Question

based on a survey, 37% of likely voters would be willing to vote by internet instead of the in - person traditional method of voting. for each of the following, assume that 15 likely voters are randomly selected. a. what is the probability that exactly 12 of those selected would do internet voting? (round to five decimal places as needed.)

Explanation:

Step1: Identify the 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, $n = 15$, $k = 12$, and $p=0.37$.

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

$C(15,12)=\frac{15!}{12!(15 - 12)!}=\frac{15!}{12!3!}=\frac{15\times14\times13}{3\times2\times1}=455$.

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

$p = 0.37$, so $p^{k}=(0.37)^{12}$. And $1 - p=1 - 0.37 = 0.63$, so $(1 - p)^{n - k}=(0.63)^{3}$.
$(0.37)^{12}\approx1.3743\times10^{-6}$ and $(0.63)^{3}=0.63\times0.63\times0.63 = 0.250047$.

Step4: Calculate the probability $P(X = 12)$

$P(X = 12)=C(15,12)\times p^{12}\times(1 - p)^{3}=455\times1.3743\times10^{-6}\times0.250047\approx0.00016$.

Answer:

$0.00016$