Chapter 5.3, Problem 40E

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

A senator is selected at random from this group.

1. What is the probability that the senator is a woman?
2. What is the probability that the senator is a Republican?
3. What is the probability that the senator is a Republican and a woman?
4. Given that the senator is a woman, what is the probability that she is a Republican?
5. Given that the senator is a Republican. what is the probability that the senator is a woman?

To determine

The probability that randomly chosen senator is a women.

Answer to Problem 40E

$0.2$

Explanation of Solution

Given:

The following table categorizes 100 senator individuals.

	Male	Female	Total
Democrat	36	16	52
Republican	42	4	46
Independent	2	0	2
Total	80	20	100

Formula Used:

Probability of an event is given by

$\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let $W$ be the event that selectedindividual senator is a woman.

$\begin{array}{c}n\left(W\right)=20\\ n\left(S\right)=100\\ P\left(W\right)=\frac{20}{100}\\ =0.2\end{array}$

Hence, the probability of the event that randomly chosen senator is a woman,is $0.2$.

To determine

The probability that randomly chosen senator is Republican.

Answer to Problem 40E

$0.52$

Explanation of Solution

Given:

The following table categorizes 100 senator individuals.

	Male	Female	Total
Democrat	36	16	52
Republican	42	4	46
Independent	2	0	2
Total	80	20	100

Formula Used:

Probability of an event is given by

$\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let $R$ be the event that selected individual senator is a Republican.

$\begin{array}{c}n\left(R\right)=52\\ n\left(S\right)=100\\ P\left(R\right)=\frac{52}{100}\\ =0.52\end{array}$

Hence, the probability of the event that randomly chosen senator is a republican, is $0.52$.

To determine

The probability that the senator is republican and women.

Answer to Problem 40E

$0.04$

Explanation of Solution

Given:

The following table categorizes 100 senator individuals.

	Male	Female	Total
Democrat	36	16	52
Republican	42	4	46
Independent	2	0	2
Total	80	20	100

Formula Used:

For any two t events $A$ and B, probability of occurring both the events is

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

Calculation:

Let $W$ be the event that selected individual senator is a woman.

Let $R$ be the event that selected individual senator is a Republican.

So, $W\text{and}R$ be the event that selected senator is republican and women.

There are 4 women who are Republican.

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

The probability that the senator is republican and women, is $0.04$.

To determine

The probability to a senator of being republican, given that she is woman.

Answer to Problem 40E

$0.238$

Explanation of Solution

Given:

The following table categorizes 100 senator individuals.

	Male	Female	Total
Democrat	36	16	52
Republican	42	4	46
Independent	2	0	2
Total	80	20	100

Formula Used:

For any two dependent events $A$ and B,

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

Where

$P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let $W$ be the event that selected individual senator is a woman.

$\begin{array}{c}n\left(W\right)=20\\ n\left(S\right)=100\\ P\left(W\right)=\frac{20}{100}\\ =0.2\end{array}$

Let $R$ be the event that selected individual senator is a Republican.

So, $W\text{and}R$ be the event that selected senator is republican and women.

There are 4 women who are Republican.

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

Therefore, $R/W$ be the event that a senator of being republican, given that she is woman.

So, the probability of the event “a senator of being republican, given that she is woman” is calculated as follows:

$\begin{array}{c}P\left(R/W\right)=\frac{P\left(R\text{and}W\right)}{P\left(W\right)}\\ =\frac{0.04}{0.2}\\ =0.2\end{array}$

The probability to a senator of being republican, given that she is woman is $0.2$.

To determine

The probability to a senator of being woman, given that individual is republican.

Answer to Problem 40E

$0.077$

Explanation of Solution

Given:

The following table categorizes 100 senator individuals.

	Male	Female	Total
Democrat	36	16	52
Republican	42	4	46
Independent	2	0	2
Total	80	20	100

Formula Used:

For any two dependent events $A$ and B,

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

Where

$P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let $R$ be the event that selected individual senator is a Republican.

$\begin{array}{c}n\left(R\right)=52\\ n\left(S\right)=100\\ P\left(R\right)=\frac{52}{100}\\ =0.52\end{array}$

Let $W$ be the event that selected individual senator is a woman.

So, $W\text{and}R$ be the event that selected senator is republican and women.

There are 4 women who are Republican.

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

Therefore, $W/R$ be the event that a senator of being republican, given that she is woman.

So, the probability of the event “a senator of being republican, given that she is woman” is calculated as follows:

$\begin{array}{c}P\left(W/R\right)=\frac{P\left(R\text{and}W\right)}{P\left(R\right)}\\ =\frac{0.04}{0.52}\\ =\frac{4}{52}\\ =0.077\end{array}$

The probability to a senator of being woman, given that individual is republican is $0.077$.