Question

Two dice are thrown sumiltaneously. Find the probability that the sum of the numbers on the faces is not divisible by $4$ or divisible by $5$

Answer 1

The sample space $S$ when two dice are thrown together is:

$(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)$.

Therefore, $n\left(S\right)=36$.

Let $A$ denote the event that sum of number is not divisible by $4$ or $5$ that is:

$(1,1),(1,2),(1,5),(1,6),(2,1),(2,4),(2,5),(3,3),(3,4),(3,6),(4,2),(4,3),(4,5),(5,1),(5,2),(5,4),(5,6),(6,1),(6,3),(6,5)$

$n\left(A\right)=20$, therefore, probability that sum of number is not divisible by $4$ or $5$ is:

$P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{20}{36}=\frac{5}{9}$

Hence, probability that sum of number on the faces is neither divisible by $4$ nor by $5$ is $\frac{5}{9}$.