find the probability of exactly five successes in seven trials of a binomial experiment in which the…

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))

[ \begin{align*} C(7,5)&=\frac{7!}{5!(7 - 5)!}\ &=\frac{7!}{5!2!}\ &=\frac{7\times6\times5!}{5!\times2\times1}\ &=\frac{7\times6}{2\times1}\ &=21 \end{align*} ]

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))

[ \begin{align*} P(X = 5)&=C(7,5)\times p^{5}\times(1 - p)^{2}\ &=21\times0.16807\times0.09\ &=21\times0.0151263\ &=0.3176523 \end{align*} ]

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%)