The National Center for Health Statistics (NCHS) reports that 70% of U.S. adults aged 65 and over have ever received a pneumococcal vaccination. Suppose that you obtain an independent sample of 20 adults aged 65 and over who visited the emergency room and the pneumococcal vaccination rate applies to the sample.

Determine the probability that exactly 12 members of the sample received a pneumococcal vaccination. The probability of obtaining exactly k successes in n trials in a binomial setting withy probability p of success is each trial is

P(X=k)=(nk)pk(1?p)n?k

where (nk) is the binomial coefficient, also represented as nCk, and is defined as

(nk)=n!k!(n?k)!

Follow the steps to compute P(x=12). First determne the binomial coefficient. Do not round your answer.

20C12=125970

Compute (.7)^12, or the probability that exactly 12 adults have been vaccinated. Round your answer to six decimal places.

(.7)^12=

Compute (.3)8, or the probability that exactly 8 adults have not been vaccinated. Report as a decimal to eight decimal places. Do not round.

(.3)^8=

Compute P(x=12)=(20C12)(.7)^12(.3)^8, or the binomial probability that exactly 12 vaccinated adults are in the random sample of 20 adults. Round your answer to four decimal places.

P(x=12)=

For each adult, there are only two possible outcomes. Either they have received the vaccination, or they have not. The probability of an adult receiving the vaccination is independent from other adults. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

70% of U.S. adults aged 65 and over have ever received a pneumococcal vaccination.

This means that [tex]p = 0.7[/tex]

Sample of 20 adults

This means that [tex]n = 20[/tex]

Determine the probability that exactly 12 members of the sample received a pneumococcal vaccination.

This is [tex]P(X = 12)[/tex].

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 12) = C_{20,12}.(0.7)^{12}.(0.3)^{8} = 0.1144[/tex]

P(x = 12) = 0.1144.