Question

find the probability of exactly four successes in five trials of a binomial experiment in which the probability of success is 40%.
p = ?%
round to the nearest tenth of a percent.

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Identify values of \(n,k,p\)

Here, \(n = 5\), \(k = 4\), \(p=0.4\), and \(1-p = 0.6\)

Step3: Calculate combination \(C(n,k)\)

\(C(5,4)=\frac{5!}{4!(5 - 4)!}=\frac{5!}{4!1!}=\frac{5\times4!}{4!×1}=5\)

Step4: Substitute into formula

\(P(X = 4)=C(5,4)\times(0.4)^{4}\times(0.6)^{5 - 4}\)
\(P(X = 4)=5\times0.4^{4}\times0.6\)
\(0.4^{4}=0.4\times0.4\times0.4\times0.4 = 0.0256\)
\(P(X = 4)=5\times0.0256\times0.6\)
\(P(X = 4)=5\times0.01536\)
\(P(X = 4)=0.0768\)

Step5: Convert to percentage

To convert to a percentage, multiply by \(100\). So \(P = 0.0768\times100=7.68\%\)

Step6: Round to nearest tenth

Rounding \(7.68\%\) to the nearest tenth gives \(7.7\%\)

Answer:

\(7.7\%\)