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 0.1779
(round to four decimal places as needed.)
(d) what is the probability that at least one has driven while under the influence of alcohol?
the probability is
(round to four decimal places as needed.)

Explanation:

Step1: Find the probability of a person not driving under influence

The probability of a 21 - 25 - year - old driving under influence \(p = 0.292\). So the probability of a 21 - 25 - year - old not driving under influence \(q=1 - p=1 - 0.292 = 0.708\)

Step2: Use the complement rule

The probability that at least one has driven while under the influence of alcohol is the complement of the event that none of them has driven while under the influence of alcohol.
Let \(P(X = 0)\) be the probability that none of the five has driven while under the influence of alcohol. We know from part (c) that \(P(X = 0)=0.1779\)
By the complement rule \(P(X\geq1)=1 - P(X = 0)\)

Step3: Calculate the probability

Substitute \(P(X = 0)=0.1779\) into the formula \(P(X\geq1)=1 - 0.1779=0.8221\)

Answer:

\(0.8221\)