Question

find the probability of exactly five successes in seven trials of a binomial experiment in which the probability of success is 70%.
p = ?%

Explanation:

Step1: Recall the 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 \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Here, \(n = 7\), \(k = 5\), \(p=0.7\), and \(1-p = 0.3\)

Step2: Calculate the combination \(C(7,5)\)

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

\(p^{k}=(0.7)^{5}=0.16807\)

\((1 - p)^{n - k}=(0.3)^{2}=0.09\)

Step4: Calculate the probability \(P(X = 5)\)

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Step5: Convert to percentage and round

\(P(X = 5)=0.3176523\times100\% = 31.8\%\) (rounded to the nearest tenth of a percent)

Answer:

\(31.8\%\)