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?

Explanation:

Step1: Identify the probability of one person driving under alcohol

The probability \(p\) that a 23 - to 28 - year - old has driven under the influence of alcohol is \(p = 0.27\).

Step2: Use the multiplication rule for independent events

Since the selections are independent, for \(n = 5\) independent events, the probability that all \(n\) events occur is \(P(X=5)=p\times p\times p\times p\times p\). Using the formula \(P(X = k)=p^{k}(1 - p)^{n - k}\), when \(k = 5\) and \(n = 5\), we have \(P(X = 5)=(0.27)^{5}\).

$$
LATEXBLOCK0
$$

Answer:

The probability that all five have driven a car while under the influence of alcohol is approximately \(0.0014\)