Rolling a 4 occurs with [tex]\dfrac16[/tex] probability. You're rolling the die three times, so the experiment involves a binomial distribution with [tex]p=\dfrac16[/tex] and [tex]n=3[/tex], which has a probability mass function
[tex]\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}[/tex]
where [tex]\dbinom nx=\dfrac{n!}{x!(n-x)!}[/tex].
19.
[tex]\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[/tex]
20.
[tex]\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[/tex]
