Chapter 4.2, Problem 43E

a.

To determine

Compute the probability that selected senator is a male Republican.

Answer to Problem 43E

The probability that selected senator is a male Republican is 0.42.

Explanation of Solution

Calculation:

There were a total of 100 senators. Senators are classified based on party affiliation and gender. The table provides idea bout number senators in each classification.

The probability of the event A can be obtained by the formula:

$P\left(A\right)=\frac{\text{Number of outcomes in }A}{\text{Number of outcomes in}\text{sample space}}$

Here, event A denotes that selected senator is a male Republican. Among the 100 senators, 42 of them are male Republican.

Substitute 42 for “number of outcomes in A” and 100 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(A\right)=\frac{42}{100}\\ =0.42\end{array}$

Thus, the probability that selected senator is a male Republican is 0.42.

b.

To determine

Compute the probability that selected senator is a Democrat or a female.

Answer to Problem 43E

The probability that selected senator is a Democrat or a female is 0.56.

Explanation of Solution

Calculation:

Event V denotes that selected senator is a Democrat, event W denote that selected senator is a female.

The probability that a randomly selected senator is a Democrat or a female can be expressed as, $P\left(V\text{or }W\right)$.

General Addition rule:

For any two events V and W the general addition rule states that

$P\left(V\text{or }W\right)=P\left(V\right)+P\left(W\right)-P\left(V\text{ and }W\right)$

From table, $P\left(V\right)=\frac{52}{100}\text{, }P\left(W\right)=\frac{20}{100},\text{and}P\left(V\text{and }W\right)=\frac{16}{100}$.

Substitute these values in the general addition rule.

Therefore,

$\begin{array}{c}P\left(V\text{or }W\right)=\frac{52}{100}+\frac{20}{100}-\frac{16}{100}\\ =\frac{56}{100}\\ =0.56\end{array}$

Thus, the probability that selected senator is a Democrat or a female is 0.56.

c.

To determine

Compute the probability that selected senator is a Republican.

Answer to Problem 43E

The probability that selected senator is a Republican is 0.46.

Explanation of Solution

Calculation:

The probability of the event E can be obtained by the formula:

$P\left(E\right)=\frac{\text{Number of outcomes in }E}{\text{Number of outcomes in}\text{sample space}}$

Here, event E denotes that selected senator is a Republican. Among the 100 senators, 46 of them are Republican.

Substitute 46 for “number of outcomes in E” and 100 for “Number of outcomes in sample space” in the probability formula.

Therefore,

$\begin{array}{c}P\left(E\right)=\frac{46}{100}\\ =0.46\end{array}$

Thus, the probability that selected senator is a Republican is 0.46.

d.

To determine

Compute the probability that selected senator is not a Republican.

Answer to Problem 43E

The probability that selected senator is not a Republican is 0.54.

Explanation of Solution

Calculation:

Complement Rule:

For any events E, $P\left(E^c\right)=1-P\left(E\right)$.

Event E denotes that selected senator is a Republican. Here events ‘selected senator is a Republican’ and ‘selected senator is not a Republican’ are complements to each other.

From part (c) it is clear that, $P\left(E\right)=0.46$. Substitute this values in the rule of complements.

Therefore,

$\begin{array}{c}P\left(\text{not a Republican}\right)=P\left(E^c\right)\\ =1-P\left(E\right)\\ =0.54\end{array}$

Thus, the probability that selected senator is not a Republican is 0.54.

e.

To determine

Compute the probability that selected senator is a Democrat.

Answer to Problem 43E

The probability that selected senator is a Democrat is 0.52.

Explanation of Solution

Calculation:

The probability of the event V can be obtained by the formula:

$P\left(V\right)=\frac{\text{Number of outcomes in }V}{\text{Number of outcomes in}\text{sample space}}$

Here, event V denotes that selected senator is a Democrat. Among the 100 senators, 52 of them are Democrat.

Substitute 52 for “number of outcomes in V” and 100 for “Number of outcomes in sample space” in the probability formula.

Therefore,

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

Thus, the probability that selected senator is a Democrat is 0.52.

f.

To determine

Compute the probability that selected senator is an independent.

Answer to Problem 43E

The probability that selected senator is an independent is 0.02.

Explanation of Solution

Calculation:

Here, event W denotes that selected senator is an independent. Among the 2 senators, 52 of them are independent.

Substitute 2 for “number of outcomes in W” and 100 for “Number of outcomes in sample space” in the probability formula.

Therefore,

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

Thus, the probability that selected senator is an independent is 0.02.

g.

To determine

Compute the probability that selected senator is a Democrat or an independent.

Answer to Problem 43E

The probability that selected senator is a Democrat or an independent is 0.54.

Explanation of Solution

Calculation:

Event V denotes that selected senator is a Democrat, event W denote that selected senator is an independent. Event V and W are mutually exclusive since the senator belongs to only one of the party affiliation.

The probability that a randomly selected senator is a Democrat or an independent can be expressed as, $P\left(V\text{or }W\right)$.

Addition rule for mutually exclusive events:

For two mutually exclusive events V and W the addition rule states that

$P\left(V\text{or }W\right)=P\left(V\right)+P\left(W\right)$

From table, $P\left(V\right)=\frac{52}{100}\text{, and }P\left(W\right)=\frac{2}{100}$.

Substitute these values in the addition rule.

Therefore,

$\begin{array}{c}P\left(V\text{or }W\right)=\frac{52}{100}+\frac{2}{100}\\ =\frac{54}{100}\\ =0.54\end{array}$

Thus, the probability that selected senator is a Democrat or an independent is 0.54.