Question

Among 21-25 year olds, 29% say they have driven while under the influence of alcohol. Suppose that three 21- to 25-year-olds are selected at random.

a.) What is the probability that all three have driven while under the influence of alcohol?
b.) What is the probability that at least one has not driven while under the influence of alcohol?
I've come up with these calculations but I'm unsure if I did the process correctly. Thank you for your assistance ahead of time.
a) .9756
b.) .0244

Answer 1

The probability that all three have driven under the influence is 2.44%, while the probability that at least one has not is 97.56%.

Step-by-step explanation:

To calculate the probability that all three 21- to 25-year-olds have driven under the influence of alcohol, given that the probability for one individual is 29%, we must assume independence between each individual's actions. We raise the probability to the third power to find the joint probability for all three:

0.293 = 0.29 × 0.29 × 0.29 = 0.0244

Therefore, the probability that all three individuals have driven under the influence is 0.0244, or 2.44%.

To determine the probability that at least one has not driven under the influence, we can calculate the complement of the probability that they all have. This is equal to 1 minus the probability that all three have driven under the influence:

1 - 0.0293 = 1 - 0.0244 = 0.9756

The probability that at least one of the three individuals has not driven under the influence of alcohol is 0.9756, or 97.56%.