Chapter 4, Problem 9RE

a.

To determine

Compute the probability that the student is female or a business major.

Answer to Problem 9RE

The probability that the student is female or a business major is 0.60.

Explanation of Solution

Calculation:

It is given that the probability that selected student is a female is 0.55, the probability that the selected student is a business major is 0.20, and probability that student is both female and business major is 0.15.

Event A denotes that selected student is a female, event B denotes that selected student is a business major, and event ‘A and B’ denotes that selected student is both female and business major.

The probability that the student is female or a business major can be expressed as, $P\left(A\text{or }B\right)$.

General Addition rule:

For any two events A and B the general addition rule states that

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

Here, $P\left(A\right)=0.55,P\left(B\right)=0.20\text{and }P\left(A\text{and }B\right)=0.15$. Substitute these values in the general addition rule.

Therefore,

$\begin{array}{c}P\left(A\text{or }B\right)=0.55+0.20-0.15\\ =0.60\end{array}$

Thus, the probability that the student is female or a business major is 0.60.

b.

To determine

Compute the probability that the student is female given that the student is a business major.

Answer to Problem 9RE

The probability that the student is female given that the student is a business major is 0.75.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

$P\left(A|B\right)=\frac{P\left(A\text{and}B\right)}{P\left(B\right)}$.

Here, $P\left(B\right)=0.20\text{and }P\left(A\text{and }B\right)=0.15$. Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(A|B\right)=\frac{0.15}{0.2}\\ =0.75\end{array}$

Thus, the probability that the student is female given that the student is a business major is 0.75.

c.

To determine

Compute the probability that the student is a business major given that the student is female.

Answer to Problem 9RE

The probability that the student is a business major given that the student is female is 0.2727.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

$P\left(B|A\right)=\frac{P\left(A\text{and}B\right)}{P\left(A\right)}$.

Here, $P\left(A\right)=0.55\text{and }P\left(A\text{and }B\right)=0.15$. Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(B|A\right)=\frac{0.15}{0.55}\\ =0.2727\end{array}$

Thus, the probability that the student is a business major given that the student is female is 0.2727.

d.

To determine

Check whether events “female” and “business major” are independent or not.

Explain the reason.

Answer to Problem 9RE

Events “female” and “business major” are not independent.

Explanation of Solution

Calculation:

Independent events:

Two or more events are said to be independent if occurrence of one does not affect the occurrence of other. For independent events A and B, $P\left(A\text{ and }B\right)=P\left(A\right)\times P\left(B\right)$.

Here, $P\left(A\right)=0.55,P\left(B\right)=0.20\text{and }P\left(A\text{and }B\right)=0.15$.

Therefore,

$\begin{array}{c}P\left(A\right)\times P\left(B\right)=\left(0.55\right)\times \left(0.20\right)\\ =0.11\\ \ne P\left(A\text{ and }B\right)\end{array}$

That is, $P\left(A\text{ and }B\right)\ne P\left(A\right)\times P\left(B\right)$

Therefore the events “female” and “business major” are not independent.

e.

To determine

Check whether events “female” and “business major” are mutually exclusive or not.

Explain the reason.

Answer to Problem 9RE

Events “female” and “business major” are not mutually exclusive.

Explanation of Solution

Mutually exclusive events:

Two or more events are said to be mutually exclusive if it is not possible to occur them together. That is, for mutually exclusive events there is no element in common. That is for mutually exclusive events $P\left(A\text{and}B\right)=0$.

Here,

$\begin{array}{c}P\left(A\text{and }B\right)=0.15\\ \ne 0\end{array}$.

Therefore the events “female” and “business major” are not mutually exclusive.