Question

a binomial probability experiment is conducted with the given parameters. compute the probability of x successes in the n independent trials of the experiment.
n = 15, p = 0.65, x = 13
p(13) =
(do not round until the final answer. then round to four decimal places as needed.)

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(x)={n\choose x}p^{x}(1 - p)^{n - x}\), where \({n\choose x}=\frac{n!}{x!(n - x)!}\)

Step2: Substitute values into formula

Here, \(n = 15\), \(p=0.65\), \(x = 13\), and \(1-p=0.35\)

First, calculate \({15\choose13}=\frac{15!}{13!(15 - 13)!}=\frac{15!}{13!2!}=\frac{15\times14}{2\times1}=105\)

Then, \(p^{x}=(0.65)^{13}\approx0.000992\) and \((1 - p)^{n - x}=(0.35)^{2}=0.1225\)

Step3: Compute \(P(13)\)

\(P(13)={15\choose13}(0.65)^{13}(0.35)^{2}\)

\(P(13)=105\times0.000992\times0.1225\)

\(P(13)=105\times0.00012152\)

\(P(13)=0.0127596\)

Answer:

\(0.0128\)