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.
$n = 15, p = 0.2, x = 4$
$p(4) = square$
(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 \(P(X = x)=\binom{n}{x}p^{x}(1 - p)^{n - x}\), where \(\binom{n}{x}=\frac{n!}{x!(n - x)!}\), \(n = 15\), \(p=0.2\), \(x = 4\), and \(1-p=0.8\).

Step2: Calculate the combination \(\binom{15}{4}\)

\(\binom{15}{4}=\frac{15!}{4!(15 - 4)!}=\frac{15!}{4!×11!}=\frac{15\times14\times13\times12}{4\times3\times2\times1}=1365\).

Step3: Calculate \(p^{x}\) and \((1 - p)^{n - x}\)

\(p^{x}=(0.2)^{4}=0.0016\), \((1 - p)^{n - x}=(0.8)^{11}\approx0.08589934592\).

Step4: Calculate \(P(X = 4)\)

\(P(X = 4)=\binom{15}{4}\times(0.2)^{4}\times(0.8)^{11}=1365\times0.0016\times0.08589934592\approx0.1876\).

Answer:

\(0.1876\)