First, we need to find the sample space for two dice. Since each dice has 6 faces, the sample space is
which consist in 36 elements.
Part A.
In this case, we need to find the combinations where the sum of the 2 dice is even. The combinations are:
for instance, in the upper left corner, the sum is 1+1=2, which is an even number and similarly for the other cases.
As we can note, there are 18 possible combinations. Then, the probability is
[tex]P(\text{ even sum)=}\frac{18}{36}=\frac{1}{2}[/tex]
Part B.
In this case, we need to find the possible combinations where the sum of the 2 dices is greater than 7. The possible combinations are:
Since there are 15 possible combinations, the probability is given by
[tex]P(\text{ Sum greater than 7)=}\frac{15}{36}[/tex]
Part C.
This case is the intersection of the two cases from above. The possible combinations are:
Since there are 9 possible combinations, the probability is
[tex]P(\text{Even sum and sum greater than 7)=}\frac{9}{36}=\frac{1}{4}[/tex]
Part D.
We need to find the possible combinations where the sum is even or the sum is greater than 7. Then, this case is the union of case A and B. Then the possible combinations are:
Since there are 18+6=24 combinations, the probability is
[tex]P(\text{Even sum or sum greater than 7)=}\frac{24}{36}=\frac{4}{6}=\frac{2}{3}[/tex]
