the probability that the san jose sharks will win any given game is 0.3694 based on a 13 - year win history of 382 wins out of 1034 games played (as of a certain date). an upcoming monthly schedule contains 12 games.
what is the probability that the san jose sharks win 8 games in that upcoming month? let ( x = ) number of games won in that upcoming month. (round your answer to four decimal places.)
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# 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 $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Determine the values of $n$, $k$, and $p$
Given $n = 12$ (number of games), $k = 8$ (number of wins), $p=0.3694$ (probability of winning a game)

## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(12,8)&=\frac{12!}{8!(12 - 8)!}\
&=\frac{12!}{8!4!}\
&=\frac{12\times11\times10\times9}{4\times3\times2\times1}\
& = 495
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$$
\begin{align*}
p^{k}&=(0.3694)^{8}\
&\approx0.000307\
(1 - p)^{n - k}&=(1 - 0.3694)^{4}\
&=(0.6306)^{4}\
&\approx0.1577
\end{align*}
$$

## Step5: Calculate the probability $P(X = 8)$
$$
\begin{align*}
P(X = 8)&=C(12,8)\times p^{8}\times(1 - p)^{4}\
&=495\times0.000307\times0.1577\
&\approx0.0241
\end{align*}
$$

# Answer:
$0.0241$

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Question

Explanation:

Step1: Identify the binomial probability formula

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

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

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

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

Answer: