Chapter 2.3, Problem 19E

The following table presents the 100 senators of the 113th U.S. Congress on January 3,2013, classified by political party affiliation and gender.

	Male	Female
Democrat	37	16
Republican	41	4
Independent	2	0

A senator is selected at random from this group. Compute the following probabilities.

1. a. The senator is a male Republican.
2. b. The senator is a Democrat or a female.
3. c. The senator is a Republican.
4. d. The senator is not a Republican.
5. e. The senator is a Democrat.
6. f. The senator is an Independent.
7. g. The senator is a Democrat or an Independent.

To determine

Find the probability that the senator is a male Republican.

Answer to Problem 19E

The probability that the senator is a male Republican is 0.41.

Explanation of Solution

Given info:

The table presents the 100 senators of the 113th U.S. Congress on January 3, 2013, classified by political party affiliation and gender. Also, a senator is selected randomly from the group.

Calculation:

Let R denote that the senator is a Republican and M denotes the senator is male respectively.

The sum of the senators of the given table is given below:

Size	Male	Female	Total
Democrat	37	16	53
Republican	41	4	45
Independent	2	0	2
Total	80	20	100

The formula for the probability that the senator is a male Republican is,

$P\left(R\cap M\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 41 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(R\cap M\right)=\frac{40}{100}\\ =0.4\end{array}$

Thus, the probability that the senator is a male Republican is 0.41.

To determine

Find the probability that the senator is a Democrat or a female.

Answer to Problem 19E

The probability that the senator is a Democrat or a female is 0.57.

Explanation of Solution

Calculation:

Addition Rule:

The formula for probability of getting event A or event B is,

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

Let D denote that the senator is a Democrat and F denotes the senator is female respectively.

The formula for probability that the senator is a Democrat or a female is,

$P\left(D\cup F\right)=P\left(D\right)+P\left(F\right)+P\left(D\cap F\right)$

The formula for probability of event D is,

$P\left(D\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 53 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(D\right)=\frac{53}{100}\\ =0.53\end{array}$

Thus, the probability of D is 0.53.

The formula for probability of event F is,

$P\left(F\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 20 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(F\right)=\frac{20}{100}\\ =0.20\end{array}$

Thus, the probability of F is 0.20.

The formula for probability of $D\cap F$ is,

$P\left(D\cap F\right)=\frac{\text{Frequency for the class in }D\text{ and }F}{\text{Total frequencies in the distribution}}$

Substitute 16 for ‘Frequency for the class A and B’ and 100 for ‘Total frequencies in the distribution’

$\begin{array}{c}P\left(D\cap F\right)=\frac{16}{60}\\ =0.16\end{array}$

Therefore, the probability of $D\cap F$ is 0.16.

Substitute 0.53 for $P\left(D\right)$, 0.20 for $P\left(F\right)$ and 0.16 for $P\left(D\cap F\right)$ in $P\left(D\cup F\right)$

$\begin{array}{c}P\left(D\cup F\right)=0.53+0.20-0.16\\ =0.57\end{array}$

Therefore, the probability that the senator is a Democrat or a female is 0.57.

To determine

Find the probability that the senator is a Republican.

Answer to Problem 19E

The probability that the senator is a Republican is 0.45.

Explanation of Solution

Calculation:

Let D denote that the red car is large.

The formula for probability that the senator is a Republican is,

$P\left(R\right)=P\left(R\cap M\right)+P\left(R\cap F\right)$

From part (a), the probability that the senator is a male Republican is 0.41.

That is, $P\left(R\cap M\right)=0.41$

The formula for probability that the senator is a female Republican is,

$P\left(R\cap F\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 4 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(R\cap F\right)=\frac{4}{100}\\ =0.04\end{array}$

Thus, the probability that the senator is a female Republican is 0.04.

Substitute 0.41 for $P\left(R\cap M\right)$ and 0.04 for $P\left(R\cap F\right)$ in $P\left(R\right)$

$\begin{array}{c}P\left(R\right)=0.41+0.04\\ =0.45\end{array}$

Thus, the probability that the senator is a Republican is 0.45.

