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 people's habits as they sneeze.

Complete parts $\left(a\right)$ through $\left(c\right)$
$\left(a\right)$ What is the probability that among $12$ randomly observed individuals, exactly $7$ do not cover their mouth when sneezing?
Using the binomial distribution, the probability is (Round to four decimal places as needed.)

Answer 1

Probability that among $12$ randomly observed individuals, exactly $7$ do not cover their mouth when sneezing:

It is given that,

Sample size, $n=12$

Probability of a randomly selected individual will not cover his or her mouth when sneezing is, $P=0.267$.

The probability of a mass function of a binomial distribution is:

$P(X=x)={}^{n}C_x{p}^x{\left(1-p\right)}^{n-x},\mathrm{for}x=0,1,2,...,n$.

Now, the probability that among $12$ randomly observed individuals, exactly $7$ do not cover their mouth when sneezing is :

$$\begin{gathered} P(X=7)={}^{12}C_7\times {\left(0.267\right)}^7{\left(1-0.267\right)}^{12-7} \\ =792\times 0.000096\times 0.2116 \\ =0.016 \end{gathered}$$

Hence, the probability that among $12$ randomly observed individuals, exactly $7$ do not cover their mouth when sneezing is $0.016$.