Chapter 12.1, Problem 31E

(a)

To determine

To find out what is the probability that this individual is a snowmobile owner, belongs to an environmental organization or owns a snowmobile and has never used a snowmobile given that the person belongs to an environmental organization.

Answer to Problem 31E

  $\begin{array}{l}P(\text{snowmobile owner)}=0.1933\\ P(\text{Yes or snowmobile owner})=0.3827\\ P(\text{Never used | Yes})=0.6951\end{array}$

Explanation of Solution

The table of the environmental clubs member and non-members are given with the numbers of person who have never used snowmobile, are snowmobiles renter, are snowmobile owners are given. Thus, the probabilities is calculated as:

1. $\begin{array}{l}P(\text{snowmobile owner)}=\frac{295}{1526}\\ =0.1933\end{array}$

2.   $\begin{array}{l}P(\text{Yes or snowmobile owner})\\ =\frac{77+212+16+279}{1526}\\ =\frac{584}{1526}\\ =0.3827\end{array}$

3. As we know the conditional probability is defined as:

  $P(A|B)=\frac{P(A\text{ and }B)}{P(B)}$

Thus, the probability is as:

  $\begin{array}{l}P(\text{Never used | Yes})\\ =\frac{P(\text{Never used and yes})}{P(\text{Yes})}\\ =\frac{\raisebox{1ex}{$212$}\!\left/ \!\raisebox{-1ex}{$1526$}\right.}{\raisebox{1ex}{$305$}\!\left/ \!\raisebox{-1ex}{$1526$}\right.}\\ =\frac{212}{305}\\ =0.6951\end{array}$

(b)

To determine

To explain are the events “is a snowmobile” and “belongs to an environmental organization” independent for the members of the sample.

Answer to Problem 31E

The events are not independent.

Explanation of Solution

The table of the environmental clubs member and non-members are given with the numbers of person who have never used snowmobile, are snowmobiles renter, are snowmobile owners are given. Thus, to check are the events “is a snowmobile” and “belongs to an environmental organization” independent for the members of the sample we have,

  $\begin{array}{l}P(\text{snowmobile owner)}=\frac{295}{1526}\\ =0.1933\end{array}$

And the conditional probability is defined as:

  $P(A|B)=\frac{P(A\text{ and }B)}{P(B)}$

So, we have,

  $\begin{array}{l}P(\text{Snowmobile owner | Yes})\\ =\frac{P(\text{Snowmobile owner and yes})}{P(\text{Yes})}\\ =\frac{\raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$1526$}\right.}{\raisebox{1ex}{$305$}\!\left/ \!\raisebox{-1ex}{$1526$}\right.}\\ =\frac{16}{305}\\ =0.0525\end{array}$

Since the two probabilities of snowmobile owner and $P(\text{Snowmobile owner | Yes})$ are not equal then the events are not independent.

(c)

To determine

To find out what is the probability that both are snowmobile owners and at least one of the two belongs to an environmental organization.

Answer to Problem 31E

  $\begin{array}{l}P(2\text{snowmobile owner)}=0.0374\\ P(\text{at least one yes})=0.3598\end{array}$

Explanation of Solution

The table of the environmental clubs member and non-members are given with the numbers of person who have never used snowmobile, are snowmobiles renter, are snowmobile owners are given. So, we have,

  $\begin{array}{l}P(\text{snowmobile owner)}=\frac{295}{1526}\\ =0.1933\end{array}$

And also,

  $\begin{array}{l}P(\text{No})=\frac{1221}{1526}\\ =0.8001\end{array}$

And as the multiplication rule and complement rule is defined as:

  $\begin{array}{l}P(A\text{ and }B)=P(A)\times P(B)\\ P(\text{Not }A)=1-P(A)\end{array}$

Thus, we will use the above to calculate the probability as:

  $\begin{array}{l}P(2\text{snowmobile owner)=}P{(}^{\text{snowmobile owner)}}\\ ={0.1933}^2\\ =0.0374\end{array}$

  $\begin{array}{l}P(2\text{No})=P{(}^{\text{No}}\\ ={(}^{\frac{1221}{1526}}\\ ={(}^{0.8001}\\ =0.6402\\ P(\text{at least one yes})=1-P(2\text{No})\\ =1-0.6402\\ =0.3598\end{array}$