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 (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, the 'success' is an individual not covering their mouth when sneezing) in a fixed number of Bernoulli trials (in this case, observing 16 individuals). This is a binomial distribution problem.

Step 2 ：For part (a), we need to calculate the probability of exactly 4 successes in 16 trials. 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 ：Given that p = 0.267, n = 16, and k = 4, we can calculate \(P_4 = 0.22251747805550204\).

Step 4 ：For part (b), we need to calculate the probability of fewer than 6 successes in 16 trials. This is the sum of the probabilities of 0, 1, 2, 3, 4, and 5 successes.

Step 5 ：Calculating this, we find that \(P_{less_6} = 0.7630050627192151\).

Step 6 ：Final Answer: (a) The probability that exactly 4 individuals do not cover their mouth is \(\boxed{0.2225}\). (b) The probability that fewer than 6 individuals do not cover their mouth is \(\boxed{0.7630}\).