frac{P\left(S\text{ and Pa}\right)}{P\left(Pa\right)}\\ =\frac{416}{580}\\ =0.717\end{array}$

The probability of sale given that passive approach is used is 0.717.

b.

To determine

Explain whether the events $S=\text{Sale}$ and $Pa=\text{Passive approach}$ are independent.

Answer to Problem 29P

The events Sale and Passive approach are not independent.

Explanation of Solution

From Part (a), the value of $P\left(S\right)$ is 0.5914 and $P\left(S/A\right)$ is 0.465.

It is noticed that $P\left(S/A\right)\ne P\left(S\right)$. Thus, the events sale and passive approach are not independent.

c.

To determine

Find the values of $P\left(A\text{ and }S\right)$ and $P\left(Pa\text{ and }S\right)$.

Answer to Problem 29P

The value of $P\left(A\text{ and }S\right)$ is 0.2327.

The value of $P\left(Pa\text{ and }S\right)$ is 0.3586

Explanation of Solution

There are 270 customers in both events sale and aggressive approach. The total number of participating customers is 1,160.

The value of $P\left(A\text{ and }S\right)$ is as follows:

$\begin{array}{c}P\left(A\text{ and }S\right)=\frac{270}{1,160}\\ =0.2327\end{array}$

Thus, the value of $P\left(A\text{ and }S\right)$ is 0.2327.

The number of customers in both events sale and passive approach is 416.

The value of $P\left(Pa\text{ and }S\right)$ is computed as follows:

$\begin{array}{c}P\left(Pa\text{ and }S\right)=\frac{416}{1,160}\\ =0.3586\end{array}$

Thus, the value of $P\left(Pa\text{ and }S\right)$ is 0.3586.

d.

To determine

Find the values of $P\left(N\right)$ and $P\left(N/A\right)$.

Answer to Problem 29P

The value of $P\left(N\right)$ is 0.408.

The value of $P\left(N/A\right)$ is 0.534.

Explanation of Solution

There are 474 customers in no sale and 1160 are participating customers.

The probability of no sale is as follows:

$\begin{array}{c}P\left(N\right)=\frac{474}{1,160}\\ =0.408\end{array}$

Thus, the probability of no sale is 0.408.

There are 270 customers in sale and aggressive approach.

The probability of no sale given that aggressive approach is used as follows:

$\begin{array}{c}P\left(N/A\right)=\frac{P\left(\text{N and }A\right)}{P\left(A\right)}\\ =\frac{310}{580}\\ =0.534\end{array}$

Thus, the probability of no sale given that aggressive approach is 0.534.

e.

To determine

Explain whether the events $N=\text{no sale}$ and $A=\text{Aggressive approach}$ are independent.

Answer to Problem 29P

The events no sale and aggressive approach are not independent.

Explanation of Solution

From Part (d), the value of $P\left(N\right)$ is 0.408 4 and $P\left(N/A\right)$ is 0.534.

It is noticed that $P\left(N/A\right)\ne P\left(N\right)$.

Therefore, the events no sale and aggressive approach are not independent.

f.

To determine

Find the value of $P\left(A\text{ or }S\right)$.

Answer to Problem 29P

The value of $P\left(A\text{ or }S\right)$ is 0.8586.

Explanation of Solution

The probability of aggressive approach or sale is as follows:

$\begin{array}{c}P\left(A\text{ or }S\right)=P\left(A\right)+P\left(S\right)-P\left(A\text{ and }S\right)\\ =\frac{580}{1,160}+\frac{686}{1,160}-\frac{270}{1,160}\\ =\frac{996}{1,160}\\ =0.8586\end{array}$