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 = 7, p = 0.7, x = 2

p(2) =
(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

Given \(n = 7\), \(p=0.7\), \(x = 2\). First, calculate \({7\choose 2}=\frac{7!}{2!(7 - 2)!}=\frac{7\times6\times5!}{2\times1\times5!}=21\)

Then, \(p^{x}=(0.7)^{2}=0.49\) and \((1 - p)^{n - x}=(1 - 0.7)^{7 - 2}=(0.3)^{5}=0.00243\)

Step3: Calculate \(P(2)\)

\(P(2)={7\choose 2}(0.7)^{2}(0.3)^{5}=21\times0.49\times0.00243\)
\(P(2)=21\times0.0011907 = 0.0250047\)

Answer:

\(0.0250\)