Chapter 3.2, Problem 15E

(a)

To determine

State whether believing that the Earth is warming and being a liberal Democrat is mutually exclusive.

Answer to Problem 15E

No.

Explanation of Solution

Calculation:

The table represents the distribution of responses by the party and ideology.

Mutually exclusive event:

An event is said to be mutually exclusive if $P\left(A\cap B\right)=P\left(A\right)\times P\left(B\right)$.

From the table, $P\left(\begin{array}{l}\text{Earth is warming}\cap \\ \text{being a liberal Democrat}\end{array}\right)=0.18$

The value is obtained as given below:

$\begin{array}{c}\left\{\begin{array}{l}P\left(\text{Earth is warming}\right)\times \\ P\left(\text{Liberal Democrat}\right)\end{array}\right\}=0.60\times 0.20\\ =0.12\\ \ne P\left(\begin{array}{l}\text{Earth is warming}\cap \\ \text{being a liberal Democrat}\end{array}\right)\end{array}$

Therefore, the Earth is warming and being a liberal Democrat are not mutually exclusive.

(b)

To determine

Obtain the probability that a randomly chosen respondent believes that the Earth is warming or is a liberal democrat.

Answer to Problem 15E

The probability that a randomly chosen respondent believes that the Earth is warming or is a liberal democrat is 0.62.

Explanation of Solution

Calculation:

Let A represent the Earth is warming and B represent the Liberal Democrat.

General Addition Rule:

For any two events A and B, the probability that at least one of them will occur is as given below:

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

Where, $P\left(A\text{ and }B\right)$ is the probability that both events occur.

The following values are obtained from the table:

$\begin{array}{l}P\left(A\right)=0.60\\ P\left(B\right)=0.20\\ P\left(A\text{ and }B\right)=0.18\end{array}$

The probability that a randomly chosen respondent believes that the Earth is warming or is a liberal democrat is as given below:

$\begin{array}{c}P\left(A\text{ or }B\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{ and }B\right)\\ =0.60+0.20-0.18\\ =0.62\end{array}$

Therefore, the probability that a randomly chosen respondent believes that the Earth is warming or is a liberal democrat is 0.62.

(c)

To determine

Obtain the probability that a randomly chosen respondent believes that the Earth is warming, given that he is a liberal democrat.

Answer to Problem 15E

The probability that a randomly chosen respondent believes that the Earth is warming, given that he is a liberal democrat is 0.9.

Explanation of Solution

Calculation:

Let A represent the Earth is warming and B represent the Liberal Democrat.

Conditional probability:

The conditional probability of outcome A, given condition  B is computed is as given below:

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

Where, $P\left(A\text{ and }B\right)$ is the probability that both events occur.

The following values are obtained from the table:

$\begin{array}{l}P\left(B\right)=0.20\\ P\left(A\text{ and }B\right)=0.18\end{array}$

The probability that a randomly chosen respondent believes that the Earth is warming, given that he is a liberal democrat is as given below:

$\begin{array}{c}P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}\\ =\frac{0.18}{0.20}\\ =0.9\end{array}$

Therefore, the probability that a randomly chosen respondent believes that the Earth is warming, given that he is a liberal democrat is 0.9.

(d)

To determine

Obtain the probability that a randomly chosen respondent believes that the Earth is warming, given that he is a conservative Republican.

Answer to Problem 15E

The probability that a randomly chosen respondent believes that the Earth is warming, given that he is a conservative Republican is 0.33.

Explanation of Solution

Calculation:

Let A represent the Earth is warming and B represent the conservative Republican.

The following values are obtained from the table:

$\begin{array}{l}P\left(B\right)=0.33\\ P\left(A\text{ and }B\right)=0.11\end{array}$

The probability that a randomly chosen respondent believes that the Earth is warming, given that he is a conservative Republican is as given below:

$\begin{array}{c}P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}\\ =\frac{0.11}{0.33}\\ =0.33\end{array}$

Therefore, the probability that a randomly chosen respondent believes that the Earth is warming, given that he is a conservative Republican is 0.33.

(e)

To determine

Delineate that it appears that whether or not a respondent believes the Earth is warning is independent of their party and ideology.

Answer to Problem 15E

No.

Explanation of Solution

From the results of Parts (c) and (d), the probability that a randomly chosen respondent believes that the Earth is warming, given that he is a liberal democrat is 0.9 and the probability that a randomly chosen respondent believes that the Earth is warming, given that he is a conservative Republican is 0.33.

If the Earth is warming is independent of their party and ideology, the results obtained in Parts (c) and (d) will be equal.

However, it is clear that the probabilities in Part (c) and Part (d) are not equal.

Therefore, the Earth is warmning is not independent of their party and ideology.

(f)

To determine

Obtain the probability that a randomly chosen respondent is a moderate/liberal Republican, given that he does not believe that the Earth is warming.

Answer to Problem 15E

The probability that a randomly chosen respondent is a moderate/liberal Republican, given that he does not believe that the Earth is warming is 0.18.

Explanation of Solution

Calculation:

Let A represent the moderate/liberal Republican and B represent that he does not believe that the Earth is warming.

The following values are obtained from the table:

$\begin{array}{l}P\left(B\right)=0.34\\ P\left(A\text{ and }B\right)=0.06\end{array}$

The probability that a randomly chosen respondent is a moderate/liberal Republican, given that he does not believe that the Earth is warming is as given below:

$\begin{array}{c}P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}\\ =\frac{0.06}{0.34}\\ =0.18\end{array}$

Therefore, the probability that a randomly chosen respondent is a moderate/liberal Republican, given that he does not believe that the Earth is warming is 0.18.