Chapter 4.5, Problem 120E

To determine

To graph: The Venn diagram that shows the relation between the events A and B in Exercise 4.112.

Explanation of Solution

Calculation: The event “A” is defined as the event that the randomly selected American household is prosperous and the event “B” is defined as the event that the randomly selected American household is educated. The probabilities are provided as $P\left(A\right)=0.138$, $P\left(B\right)=0.261,$ and $P\left(A\text{and }B\right),$ which can be represented as $P\left(A\cap B\right),$ is 0.082.

Graph: The Venn diagram indicating the above probabilities and depicting the events $\left\{A\right\},\left\{B\right\},\left\{A\cap B\right\},\left\{A^C\cap B\right\},\left\{A\cap B^C\right\},\text{and}\left\{A^C\cap B^C\right\}$ in the universal set $U$ is as follows:

(a)

To determine

The probability for the event $\left\{A\text{and}B\right\}$.

Answer to Problem 120E

Solution: The required probability is 0.082.

Explanation of Solution

From the Venn diagram, the probability of event $\left\{A\text{and}B\right\}$ is ascertained as follows.

$\begin{array}{c}P\left(A\text{and }B\right)=P\left(A\cap B\right)\\ =0.082\end{array}$

Hence, the required probability is 0.082.

To determine

To explain: THE event $\left\{A\text{and}B\right\}$ in words.

Answer to Problem 120E

Solution: The event $\left\{A\text{and}B\right\}$ means that the household is prosperous $\left(A\right)$ as well as educated $\left(B\right)$.

Explanation of Solution

The event $\left\{A\text{and}B\right\}$ means the intersection of $A$ and $B$, that is, the household is prosperous $\left(A\right)$ as well as educated $\left(B\right)$.

(b)

To determine

The probability for the event $\left\{A^C\text{and}B\right\}$.

Answer to Problem 120E

Solution: The required probability is 0.179.

Explanation of Solution

From the Venn diagram, the probability of event $\left\{A^C\text{and}B\right\}$ is ascertained as follows.

$\begin{array}{c}P\left(A^C\text{and }B\right)=P\left(A^C\cap B\right)\\ =P\left(B\right)-P\left(A\cap B\right)\\ =0.261-0.082\\ =0.179\end{array}$

Hence, the required probability is 0.179.

To determine

To explain: The event $\left\{A^C\text{and}B\right\}$ in words.

Answer to Problem 120E

Solution: The event $\left\{A^C\text{and}B\right\}$ means that the household is educated $\left(B\right)$ but not prosperous $\left(A\right)$.

Explanation of Solution

The event $\left\{A^C\text{and}B\right\}$ means the intersection of $B$ and the complement of $A$, that is, the household is educated $\left(B\right)$ but not prosperous $\left(A\right)$.

(c)

To determine

The probability for the event $\left\{A\text{and}{B}^C\right\}$.

Answer to Problem 120E

Solution: The required probability is 0.056.

Explanation of Solution

From the Venn diagram, the probability of event $\left\{A\text{and}{B}^C\right\}$ is ascertained as follows.

$\begin{array}{c}P\left(A\text{and }{B}^C\right)=P\left(A\cap B^C\right)\\ =P\left(A\right)-P\left(A\cap B\right)\\ =0.138-0.082\\ =0.056\end{array}$

Hence, the required probability is 0.056.

To determine

To explain: The event $\left\{A\text{and}{B}^C\right\}$ in words.

Answer to Problem 120E

Solution: The event $\left\{A\text{and}{B}^C\right\}$ means that the household is prosperous $\left(A\right)$ but not educated $\left(B\right)$.

Explanation of Solution

The event $\left\{A\text{and}{B}^C\right\}$ means the intersection of $A$ and the complement of $B$, that is, the household is prosperous $\left(A\right)$ but not educated $\left(B\right)$.

(d)

To determine

The probability for the event $\left\{A^C\text{and}{B}^C\right\}$.

Answer to Problem 120E

Solution: The required probability is 0.683.

Explanation of Solution

From the Venn diagram, the probability of event $\left\{A^C\text{and}{B}^C\right\}$ is ascertained as follows.

$\begin{array}{c}P\left(A^C\text{and }{B}^C\right)=P\left(A^C\cap B^C\right)\\ =1-P\left(A\text{or }B\right)\\ =1-P\left(A\cup B\right)\\ =1-0.317\end{array}$

$=0.683$

Hence, the required probability is 0.683.

To determine

To explain: The event $\left\{A^C\text{and}{B}^C\right\}$ in words.

Answer to Problem 120E

Solution: The event $\left\{A^C\text{and}{B}^C\right\}$ means that the household is neither prosperous $\left(A\right)$ nor educated $\left(B\right)$.

Explanation of Solution

The event $\left\{A^C\text{and}{B}^C\right\}$ means the intersection of complement of $A$ and the complement of $B$, that is, the household is neither prosperous $\left(A\right)$ nor educated $\left(B\right)$.