Problem

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 (Round to four decimal places as needed.)

Answer

Expert–verified
Hide Steps
Answer

Final Answer: The probability that among 16 randomly observed individuals exactly 4 do not cover their mouth when sneezing is \(\boxed{0.2225}\).

Steps

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 \(\boxed{0.2225}\).

link_gpt