find the probability of no successes in eight 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 $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 8$, $k = 0$, and $p=0.07$.

Step2: Calculate the combination $C(n,k)$

When $k = 0$ and $n = 8$, $C(8,0)=\frac{8!}{0!(8 - 0)!}=\frac{8!}{8!}=1$.

Step3: Calculate $(1 - p)^{n - k}$

Since $p = 0.07$, then $1-p=1 - 0.07 = 0.93$, and $n - k=8-0 = 8$. So, $(1 - p)^{n - k}=(0.93)^{8}$.

Step4: Calculate the probability $P(X = 0)$

$P(X = 0)=C(8,0)\times(0.07)^{0}\times(0.93)^{8}$. Since $(0.07)^{0}=1$ and $C(8,0)=1$, then $P(X = 0)=(0.93)^{8}\approx0.5596$.

Answer:

$55.96%$