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

Answer:

$7.3%$

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 $n = 5$, $k = 3$, and $p=0.9$. Then $1 - p = 1-0.9 = 0.1$.

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

$C(5,3)=\frac{5!}{3!(5 - 3)!}=\frac{5!}{3!2!}=\frac{5\times4\times3!}{3!\times2\times1}=\frac{5\times4}{2\times1}=10$.

Step4: Calculate the probability $P(X = 3)$

$P(X = 3)=C(5,3)\times p^{3}\times(1 - p)^{5 - 3}=10\times(0.9)^{3}\times(0.1)^{2}$. $(0.9)^{3}=0.9\times0.9\times0.9 = 0.729$ and $(0.1)^{2}=0.01$. So $P(X = 3)=10\times0.729\times0.01=0.0729$.

Step5: Convert to percentage and round

To convert to a percentage, multiply by 100: $0.0729\times100 = 7.29%$. Rounding to the nearest tenth of a percent gives $7.3%$.