Question

a die is rolled 9 times. find the probability of rolling exactly 1 five.
the probability is
(round to four 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 in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

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

Here, \(n = 9\) (number of die - rolls), \(k = 1\) (number of fives), and \(p=\frac{1}{6}\) (probability of rolling a five in a single die - roll), \(1-p = 1-\frac{1}{6}=\frac{5}{6}\)

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

\(C(9,1)=\frac{9!}{1!(9 - 1)!}=\frac{9!}{1!8!}=\frac{9\times8!}{8!}=9\)

Step4: Substitute the values into the binomial formula

\(P(X = 1)=C(9,1)\times(\frac{1}{6})^{1}\times(\frac{5}{6})^{9 - 1}\)
\(=9\times\frac{1}{6}\times(\frac{5}{6})^{8}\)
\(=\frac{9}{6}\times(\frac{5}{6})^{8}\)
\(=\frac{3}{2}\times(\frac{5}{6})^{8}\)
\((\frac{5}{6})^{8}=\frac{5^{8}}{6^{8}}=\frac{390625}{1679616}\approx0.23256\)
\(\frac{3}{2}\times0.23256 = 0.34884\)

Answer:

\(0.3488\)