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 localid="1649516803996" $67$members of the US Senate In June localid="1649516807155" $2012$. and when they are up freelection.

The events "Other" and "Up for reelection in November localid="1649516811372" $2016$' are

a. mutually exclusive.

b. independent.

c. both mutually exclusive and independent.

d. neither mutually exclusive nor independent

Short answer

In November $2016$, the events Other and Up for reelection are mutually exclusive.

As a result, mutually exclusive is the correct answer.

Step-by-step solution

Given

The table shows the political party affiliation of each of $67$members of the US Senate In June $2012$. and when they are up reelection.

Finding the probability of event Other and Is up for reelection in 2016

The total as:

Events A and B are mutually exclusive if the probability of event $A$AND $B$equals $0$.

Finding the probability of event Other and Is up for reelection in $2016$:

$P=\frac{0}{67}=0$

Thus, we can conclude that events are mutually exclusive.

If events A and B are mutually exclusive, then they cannot be independent. That is because, if they are mutually exclusive, they cannot happen at the same time. If we know that B happened, then we immediately know that A didn't. Thus:

$P(A\mid B)=0\ne P\left(A\right)$