Chapter 0.5, Problem 5E

(a)

To determine

Whether the event “blue or black” is mutually exclusive or not mutually exclusive, also find its probability.

Answer to Problem 5E

The event “blue or black” is mutually exclusive.

Probability of the event is 0.55.

Explanation of Solution

Given information:

40 vehicles are there on a rental car lot.

3 of the 18 red vehicles are sedan.

9 of the 15 blue vehicles are SUVs.

Rest are black including 2 SUVs.

Calculations:

Out of total 40 cars,

15 are blue, 18 are red, and the remaining black cars must be 7.

Now,

Probability for a blue car,

  $P\left(\text{Blue}\right)=\frac{15}{40}$

Probability for a black car,

  $P\left(\text{Black}\right)=\frac{7}{40}$

Since a car cannot be both blue and black, the event will be mutually exclusive.

Apply addition rule for mutually exclusive events:

  $\begin{array}{l}P\left(\text{Blue}\text{or}\text{Black}\right)=P\left(\text{Blue}\right)+P\left(\text{Black}\right)\\ =\frac{15}{40}+\frac{7}{40}\\ =\frac{22}{40}\\ =\frac{11}{20}\\ =0.55\end{array}$

The event is mutually exclusive with the probability of 0.55.

(b)

To determine

Whether the event “red or SUV” is mutually exclusive or not mutually exclusive, also find its probability.

Answer to Problem 5E

The event “red or SUV” is mutually exclusive.

Probability of the event is 0.725.

Explanation of Solution

Given information:

40 vehicles are there on a rental car lot.

3 of the 18 red vehicles are sedan.

9 of the 15 blue vehicles are SUVs.

Rest are black including 2 SUVs.

Calculations:

In the parking lot, there are total 40 cars.

18 are red cars, 3 of which are sedans and the remaining 15 must be SUVs.

Thus,

In the parking lot, there are total 26 SUVs.

Now,

Probability for a red car,

  $P\left(\text{Red}\right)=\frac{18}{40}$

Probability for SUV,

  $P\left(\text{SUV}\right)=\frac{26}{40}$

Probability for red SUV,

  $P\left(\text{Red}\text{and}\text{SUV}\right)=\frac{15}{40}$

Since a car can be both red and an SUV, the event will not be mutually exclusive.

Thus,

The probability for “red or SUV”:

  $\begin{array}{l}P\left(\text{Red}\text{or}\text{SUV}\right)=P\left(\text{Red}\right)+P\left(\text{SUV}\right)-P\left(\text{Red}\text{and}\text{SUV}\right)\\ =\frac{18}{40}+\frac{26}{40}-\frac{15}{40}\\ =\frac{29}{40}\\ =0.725\end{array}$

The event is not mutually exclusive with the probability of 0.725.

(c)

To determine

Whether the event “black or sedan” is mutually exclusive or not mutually exclusive, also find its probability.

Answer to Problem 5E

The event “black or sedan” is mutually exclusive.

Probability of the event is 0.4.

Explanation of Solution

Given information:

40 vehicles are there on a rental car lot.

3 of the 18 red vehicles are sedan.

9 of the 15 blue vehicles are SUVs.

Rest are black including 2 SUVs.

Calculations:

In the parking lot, there are total 40 cars.

15 are blue, 18 are red, and the remaining black cars must be 7.

There are 3 red sedans, 6 blue sedans and 5 black sedans.

Thus,

In the parking lot, there are total 14 sedans.

Now,

Probability for a black car,

  $P\left(\text{Black}\right)=\frac{7}{40}$

Probability for sedan,

  $P\left(\text{sedan}\right)=\frac{14}{40}$

Probability for black sedan,

  $P\left(\text{Black}\text{and}\text{sedan}\right)=\frac{5}{40}$

Since a car can be both black and a sedan, the event will not be mutually exclusive.

Thus,

The probability for “black or sedan”:

  $\begin{array}{l}P\left(\text{Black}\text{or}\text{sedan}\right)=P\left(\text{Black}\right)+P\left(\text{sedan}\right)-P\left(\text{Black}\text{and}\text{sedan}\right)\\ \text{=}\frac{7}{\text{40}}\text{+}\frac{14}{\text{40}}-\frac{\text{5}}{\text{40}}\\ =\frac{16}{40}\\ =0.4\end{array}$

The event is not mutually exclusive with the probability of 0.4.