Question

find the probability of no successes in six trials of a binomial experiment in which the probability of success is 30%.
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 \(C(n,k)=\frac{n!}{k!(n - k)!}\). Here, \(n = 6\), \(k = 0\), \(p=0.3\), and \(1-p = 0.7\).

Step2: Calculate \(C(n,k)\)

When \(k = 0\) and \(n = 6\), \(C(6,0)=\frac{6!}{0!(6-0)!}=\frac{6!}{6!×1}=1\).

Step3: Substitute values into formula

\(P(X = 0)=C(6,0)\times(0.3)^{0}\times(0.7)^{6}\). Since \((0.3)^{0}=1\), then \(P(X = 0)=1\times1\times(0.7)^{6}\).
\((0.7)^{6}=0.7\times0.7\times0.7\times0.7\times0.7\times0.7 = 0.117649\).

Step4: Convert to percentage

Multiply by \(100\) to get the percentage: \(0.117649\times100 = 11.7649\%\).

Step5: Round to the nearest tenth

Rounding \(11.7649\%\) to the nearest tenth gives \(11.8\%\).

Answer:

\(11.8\)