a recent survey found that 82% of first - year college students were involved in volunteer work at least occasionally. suppose a random sample of 11 college students is taken. find the probability that exactly 8 students volunteered at least occasionally.
the probability is
(simplify your answer. round to four decimal places as needed.)

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Accepted Answer

# 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 in a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Determine the values of $n$, $k$, and $p$
Here, $n = 11$, $k = 8$, $p=0.82$, and $1 - p = 0.18$

## Step3: Calculate the combination $C(11,8)$
$$
\begin{align*}
C(11,8)&=\frac{11!}{8!(11 - 8)!}\
&=\frac{11!}{8!3!}\
&=\frac{11\times10\times9}{3\times2\times1}\
& = 165
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.82)^{8}\approx0.2004$
$(1 - p)^{n - k}=(0.18)^{3}\approx0.0058$

## Step5: Calculate the probability $P(X = 8)$
$$
\begin{align*}
P(X = 8)&=C(11,8)\times p^{8}\times(1 - p)^{3}\
&=165\times0.2004\times0.0058\
&=165\times0.00116232\
&\approx0.1918
\end{align*}
$$

# Answer:
$0.1918$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial probability formula

Step2: Determine the values of \(n\), \(k\), and \(p\)

Step3: Calculate the combination \(C(11,8)\)

Step4: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Step5: Calculate the probability \(P(X = 8)\)

Answer: