According to a study done by a university student, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267 . Suppose you sit on a bench in a mall and observe people's habits as they sneeze. (a) What is the probability that among 16 randomly observed individuals exactly 4 do not cover their mouth when sneezing? (b) What is the probability that among 16 randomly observed individuals fewer than 6 do not cover their mouth when sneezing? (c) Would you be surprised if, after observing 16 individuals, fewer than half covered their mouth when sneezing? Why? (a) The probability that exactly 4 individuals do not cover their mouth is 2225 . (Round to four decimal places as needed.) (b) The probability that fewer than 6 individuals do not cover their mouth is .7630 (Round to four decimal places as needed.) (c) Fewer than half of 16 individuals covering their mouth be surprising because the probability of observing fewer than half covering their mouth when sneezing is which an unusual event. (Round to four decimal places as needed.)

Step 1 ：The problem is asking for the probability of a certain number of successes (in this case, people not covering their mouth when sneezing) in a fixed number of Bernoulli trials (in this case, observing 16 people). This is a binomial distribution problem.

Step 2 ：For part (a), we need to find the probability that exactly 4 out of 16 people do not cover their mouth when sneezing. The formula for the probability mass function of a binomial distribution is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability of k successes, \(C(n, k)\) is the number of combinations of n items taken k at a time, p is the probability of success on a single trial, and n is the number of trials.

Step 3 ：For part (b), we need to find the probability that fewer than 6 out of 16 people do not cover their mouth when sneezing. This is the cumulative probability for k=0 to 5.

Step 4 ：For part (c), we need to find the probability that fewer than half of 16 individuals (i.e., fewer than 8) do not cover their mouth when sneezing. This is the cumulative probability for k=0 to 7.

Step 5 ：Given that p = 0.267, n = 16, k = 4, the calculated probabilities are: prob_a = 0.2225, prob_b = 0.7630, prob_c = 0.9605.

Step 6 ：Final Answer: (a) The probability that exactly 4 out of 16 individuals do not cover their mouth when sneezing is approximately \(\boxed{0.2225}\). (b) The probability that fewer than 6 out of 16 individuals do not cover their mouth when sneezing is approximately \(\boxed{0.7630}\). (c) The probability that fewer than half of 16 individuals (i.e., fewer than 8) do not cover their mouth when sneezing is approximately \(\boxed{0.9605}\). This means that it would be quite unusual (with a probability of only about 0.0395) to observe fewer than half of the individuals covering their mouth when sneezing.