Question

Find the binomial probability of rolling 5 dice and getting exactly 4 even numbers. 11.6%

Answer 1

The probability of getting an even number on any one die in a single roll is $\dfrac12$.

For 5 dice, the probability of rolling $x$ even numbers (where $0\le x\le5$) is

$p(x)=\dbinom5x\left(\dfrac12\right)^x\left(1-\dfrac12\right)^{5-x}$
$p(x)=\dbinom5x\left(\dfrac12\right)^5$


You then have

$p(4)=\dbinom54\left(\dfrac12\right)^5=\dfrac{5!}{2^54!(5-4)!}=\dfrac5{2^5}\approx0.156=15.6\%$