Snowmobiles

(a) If we choose a survey respondent at random, what’s the probability that this individual (i) is a snowmobile owner? (ii) belongs to an environmental organization or owns a snowmobile? (iii) has never used a snowmobile given that the person belongs to an environmental organization?

(b) Are the events “is a snowmobile owner” and “belongs to an environmental organization” independent for the members of the sample? Justify your answer.

(c) If we choose two survey respondents at random, what’s the probability that (i) both are snowmobile owners? (ii) at least one of the two belongs to an environmental organization?

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$\text{(i)}P\left(\text{snowmobile owner}\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$=\frac{295}{1526}$

$\approx 0.1933$

$\text{(ii)}P\left(\text{Yes or snowmobile owner}\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$=\frac{77+212+16+279}{1526}$

$=\frac{584}{1526}$

$\approx 0.3827$

(iii) Definition conditional probability:

$P(A\mid B)=\frac{P\left(A\text{and}B\right)}{P\left(B\right)}$

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P(\text{Never used}\mid \text{Yes})=\frac{P\left(\text{Never used and Yes}\right)}{P\left(\text{Yes}\right)}$

$=\frac{212/1526}{305/1526}$

$=\frac{212}{305}$

$\approx 0.6951$

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P\left(\text{snowmobile owner}\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$=\frac{295}{1526}$

$\approx 0.1933$

Definition of conditional probability:

$P(A\mid B)=\frac{P\left(A\text{and}B\right)}{P\left(B\right)}$

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P(\text{snowmobile owner}\mid \mathrm{Yes})=\frac{P\left(\text{snowmobile owner and}\mathrm{Yes}\right)}{P\left(\mathrm{Yes}\right)}$

$=\frac{16/1526}{305/1526}$

$=\frac{16}{305}$

$\approx 0.0525$

Since the two probabilities $P$(snowmobile owner ) and $P$$(snowmobileowner|Yes)$are not equal, the events are not independent.

The probability is the number of favorable outcomes divided by the number of possible outcomes:

$P\left(\text{snowmobile owner}\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$=\frac{295}{1526}$

$\approx 0.1933$

$P\left(\text{No}\right)=\frac{\#\text{of favorable outcomes}}{\#\text{of possible outcomes}}$

$=\frac{1221}{1526}$

$\approx 0.8001$

Multiplication rule and complement rule:

$P\left(A\text{and}B\right)=P\left(A\right)\times P\left(B\right)$

$P(notA=1-P(A)$

Determine the probabilities using the multiplication and complement rule:

$\text{(i)}P\left(2\text{snowmobile owners}\right)=\left(P\right(\text{snowmobile owner}){)}^2$

$=0.{1933}^2$

$\approx 0.0374$

$\text{(ii)}P\left(2\mathrm{No}\right)=\left(P\right(\mathrm{No}){)}^2$

$=0.{8001}^2$

$\approx 0.6402$

$P\left(\text{at least one Yes}\right)=1-P\left(2\text{No}\right)$

$=1-0.6402$

$=0.3598$