Question

Reliability of gas station air gauges. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note(Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air gauges. The next table gives the percentage of gas stations that provide air gauges that over-report the pressure level in the tire.

a. If the gas station air pressure gauge reads 35 psi, what is the probability that the pressure is over-reported by 6 psi or more?

b. If the gas station air pressure gauge reads 55 psi, what is the probability that the pressure is over-reported by 8 psi or more?

c. If the gas station air pressure gauge reads 25 psi, what is the probability that the pressure is not over-reported by 4 psi or more?

d. Are the events A= {over report by 4 psi or more} and B= {over report by 6 psi or more} mutually exclusive?

e.Based on your answer to part d, why do the probabilities in the table not sum to 1?

Short answer

1. The probability is 0.09.
2. The probability is 0.09.
3. The probability is 0.84.
4. The events are not mutually exclusive.
5. Because column events are not mutually exclusive.

Step-by-step solution

Important formula

The formula for probability is$\mathbf{P}\mathbf{=}\frac{\mathrm{Favourable}\mathrm{out}\mathrm{comes}}{\mathrm{Total}\mathrm{out}\mathrm{comes}}$

(a) The probability that the pressure is over-reported by 6 psi or more 

According to the table, the probability is 0.09.

(b) The probability that the pressure is over-reported by 8 psi or more

A. To the table, the probability is 0.09.

(c) The probability that the pressure is not over-reported by 4 psi or more 

The probability of over-reported is 0.16.

$\begin{array}{l}\mathbf{P}\mathbf{(}\mathbf{not}\mathbf{}\mathbf{more}\mathbf{}\mathbf{6}\mathbf{}\mathbf{psi}\mathbf{}\mathbf{or}\mathbf{}\mathbf{more}\mathbf{\%}\mathbf{)}\mathbf{=}\mathbf{1}-\mathbf{P}\mathbf{(}\mathbf{over}\mathbf{}\mathbf{reported}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}-\mathbf{0}\mathbf{.}\mathbf{16}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{84}\end{array}$

So, the probability is 0.84.

(d) Event A and B are mutually exclusive

No. The events are not mutually exclusive.

(e) The probabilities in the table do not sum to 1

The probabilities in the table did not sum to 1 because column events are not mutually exclusive. Therefore, column events are not mutually exclusive.