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
(round to four decimal places as needed.)

Explanation:

Step1: Find the probability that all five have driven under the influence

Let \(p = 0.292\) (probability that a 21 - 25 year - old has driven under the influence). The probability that all five have driven under the influence is \(P(X = 5)=p^{5}\) (using the multiplication rule for independent events). We know \(P(X = 5)=0.292^{5}\approx0.0021\) (from part (a)).

Step2: Use the complement rule

The event “at least one has not driven while under the influence” is the complement of the event “all five have driven while under the influence”.
Let \(A\) be the event “all five have driven while under the influence” and \(B\) be the event “at least one has not driven while under the influence”. Then \(P(B)=1 - P(A)\)
Since \(P(A)=0.292^{5}\approx0.0021\), then \(P(B)=1 - 0.0021\)

Answer:

\(0.9979\)