Final answer:
The probability that all three young adults have driven under the influence is 0.019683, that at least one has not is 0.980317, and that at least one has is 0.610983.
Step-by-step explanation:
The question asks us to calculate the probability of certain outcomes when selecting 3 young adults at random concerning their behavior with driving under the influence of alcohol. Here's how we solve each part:
Probability all 3 have driven under the influence of alcohol
We know that 27% (or 0.27 as a decimal) of young adults have driven under the influence. To find the probability that all three have done so, we multiply the individual probabilities together:
0.27 * 0.27 * 0.27 = 0.019683
Probability at least one has not driven under the influence of alcohol
First, we find the probability that a young adult has not driven under the influence, which is 1 - 0.27 = 0.73. To find the probability that at least one has not driven under the influence, we calculate the complement of the probability that all three have driven under the influence:
1 - (0.27 * 0.27 * 0.27) = 1 - 0.019683 = 0.980317
Probability at least one has driven under the influence of alcohol
This is the complement of the probability that none of them have driven under the influence. The probability that one young adult has not driven under the influence is 0.73, so for three young adults, it is 0.73 * 0.73 * 0.73. Thus:
1 - (0.73 * 0.73 * 0.73) = 1 - 0.389017 = 0.610983