Question

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.)

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 \(p\)

Here, \(n = 12\) (number of games), \(k = 8\) (number of wins), \(p=0.3694\) (probability of winning a single game), and \(1-p = 1 - 0.3694=0.6306\)

Step3: Calculate the combination \(C(12,8)\)

$$
LATEXBLOCK0
$$

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

\(p^{k}=(0.3694)^{8}\approx0.0003\)
\((1 - p)^{n - k}=(0.6306)^{4}\approx0.1577\)

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

$$
LATEXBLOCK1
$$

Answer:

\(0.0237\)