Question

according to a study done by nick wilson of otago university wellington, 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 peoples habits as they sneeze. complete parts (a) through (c). (a) what is the probability that among 12 randomly observed individuals, exactly 4 do not cover their mouth when sneezing? using the binomial distribution, the probability is (round to four decimal places as needed.)

Explanation:

Step1: Identify the binomial formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Assign values to the formula

Here, \(n = 12\), \(k = 4\), \(p=0.267\), and \(1-p = 1 - 0.267=0.733\)

First, calculate \(C(12,4)=\frac{12!}{4!(12 - 4)!}=\frac{12!}{4!8!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495\)

Then, \(p^{k}=(0.267)^{4}\approx0.0050\) and \((1 - p)^{n - k}=(0.733)^{8}\approx0.0673\)

Step3: Calculate the probability

\(P(X = 4)=C(12,4)\times(0.267)^{4}\times(0.733)^{8}\)

\(P(X = 4)=495\times0.0050\times0.0673\)

\(P(X = 4)\approx495\times0.0003365\)

\(P(X = 4)\approx0.1666\)

Answer:

\(0.1666\)