Question

a die is rolled 12 times. find the probability of rolling the following. exactly 12 fives. use the formula for binomial probability to calculate the probability of rolling exactly 12 fives. assume success in this case is rolling a five. p(x successes in n trials) = c(n,x)·p^x·(1 - p)^{n - x}=c(12,12)·□^□·(1 - □)^□ (type integers or fractions.)

Explanation:

Step1: Determine the values of \(n\), \(x\), and \(p\)

• \(n = 12\) (number of trials, i.e., die - rolls)
• \(x = 12\) (number of successes, i.e., rolling fives)
• \(p=\frac{1}{6}\) (probability of rolling a five on a single die - roll)

Step2: Substitute the values into the binomial probability formula

The binomial probability formula is \(P(x)=\binom{n}{x}\cdot p^{x}\cdot(1 - p)^{n - x}\)

• \(\binom{n}{x}=\binom{12}{12}=\frac{12!}{12!(12 - 12)!}=1\)
• \(p^{x}=(\frac{1}{6})^{12}\)
• \(1-p = 1-\frac{1}{6}=\frac{5}{6}\) and \(n - x=12-12 = 0\), so \((1 - p)^{n - x}=(\frac{5}{6})^{0}=1\)

Answer:

\(P(12)=\binom{12}{12}\cdot(\frac{1}{6})^{12}\cdot(1-\frac{1}{6})^{0}\)
So the blanks are filled as \(C(12,12)\cdot(\frac{1}{6})^{12}\cdot(1-\frac{1}{6})^{0}\)