Question

Among 21- to 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 at least 1 has driven while under the influence of alcohol?
b. Among 21- to 25-year-olds, 29% say they have driven while under the influence of alcohol.

Answer 1

Answer:

0.6421

Step-by-step explanation:

In this case we have 3 trials and we have 2 options for each one. The driver has or hasn't been under alcohol influence. The probability that the driver has is 0.29 and the probabiility that the driver hasn't is 1 - 0.29 = 0.71

each trial is independent because we are assuming that the population of drivers in between 21 and 25 years old is very big.

The probability that one of them was under alcohol influence can be found by finding the probability that non of them was under alcohol influence because:

1 = p(x = 0) + p(x ≥ 1)

p(x ≥ 1) = 1 - p(0)

The probability that none of them was under alcohol influence is going to be:

0.71×0.71×0.71 = 0.3579

The probability of finding at least one driver that has been under alcohol influence is:

0.6421

Answer 2

Using the binomial distribution, it is found that there is a 0.6421 = 64.21% probability that at least 1 has driven while under the influence of alcohol.

For each person, there are only two possible outcomes, either they have driven while under the influence of alcohol, or they have not. The answer of each person is independent of any other person, hence the binomial distribution is used to solve this question.

What is the binomial distribution formula?

The formula is:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

In this problem:

• Among 21- to 25-year-olds, 29% say they have driven while under the influence of alcohol, hence p = 0.29.

• Three people are chosen, hence n = 3.

The probability that at least 1 has driven while under the influence of alcohol is given by:

$P(X \geq 1) = 1 - P(X = 0)$

In which:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{3,0}.(0.29)^{0}.(0.71)^{3} = 0.3579$

Then:

$P(X \geq 1) = 1 - P(X = 0) = 1 - 0.3579 = 0.6421$

0.6421 = 64.21% probability that at least 1 has driven while under the influence of alcohol.

More can be learned about the binomial distribution at https://brainly.com/question/14424710