a binomial probability experiment is conducted with the given parameters. compute the probability of x…

a binomial probability experiment is conducted with the given parameters. compute the probability of x successes in the n independent trials of the experiment. n = 40, p = 0.05, x = 2 p(2)= (do not round until the final answer. then round to four decimal places as needed.)

a binomial probability experiment is conducted with the given parameters. compute the probability of x successes in the n independent trials of the experiment. n = 40, p = 0.05, x = 2 p(2)= (do not round until the final answer. then round to four decimal places as needed.)

Answer

Explanation:

Step1: Recall binomial - probability formula

The binomial - probability formula is $P(X = x)=C(n,x)\times p^{x}\times(1 - p)^{n - x}$, where $C(n,x)=\frac{n!}{x!(n - x)!}$, $n$ is the number of trials, $p$ is the probability of success on a single trial, and $x$ is the number of successes.

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

First, calculate $C(40,2)=\frac{40!}{2!(40 - 2)!}=\frac{40!}{2!×38!}=\frac{40\times39\times38!}{2\times1\times38!}=780$.

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

Given $p = 0.05$, then $1-p=0.95$, and $n - x=40 - 2 = 38$. So, $(1 - p)^{n - x}=(0.95)^{38}$.

Step4: Calculate $p^{x}$

Given $p = 0.05$ and $x = 2$, then $p^{x}=(0.05)^{2}=0.0025$.

Step5: Calculate $P(X = 2)$

$P(X = 2)=C(40,2)\times p^{2}\times(1 - p)^{38}=780\times0.0025\times(0.95)^{38}$. $(0.95)^{38}\approx0.14127$. $P(X = 2)=780\times0.0025\times0.14127 = 780\times0.000353175\approx0.2755$.

Answer:

$0.2755$