Rolling a 4 occurs with \(\dfrac16\) probability. You're rolling the die three times, so the experiment involves a binomial distribution with \(p=\dfrac16\) and \(n=3\), which has a probability mass function
\(\mathbb P(X=x)=\begin{cases}\dbinom3x\left(\dfrac16\right)^x\left(\dfrac56\right)^{3-x}&\text{for }x\in\{0,1,2,3\}\\\\0&\text{otherwise}\end{cases}\)
where \(\dbinom nx=\dfrac{n!}{x!(n-x)!}\).
19.
\(\mathbb P(X=0)=\dbinom30\left(\dfrac16\right)^0\left(\dfrac56\right)^{3-0}=1\times1\times\left(\dfrac56\right)^3=\dfrac{125}{216}\approx0.58\)
20.
\(\mathbb P(X=1)=\dbinom31\left(\dfrac16\right)^1\left(\dfrac56\right)^{3-1}=3\times\dfrac16\times\left(\dfrac56\right)^2=\dfrac{25}{72}\approx0.35\)
