Chapter 15, Problem 20E

(a)

To determine

To find: the probability he has neither retirement plan nor employer-sponsored health insurance.

Answer to Problem 20E

0.25

Explanation of Solution

Given:

  $\begin{array}{l}P\left(R\right)=0.56\\ P\left(H\right)=0.68\\ P\left(R\cap H\right)=0.49\end{array}$

Formula used:

  $P\left(R^{\text{'}}\cap H^{\text{'}}\right)=1-P\left(R\cup H\right)$

Calculation:

R = American worker has retirement plan

H = American worker has health insurance plan

The required probability is

Calculated using complements

  $\begin{array}{c}P\left(R^{\text{'}}\cap H^{\text{'}}\right)=1-P\left(R\cup H\right)\\ =1-\{P\left(R\right)+P\left(H\right)-P\left(R\cap H\right)\}\\ =1-\{0.56+0.68-0.49\}\\ =1-0.75\\ =0.25\end{array}$

About 25% of American worker has neither retirement plan nor health insurance plan.

(b)

To determine

To find: the probability that he has health insurance if he has a retirement plan.

Answer to Problem 20E

0.875

Explanation of Solution

Given:

  $\begin{array}{l}P\left(R\right)=0.56\\ P\left(H\right)=0.68\\ P\left(R\cap H\right)=0.49\end{array}$

Formula used:

  $P\left(H|R\right)=\frac{P\left(R\cap H\right)}{P\left(R\right)}$

Calculation:

Probability that he has health insurance plan if he retirement plan is already exists.

  $\begin{array}{c}P\left(H|R\right)=\frac{P\left(R\cap H\right)}{P\left(R\right)}\\ =\frac{0.49}{0.56}\\ =0.875\end{array}$

There is 87.5% possibility that he has health insurance plan if he has retirement plan.

(c)

To determine

To Explain: the health insurance and a retirement plan are having independent events.

Answer to Problem 20E

  $P\left(R\cap H\right)\ne P\left(R\right)P\left(H\right)$

Not independent events

Explanation of Solution

Given:

  $\begin{array}{l}P\left(R\right)=0.56\\ P\left(H\right)=0.68\\ P\left(R\cap H\right)=0.49\end{array}$

Calculation:

Test for independent events

  $\begin{array}{c}P\left(R\cap H\right)=P\left(R\right)P\left(H\right)\\ =0.56\times 0.68\\ =0.3808\end{array}$

But from the given information $P\left(R\cap H\right)=0.49$

Therefore, $P\left(R\cap H\right)\ne P\left(R\right)P\left(H\right)$

Hence, having health insurance and retirement plan are not independent events.

(d)

To determine

To Explain: these two benefits are having mutually exclusive.

Answer to Problem 20E

Not mutually exclusive

Explanation of Solution

Given:

  $\begin{array}{l}P\left(R\right)=0.56\\ P\left(H\right)=0.68\\ P\left(R\cap H\right)=0.49\end{array}$

Since $P\left(R\cap H\right)\ne 0$ the two events are not mutually exclusive. Therefore it is concluded that having retirement plan and health insurance are not mutually exclusive.