Question

1. among 23 - to 28 - year - olds, 27% say they have driven a car while under the influence of alcohol. suppose five 23 - to 28 - year - olds are selected at random.

a. what is the probability that all five have driven a car while under the influence of alcohol?
b. what is the probability that at least one has not driven a car while under the influence of alcohol?
c. what is the probability that none of the five have driven a car while under the influence of alcohol?
d. what is the probability that at least one has driven a car while under the influence of alcohol?

Explanation:

Step1: Calculate the probability that none of the five have driven a car while under the influence of alcohol

Let \(p = 0.27\) (probability that a single 23 - 28 - year - old has driven a car while under the influence of alcohol). Then the probability that a single 23 - 28 - year - old has not driven a car while under the influence of alcohol is \(q=1 - p=1 - 0.27 = 0.73\).
The probability that none of the five (\(n = 5\)) have driven a car while under the influence of alcohol is \(P(\text{none})=(0.73)^{5}\)

$$P(\text{none})=(0.73)^{5}=0.73\times0.73\times0.73\times0.73\times0.73\approx0.2073$$

Step2: Calculate the probability that at least one has driven a car while under the influence of alcohol

We use the formula \(P(X\geq1)=1 - P(X = 0)\) (where \(X\) is the number of people who have driven a car while under the influence of alcohol).
Since \(P(X = 0)=(0.73)^{5}\approx0.2073\), then \(P(X\geq1)=1-(0.73)^{5}\)

$$P(X\geq1)=1 - 0.2073=0.7927$$

Answer:

The probability that at least one has driven a car while under the influence of alcohol is approximately \(0.7927\)