Chapter 5, Problem 9RE

(a)

To determine

The probability that the student is female and business major.

Answer to Problem 9RE

The probability of student is female and business major is $0.6$ .

Explanation of Solution

Given:

Probability of female is 0.55, probability of business major is 0.2 and the probability that the student is female and business major is 0.15.

Calculation:

The probability that the student is female and business major is calculated as follows:

  $\begin{array}{l}P\left(F\cup B\right)=P\left(F\right)+P\left(B\right)-P\left(F\cap B\right)\\ P\left(F\cup B\right)=0.55+0.2-0.15\\ P\left(F\cup B\right)=0.6\end{array}$

Thus, the probability of student is female and business major is $0.6$ .

(b)

To determine

The probability that the student is female if it is given that the student is business major.

Answer to Problem 9RE

The probability that the student is female if it is given that the student is business major is $0.75$ .

Explanation of Solution

Given:

Probability of female is 0.55, probability of business major is 0.2 and the probability that the student is female and business major is 0.15.

Calculation:

The probability that the student is female if it is given that the student is business major is calculated as follows:

  $\begin{array}{l}P\left(F/B\right)=\frac{P\left(F\cap B\right)}{P\left(B\right)}\\ P\left(F/B\right)=\frac{0.15}{0.2}\\ P\left(F/B\right)=0.75\end{array}$

Thus, the probability that the student is female if it is given that the student is business major is $0.75$ .

(c)

To determine

The probability that the student is business major if it is given that student is female.

Answer to Problem 9RE

The probability that the student is business major if it is given that student is female is $0.2727$ .

Explanation of Solution

Given:

Probability of female is 0.55, probability of business major is 0.2 and the probability that the student is female and business major is 0.15.

Calculation:

The probability that the student is business major if it is given that student is female is calculated as follows:

  $\begin{array}{l}P\left(B/F\right)=\frac{P\left(F\cap B\right)}{P\left(F\right)}\\ P\left(F/B\right)=\frac{0.15}{0.55}\\ P\left(F/B\right)=0.2727\end{array}$

Thus, the probability that the student is business major if it is given that student is female is $0.2727$ .

(d)

To determine

Whether the events are independent or not.

Answer to Problem 9RE

The events are not independent.

Explanation of Solution

Given:

Probability of female is 0.55, probability of business major is 0.2 and the probability that the student is female and business major is 0.15.

Independent events are those events that are not depend upon each other. The expression for the independent event is shown below:

  $P\left(F\cap B\right)=P\left(F\right)\times P\left(B\right)$

Apply the condition for independent events for the given condition as follows:

  $\begin{array}{l}\Rightarrow P\left(F\right)\times P\left(B\right)\\ \Rightarrow 0.55\times 0.2\\ \Rightarrow 0.11\left(\ne 0.15\right)\end{array}$

Thus, the events are not independent.

(e)

To determine

Whether the events are mutual exclusive or not.

Answer to Problem 9RE

The events are not mutual exclusive.

Explanation of Solution

Given:

Probability of female is 0.55, probability of business major is 0.2 and the probability that the student is female and business major is 0.15.

Mutual exclusive events are those events that cannot happen together. The expression for the mutual exclusive event is shown below:

  $P\left(F\cap B\right)=0$

Apply the condition for mutual exclusive event for the given condition as follows:

  $P\left(F\cap B\right)=0.15\left(\ne 0\right)$

Thus, the events are not mutual exclusive.