Chapter 4.3, Problem 28E

Congressional Terms Below is given the summary from the 112th Congress of Senators whose terms end in 2013, 2015, or 2017.

2013	2015	2017
Democrat	21	20	1
Republican	8	15	13

Choose one of these Senators at random and find

a. P(Democrat and term expires in 2015)

b. P(Republican or term expires in 2013)

c. P(Republican given term expires in 2017)

Are the events “Republican” and “term expires in 2015” independent? Explain.

To determine

To obtain: The P(Democrat and term expires in 2015).

Answer to Problem 28E

The P(Democrat and term expires in 2015) is 0.256.

Explanation of Solution

Given info:

The data shows that the results Democrat and Republican Senators and terms expire which are 2013, 2015, or 2017.

Calculation:

The total number of Democrat and Republican obtained as shown in the Table (1).

	2013	2015	2017	Total
Democrat	21	20	1	42
Republican	8	15	13	36
Total	29	35	14	78

Table (1)

Let event A denote that theSenator is Democrat and event B denote that theSenator term expires in 2015.

The formula for probability of event A and B is,

$P\left(A\text{ and }B\right)=\frac{\text{Number of outcomes in }A\text{ and }B}{\text{Total number of outcomes in the sample space}}$

Substitute 20 for ‘Number of outcomes in A and B’ and78 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(A\text{ and }B\right)=\frac{20}{78}\\ =0.256\end{array}$

Thus, the P(Democrat and term expires in 2015) is 0.256.

To determine

To obtain:The P(Republican or term expires in 2013).

Answer to Problem 28E

The P(Republican or term expires in 2013)is 0.731.

Explanation of Solution

Calculation:

Let event C denote that theSenator is Republican and event D denote that theSenator term expires in 2013.

The probability of event C is,

$P\left(C\right)=\frac{\text{Number of outcomes in }C}{\text{Total number of outcomes in the sample space}}$

Substitute 36 for ‘Number of outcomes in C’ and78 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(C\right)=\frac{36}{78}\\ =0.462\end{array}$

The formula for probability of event D is,

$P\left(D\right)=\frac{\text{Number of outcomes in }D}{\text{Total number of outcomes in the sample space}}$

Substitute 29 for ‘Number of outcomes in D’ and 78 for‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(D\right)=\frac{29}{78}\\ =0.372\end{array}$

The formula for probability of event C and D is,

$P\left(C\text{ and }D\right)=\frac{\text{Number of outcomes in }C\text{ and }D}{\text{Total number of outcomes in the sample space}}$

Substitute 8 for ‘Number of outcomes inC and D’ and78 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(C\text{ and }D\right)=\frac{8}{78}\\ =0.103\end{array}$

Addition Rule:

The formula for probability of getting event C or event D is,

$P\left(C\text{ or }D\right)=P\left(C\right)+P\left(D\right)-P\left(C\text{ and }D\right)$

Substitute 0.462 for ‘ $P\left(C\right)$’, 0.372 for ‘ $P\left(D\right)$’ and 0.103 for ‘ $P\left(C\text{ and }D\right)$’

$\begin{array}{c}P\left(C\text{ or }D\right)=0.462+0.372-0.103\\ =0.731\end{array}$

Thus, theP(Republican or term expires in 2013)is 0.731.

To determine

To obtain:The P(Republican given terms expires in 2017).

Answer to Problem 28E

The P(Republicangiven terms expires in 2017)is 0.929.

Explanation of Solution

Calculation:

Let event E denote that theSenator term expires in 2017.

The probability of event E is,

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

Substitute 14 for ‘Number of outcomes in C’ and78 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(E\right)=\frac{14}{78}\\ =0.1795\end{array}$

The formula for probability of event C and E is,

$P\left(C\text{ and }E\right)=\frac{\text{Number of outcomes in }C\text{ and }E}{\text{Total number of outcomes in the sample space}}$

Substitute 13 for ‘Number of outcomes inC and E’ and78 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(C\text{ and }D\right)=\frac{13}{78}\\ =0.1667\end{array}$

Conditional rule:

The formula for probability of C given E is, $P\left(C|E\right)=\frac{P\left(C\text{ and }E\right)}{P\left(E\right)}$

Then, the probability that the Senator is Republican given that theSenator term expires in 2017 is,

$\begin{array}{c}P\left(C|E\right)=\frac{P\left(C\text{ and }E\right)}{P\left(E\right)}\\ =\frac{0.1667}{0.1795}\\ =0.929\end{array}$

Therefore, the P(Republicangiven terms expires in 2017)is 0.929.

To determine

To explain: Whether the events “Republican” and “term expires in 2015”are independent or not.

Answer to Problem 28E

The events “Republican” and “term expires in 2015”are not independent.

Explanation of Solution

Calculation:

Let event C denote that theSenator is Republican and event B denote that theSenator term expires in 2015.

From, part (b) it is observed that theprobability that the Senator is Republican is 0.462.

The formula for probability of event B is,

$P\left(B\right)=\frac{\text{Number of outcomes in }B}{\text{Total number of outcomes in the sample space}}$

Substitute 35 for ‘Number of outcomes in B’ and 78 for‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(B\right)=\frac{35}{78}\\ =0.449\end{array}$

The formula for probability of event C and B is,

$P\left(C\text{ and }B\right)=\frac{\text{Number of outcomes in }C\text{ and }B}{\text{Total number of outcomes in the sample space}}$

Substitute 15 for ‘Number of outcomes inC and B’ and78 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(C\text{ and }B\right)=\frac{15}{78}\\ =0.192\end{array}$

Independent events:

If $P\left(C\text{ and }B\right)=P\left(C\right)\times P\left(B\right)$ then the events C and B are independent.

Substitute 0.462 for $P\left(C\right)$, 0.449 for $P\left(B\right)$ in $P\left(C\right)\times P\left(B\right)$

$\begin{array}{c}P\left(C\right)\times P\left(B\right)=0.462\times 0.449\\ =0.207\end{array}$

Here, $P\left(C\right)\times P\left(B\right)$ is not equal to $P\left(C\text{ and }B\right)$. That is, $0.207\ne 0.192$. Thus, the events “Republican” and “term expires in 2015”are not independent.