Question

among 21 - to 25 - year - olds, 29.2% say they have driven while under the influence of alcohol. suppose that five 21 - to 25 - year - olds are selected at random. complete parts (a) through (d)
(a) what is the probability that all five have driven while under the influence of alcohol?
the probability is 0.0021
(round to four decimal places as needed.)
(b) what is the probability that at least one has not driven while under the influence of alcohol?
the probability is 0.9979
(round to four decimal places as needed.)
(c) what is the probability that none of the five have driven while under the influence of alcohol?
the probability is
(round to four decimal places as needed.)

Explanation:

Step1: Determine the probability of not driving under influence

The probability of driving under influence \(p = 0.292\). So the probability of not driving under influence \(q=1 - p=1 - 0.292 = 0.708\)

Step2: Use the formula for independent events

Since we are selecting 5 people and we want the probability that none of them have driven under influence. For independent events, the probability that \(n = 5\) events all occur (in this case, all 5 not driving under influence) is \(q\times q\times q\times q\times q=q^{n}\)
Substitute \(n = 5\) and \(q = 0.708\) into the formula: \(P(X = 0)=0.708^{5}\)
Calculate \(0.708^{5}=0.708\times0.708\times0.708\times0.708\times0.708\approx0.1772\)

Answer:

\(0.1772\)