To determine

Find the probability that the senator is not a Republican.

Answer to Problem 19E

The probability that the senator is not a Republican is 0.55.

Explanation of Solution

Calculation:

Let R^c denote that the senator is not a Republican.

From part (c), the probability that the senator is a Republican $P\left(R\right)$ is 0.45.

The formula for probability that the senator is not a Republican is,

$P\left(R^c\right)=1-P\left(R\right)$

Substitute 0.45 for $P\left(R\right)$,

$\begin{array}{c}P\left(R^c\right)=1-0.45\\ =0.55\end{array}$

Thus, the probability that the senator is not a Republican is 0.55.

To determine

Find the probability that the senator is a Democrat.

Answer to Problem 19E

The probability that the senator is a Democrat is 0.53.

Explanation of Solution

Calculation:

The formula for probability that the senator is a Democrat is,

$P\left(D\right)=P\left(D\cap M\right)+P\left(D\cap F\right)$

From part (b), the probability that the senator is a female Democrat is 0.16.

That is, $P\left(D\cap F\right)=0.16$

The formula for probability that the senator is a male Democrat is,

$P\left(D\cap M\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 37 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(D\cap M\right)=\frac{37}{100}\\ =0.37\end{array}$

Thus, the probability that the senator is a male Democrat is 0.37.

Substitute 0.37 for $P\left(D\cap M\right)$ and 0.16 for $P\left(D\cap F\right)$ in $P\left(D\right)$

$\begin{array}{c}P\left(D\right)=0.37+0.16\\ =0.53\end{array}$

Thus, the probability that the senator is a Democrat is 0.53.

To determine

Find the probability that the senator is an Independent.

Answer to Problem 19E

The probability that the senator is an Independent is 0.02.

Explanation of Solution

Calculation:

Let I denote the senator is an Independent.

The formula for probability that the senator is an Independent is,

$P\left(I\right)=P\left(I\cap M\right)+P\left(I\cap F\right)$

The formula for probability that the senator is a male Independent is,

$P\left(I\cap M\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 2 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(I\cap M\right)=\frac{2}{100}\\ =0.02\end{array}$

Thus, the probability that the senator is a male Independent is 0.02.

The formula for probability that the senator is a female Independent is,

$P\left(I\cap F\right)=\frac{\text{Frequency for the class}}{\text{Total frequencies in the distribution}}$

Substitute 0 for ‘Frequency for the class’ and 100 for ‘Total frequencies in the distribution’,

$\begin{array}{c}P\left(I\cap F\right)=\frac{0}{100}\\ =0\end{array}$

Thus, the probability that the senator is a female Independent is 0.

Substitute 0.02 for $P\left(I\cap M\right)$ and 0 for $P\left(I\cap F\right)$ in $P\left(I\right)$

$\begin{array}{c}P\left(I\right)=0.02+0\\ =0.02\end{array}$

Thus, the probability that the senator is an Independent is 0.02.

To determine

Find the probability that the senator is a Democrat or an Independent.

Answer to Problem 19E

The probability that the senator is a Democrat or an Independent is 0.55.

Explanation of Solution

Calculation:

Mutually exclusive:

The events A and B are mutually exclusive if they have no common outcomes. That is,

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$ and $P\left(A\cap B\right)=0$

From part (e), $P\left(D\right)=0.53$ and from part (f), $P\left(I\right)=0.02$.

Here, it is observed that Democrat and Independent are mutually exclusive events. Therefore, the formula for probability that the senator is a Democrat or an Independent is,

$P\left(D\cup I\right)=P\left(D\right)+P\left(I\right)$

Substitute 0.53 for $P\left(D\right)$ and 0.02 for $P\left(I\right)$ in $P\left(D\cup I\right)$

$\begin{array}{c}P\left(D\cup I\right)=0.53+0.02\\ =0.55\end{array}$

Therefore, the probability that the senator is a Democrat or a female is 0.57.