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Problem 93: A fair die is rolled eight times. Calculate the probability that there are:
a. Exactly five even numbers
b. Exactly one 6
c. No 4s

Problem 93: A fair die is rolled eight times. Calculate the probability that there are:
a. Exactly five even numbers
b. Exactly one 6
c. No 4s

Solution:


a. Exactly five even numbers
Given that $x = 5, n=8$
$p(even\; numbers) = \frac{3}{6} = \frac{1}{2} = 0.5$
$P(x= 5) = b(5;8,0.5)$
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$P(x= 5) = b(5;8,0.5) = \left(\begin{array}{c}8\\ 5\end{array}\right){0.5}^{5}(1-0.5)^{8-5} = 0.21875$
b. Exactly one 6
Given that $x = 5, n=8$
$p(6) = \frac{1}{6}$ $P(x= 1) = b(1;8,\frac{1}{6})$
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$P(x= 1) = b(1;8,\frac{1}{6}) = \left(\begin{array}{c}8\\ 1\end{array}\right){\frac{1}{6}}^{1}(1-\frac{1}{6})^{8-1} = 0.372109$
c. No 4s
Given that $x = 5, n=8$
$p(4) = \frac{1}{6}$ $P(x= 0) = b(0;8,\frac{1}{6})$
$b(x;n,p)=\left(\begin{array}{c}n\\ x\end{array}\right){p}^{x}(1-p)^{n-x}$
$P(x= 0) = b(0;8,\frac{1}{6}) = \left(\begin{array}{c}8\\ 0\end{array}\right){\frac{1}{6}}^{0}(1-\frac{1}{6})^{8-0} = 0.232568$

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