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)•(\frac{1}{6})^{12}•(1 - \frac{1}{6})^0
(type integers or fractions.)
the probability is □
(use scientific notation. round to three decimal places as needed.)

Explanation:

Step1: Calculate combination

The combination formula \(C(n,x)=\frac{n!}{x!(n - x)!}\). Here \(n = 12\), \(x=12\), so \(C(12,12)=\frac{12!}{12!(12 - 12)!}=1\)

Step2: Calculate \(p^{x}\)

Given \(p=\frac{1}{6}\), \(x = 12\), so \(p^{x}=(\frac{1}{6})^{12}\)

Step3: Calculate \((1 - p)^{n - x}\)

Given \(p=\frac{1}{6}\), \(n = 12\), \(x = 12\), then \(1-p=\frac{5}{6}\), \(n - x=0\), so \((1 - p)^{n - x}=(\frac{5}{6})^{0}=1\)

Step4: Calculate the probability

\(P(x = 12)=C(12,12)\times(\frac{1}{6})^{12}\times(\frac{5}{6})^{0}=1\times(\frac{1}{6})^{12}\times1\)
\(P(x = 12)=\frac{1}{6^{12}}=\frac{1}{2176782336}\approx4.59\times10^{-10}\)

Answer:

\(4.59\times 10^{-10}\)