Question

I need help with binomial probability applied to rolling dice

on numbers 19 & 20 that my professor assigned for me to do
(Show your work, if necessary)

Answer 1

Rolling a 4 occurs with
`\frac16` probability. You're rolling the die three times, so the experiment involves a binomial distribution with
`p=\frac16` and
`n=3`, which has a probability mass function


`\mathbb P(X=x)=\begin{cases}\dbinom3x\left(\frac16\right)^x\left(\frac56\right)^(3-x)&\text{for }x\in\{0,1,2,3\}\\\\0&\text{otherwise}\end{cases}`

where
`\dbinom nx=(n!)/(x!(n-x)!)`.

19.


`\mathbb P(X=0)=\dbinom30\left(\frac16\right)^0\left(\frac56\right)^(3-0)=1*1*\left(\frac56\right)^3=(125)/(216)\approx0.58`

20.


`\mathbb P(X=1)=\dbinom31\left(\frac16\right)^1\left(\frac56\right)^(3-1)=3*\frac16*\left(\frac56\right)^2=(25)/(72)\approx0.35`