1. among 23 - to 28 - year - olds, 27% say they have driven a car while under the influence of alcohol…

Explanation:

Step1: Identify the probability of an individual driving under alcohol influence

Let (p = 0.27) (probability that a 23 - 28 - year - old has driven under alcohol influence) and (q=1 - p= 1 - 0.27 = 0.73) (probability that a 23 - 28 - year - old has not driven under alcohol influence). Since the selections are independent, for (n = 5) people.

Step2: Solve part (a)

The probability that all five have driven under alcohol influence is (P(X = 5)). Using the formula for independent events (P(X = 5)=p\times p\times p\times p\times p). Substitute (p = 0.27), we get (P(X = 5)=(0.27)^{5}\approx0.0014)

Step3: Solve part (b)

The probability that at least one has not driven under alcohol influence is the complement of the event that all have driven under alcohol influence. Let (A) be the event that all have driven under alcohol influence. Then (P(\text{at least one has not})=1 - P(A)). From part (a), (P(A)=(0.27)^{5}), so (P(\text{at least one has not})=1-(0.27)^{5}\approx1 - 0.0014=0.9986)

Step4: Solve part (c)

The probability that none of the five have driven under alcohol influence is (P(X = 0)). Using the formula for independent events (P(X = 0)=q\times q\times q\times q\times q). Substitute (q = 0.73), we get (P(X = 0)=(0.73)^{5}\approx0.2073)

Step5: Solve part (d)

The probability that at least one has driven under alcohol influence is the complement of the event that none have driven under alcohol influence. Let (B) be the event that none have driven under alcohol influence. Then (P(\text{at least one has})=1 - P(B)). From part (c), (P(B)=(0.73)^{5}), so (P(\text{at least one has})=1-(0.73)^{5}\approx1 - 0.2073 = 0.7927)

Answer:

a. (\approx0.0014) b. (\approx0.9986) c. (\approx0.2073) d. (\approx0.7927)