College Algebra, Chapter 10, Review Exercises, Section Review Exercises, Problem 40

A fair die is rolled eight times. Find the probability of each event.

Recall that the binomial probability is represented by the equation

$c(n,r) (p)^r (q)^{n-r}$

a.) A six occurs four times.

The probability that a dies shows a six or the probability of success $p$ is $\displaystyle \frac{1}{6}$. On the other hand, the probability that a die do not show a $6$ or the probability of failure is $\displaystyle q=1-p = \frac{5}{6}$. Thus, the probability in this case is

$\displaystyle = C(8,4) \left( \frac{1}{6} \right)^4 \left( \frac{5}{6} \right)^{8-4}$

$= 0.026$

b.) An even number occurs two or more times.

The probability that a die shows an even number or the probability of success $p$ is $\displaystyle \frac{3}{6} = \frac{1}{2}$. On the other hand, the probability that a die do not show an even number is $\displaystyle q=1-p = \frac{1}{2}$. To solve this in a faster way, we can apply the compliment to the probability that even number occurs once or never. In this case, we have


$
\begin{equation}
\begin{aligned}

=& 1 - \left[ C(8,11) \left( \frac{1}{2} \right)^1 \left( \frac{1}{2} \right)^{8-1} + C(8,0) \left( \frac{1}{2} \right)^0 \left( \frac{1}{2} \right)^{8-0} \right]
\\
\\
=& 1 - [0.3721 + 0.1667]
\\
\\
=& 0.4612

\end{aligned}
\end{equation}
$