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 0.250. 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 10 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, \(n = 10\), \(k = 4\), \(p=0.25\), and \(1-p = 0.75\)

First, calculate \(C(10,4)=\frac{10!}{4!(10 - 4)!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210\)

Then, \(p^{k}=(0.25)^{4}=0.00390625\)

And \((1 - p)^{n - k}=(0.75)^{6}\approx0.1779785\)

Step3: Calculate the probability

\(P(X = 4)=C(10,4)\times p^{4}\times(1 - p)^{6}\)

\(P(X = 4)=210\times0.00390625\times0.1779785\)

\(P(X = 4)\approx0.1459\)

Answer:

\(0.1459\)