Question

a die is rolled 8 times. find the probability of rolling the following. exactly 8 twos. the probability is (use scientific notation. round to the nearest whole number as needed.)

Explanation:

Step1: Identify the probability of rolling a two in a single roll

The probability of rolling a two on a fair six - sided die is \(p=\frac{1}{6}\). The probability of not rolling a two is \(q = 1-\frac{1}{6}=\frac{5}{6}\).

Step2: Use the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times q^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successful trials, \(p\) is the probability of success in a single trial, \(q\) is the probability of failure in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\).
Here, \(n = 8\), \(k = 8\), \(p=\frac{1}{6}\), and \(q=\frac{5}{6}\).
First, calculate \(C(8,8)=\frac{8!}{8!(8 - 8)!}=\frac{8!}{8!0!}=1\) (since \(0!=1\)).
Then, \(p^{k}=(\frac{1}{6})^{8}\) and \(q^{n - k}=(\frac{5}{6})^{0}=1\).
So, \(P(X = 8)=C(8,8)\times(\frac{1}{6})^{8}\times(\frac{5}{6})^{0}\).
Substitute the values: \(P(X = 8)=1\times(\frac{1}{6})^{8}\times1\).
\((\frac{1}{6})^{8}=\frac{1}{6^{8}}=\frac{1}{1679616}\approx6\times10^{-7}\)

Answer:

\(6\times 10^{-7}\)