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

Explanation:

Step1: Identify 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: Define the values of $n$, $k$, and $p$

Given $n = 8$, $k = 3$, and $p=0.45$ (since $45%=0.45$), then $1 - p = 1-0.45 = 0.55$.

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

$C(8,3)=\frac{8!}{3!(8 - 3)!}=\frac{8!}{3!5!}=\frac{8\times7\times6\times5!}{3\times2\times1\times5!}=56$.

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

$P(X = 3)=C(8,3)\times p^{3}\times(1 - p)^{8 - 3}=56\times(0.45)^{3}\times(0.55)^{5}$. $(0.45)^{3}=0.45\times0.45\times0.45 = 0.091125$. $(0.55)^{5}=0.55\times0.55\times0.55\times0.55\times0.55\approx0.050328$. $P(X = 3)=56\times0.091125\times0.050328\approx0.2568$.

Answer:

$0.2568$