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 10 randomly observed individuals, exactly 4 do not cover their mouth when sneezing?
using the binomial distribution, the probability is 0.1655
(round to four decimal places as needed.)
(b) what is the probability that among 10 randomly observed individuals, fewer than 6 do not cover their mouth when sneezing?
using the binomial distribution, the probability is □
(round to four decimal places as needed.)

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 10\), \(p=0.267\), and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Calculate \(P(X<6)\)

\(P(X<6)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)+P(X = 5)\)

For \(k = 0\):
\(C(10,0)=\frac{10!}{0!(10-0)!}=1\)
\(P(X = 0)=C(10,0)\times(0.267)^{0}\times(1 - 0.267)^{10-0}=1\times1\times(0.733)^{10}\approx0.0317\)

For \(k = 1\):
\(C(10,1)=\frac{10!}{1!(10 - 1)!}=10\)
\(P(X = 1)=C(10,1)\times(0.267)^{1}\times(0.733)^{9}=10\times0.267\times(0.733)^{9}\approx0.1213\)

For \(k = 2\):
\(C(10,2)=\frac{10!}{2!(10-2)!}=45\)
\(P(X = 2)=C(10,2)\times(0.267)^{2}\times(0.733)^{8}=45\times0.0713\times(0.733)^{8}\approx0.2293\)

For \(k = 3\):
\(C(10,3)=\frac{10!}{3!(10 - 3)!}=120\)
\(P(X = 3)=C(10,3)\times(0.267)^{3}\times(0.733)^{7}=120\times0.0190\times(0.733)^{7}\approx0.2750\)

For \(k = 4\):
\(C(10,4)=\frac{10!}{4!(10-4)!}=210\)
\(P(X = 4)=C(10,4)\times(0.267)^{4}\times(0.733)^{6}=210\times0.0051\times(0.733)^{6}\approx0.2140\)

For \(k = 5\):
\(C(10,5)=\frac{10!}{5!(10 - 5)!}=252\)
\(P(X = 5)=C(10,5)\times(0.267)^{5}\times(0.733)^{5}=252\times0.0014\times(0.733)^{5}\approx0.1040\)

Step3: Sum up the probabilities

\(P(X<6)=0.0317 + 0.1213+0.2293+0.2750+0.2140+0.1040\approx0.9753\)

Answer:

\(0.9753\)