Step 1 :Given that the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. We are asked to find the probability that among 16 randomly observed individuals exactly 4 do not cover their mouth when sneezing.
Step 2 :This is a binomial distribution problem. The probability of success (not covering mouth when sneezing) is given as 0.267. We are asked to find the probability of exactly 4 successes in 16 trials.
Step 3 :The formula for 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 in n trials, \(C(n, k)\) is the combination of n items taken k at a time, p is the probability of success, and n is the number of trials.
Step 4 :We can plug in the given values into this formula to find the answer. Where p = 0.267, n = 16, and k = 4.
Step 5 :Using the combination formula, we find that \(C(n, k)\) equals 1820.
Step 6 :Substituting these values into the binomial distribution formula, we find that the probability is approximately 0.22251747805550204.
Step 7 :Final Answer: The probability that among 16 randomly observed individuals exactly 4 do not cover their mouth when sneezing is \(\boxed{0.2225}\).