find the probability of exactly 4 successes in 5 trials of a binomial experiment in which the probability of…

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 on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Identify the values of $n$, $k$, and $p$

Given that $n = 5$, $k = 4$, and $p=0.6$. Then $1 - p = 1-0.6 = 0.4$.

Step3: Calculate the combination $C(n,k)$

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

Step4: Calculate the probability $P(X = 4)$

$P(X = 4)=C(5,4)\times p^{4}\times(1 - p)^{5 - 4}=5\times(0.6)^{4}\times0.4$. $(0.6)^{4}=0.6\times0.6\times0.6\times0.6 = 0.1296$. $5\times0.1296\times0.4=5\times0.05184 = 0.2592$.

Step5: Convert to percentage

To convert the decimal to a percentage, multiply by 100. So $0.2592\times100 = 25.92%$.

Answer:

$25.92$