Question

Use the information in the Table $3.19$to answer the next eight exercises. The table shows the political party affiliation of each of $67$members of the US Senate In June $2012$. and when they are up freelection.

The events "Republican' and "Up for reelection in $2016$" are

a. mutually exclusive.

b. independent.

c. both mutually exclusive and independent.

d. neither mutually exclusive nor independent.

Short answer

The events republican and up for reelection in $2016$is neither mutually exclusive nor independent.So answer d is correct.

Step-by-step solution

Given information

Total no of persons in political party affiliation is $67$.

In $2016$number of persons in republican party is$24$.

Calculation for mutually exclusive

For mutually exclusive ,

If the probability of both events are equals to zero means ,they are mutually exclusive.

Calculating the probability of the following events:

$\mathrm{P}=\frac{24}{67}$

$\mathrm{P}=\frac{24}{67}\ne 0$

Thus, we can conclude that events are not mutually exclusive.

Calculation for independent

For independent it is satisfy the following condition,

$\mathrm{P}\left(\mathrm{R}\right|\mathrm{U})=\mathrm{P}(\mathrm{U})$

Where,

$\mathrm{P}\left(\mathrm{R}\right)$is probability of republican.

$\mathrm{P}\left(\mathrm{U}\right)$is probability of members in political party.

So,

localid="1649672743221" $\mathrm{P}(\mathrm{R}\mathrm{AND}\mathrm{U})=\mathrm{P}\left(\mathrm{R}\right)\times \mathrm{P}\left(\mathrm{U}\right)$

localid="1649673061264" $\mathrm{P}(\mathrm{R}\mathrm{AND}\mathrm{U})=\mathrm{P}\left(\mathrm{R}\right)+\mathrm{P}\left(\mathrm{U}\right)-\mathrm{P}(\mathrm{R}\mathrm{OR}\mathrm{U})$

localid="1649672629690" $=\frac{24}{37}+\frac{1}{67}+\left(\frac{24}{37}\times \frac{1}{67}\right)$

$=0.3828$

$\mathrm{P}\left(\mathrm{R}\right)\times \mathrm{P}\left(\mathrm{U}\right)=\frac{24}{37}\times \frac{1}{67}$

$=0.009$

$\mathrm{P}(\mathrm{R}\mathrm{AND}\mathrm{U})=\mathrm{P}\left(\mathrm{R}\right)\times \mathrm{P}\left(\mathrm{U}\right)$is not satisfied.