Question 7 $10 \mathrm{pts}$ Three out of every eight valley teenagers admit to having experimented with some type of illicit drug. 10 teens are chosen at.random in the valley. What is the probability that exactly 5 of them admit to using illicit drugs? Upload Choose a File

Step 1 ：This problem is a binomial probability problem. The binomial distribution model is appropriate for a statistical experiment if the following conditions are met: 1. The experiment consists of n repeated trials. 2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. 3. The probability of success, denoted by P, is the same on every trial. 4. The trials are independent; the outcome on one trial does not affect the outcome on other trials.

Step 2 ：In this case, the experiment is choosing 10 teenagers at random. Each trial can result in two possible outcomes: the teenager admits to using illicit drugs (success) or does not admit to using illicit drugs (failure). The probability of success is 3/8, and this probability is the same for each teenager chosen. The teenagers are chosen independently; choosing one teenager does not affect the probability of what the next teenager chosen will admit to.

Step 3 ：The formula for the binomial probability is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: - \(P(X=k)\) is the probability we are trying to calculate, - \(C(n, k)\) is the number of combinations of n items taken k at a time, - p is the probability of success, - n is the number of trials, and - k is the number of successes we want.

Step 4 ：In this case, n=10, k=5, and p=3/8. We can plug these values into the formula to find the probability.

Step 5 ：n = 10, k = 5, p = 0.375, combinations = 252, p_k = 0.007415771484375, p_n_k = 0.095367431640625, probability = 0.1782202161848545

Step 6 ：The probability that exactly 5 out of 10 randomly chosen teenagers in the valley admit to using illicit drugs is approximately \(\boxed{0.178}\).