find the probability of exactly 6 successes in 7 trials of a binomial experiment in which the probability of…

Explanation:

Step1: Identify 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)!})

Step2: Substitute the values into the formula

Here, (n = 7), (k = 6), (p=0.8), and (1 - p = 0.2)

First, calculate (C(7,6)): (C(7,6)=\frac{7!}{6!(7 - 6)!}=\frac{7!}{6!1!}=\frac{7\times6!}{6!×1}=7)

Then, (p^{k}=(0.8)^{6}) and ((1 - p)^{n - k}=(0.2)^{1})

(P(X = 6)=C(7,6)\times(0.8)^{6}\times(0.2)^{1})

(P(X = 6)=7\times0.262144\times0.2)

(P(X = 6)=7\times0.0524288)

(P(X = 6)=0.3670016)

Step3: Convert to percentage and round

To convert to a percentage, multiply by (100): (0.3670016\times100 = 36.70016%)

Rounding to the nearest tenth of a percent gives (36.7%)

Answer:

(36.7%)