Question

the senators have won 75% of the games played against the canadiens over the past few seasons. what is the probability that the canadiens will win exactly 4 of the 8 games they play against the senators this season?
0.069
0.25
0.087
0.011

Explanation:

Step1: Determine the probability of Senators winning and Canadiens winning

The Senators have a \(75\% = 0.75\) chance of winning a game. So the Canadiens have a \(p=1 - 0.75=0.25\) chance of winning a game. The number of trials \(n = 8\), and the number of successes \(k = 4\).

Step2: Use the binomial probability formula

The binomial probability formula is \(P(X=k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\).
First, calculate \(\binom{8}{4}=\frac{8!}{4!(8 - 4)!}=\frac{8\times7\times6\times5}{4\times3\times2\times1}=70\).
Then, \(p = 0.25\), \(n = 8\), \(k = 4\), and \(1-p=0.75\).
\(P(X = 4)=\binom{8}{4}(0.25)^{4}(0.75)^{8 - 4}\)
\(P(X = 4)=70\times(0.25)^{4}\times(0.75)^{4}\)
\(P(X = 4)=70\times\frac{1}{256}\times\frac{81}{256}\)
\(P(X = 4)=70\times\frac{81}{65536}\approx0.087\)

Answer:

\(0.087\)