Please someone help me with this Algebra 2 problem?

Question

An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment. Compute the probability of each of the following events: Event A : The sum is greater than 5 . Event B : The sum is an odd number. Write your answers as exact fractions.

Answer 1

First, you'll want to find the sample space of the experiment. That is, you want to find all the possible sums after two rolls. You could roll 1 and then 1 for a sum of 2, 1 and then 2 for a sum of 3, etc. It's easiest to make a list:

(1,1) = 2
(1,2) = 3
(1,3) = 4
(1,4) = 5
(1,5) = 6
(1,6) = 7
(2,1) = 3
(2,2) = 4
(2,3) = 5
(2,4) = 6
(2,5) = 7
(2,6) = 8
(3,1) = 4
(3,2) = 5
(3,3) = 6
(3,4) = 7
(3,5) = 8
(3,6) = 9
(4,1) = 5
(4,2) = 6
(4,3) = 7
(4,4) = 8
(4,5) = 9
(4,6) = 10
(5,1) = 6
(5,2) = 7
(5,3) = 8
(5,4) = 9
(5,5) = 10
(5,6) = 11
(6,1) = 7
(6,2) = 8
(6,3) = 9
(6,4) = 10
(6,5) = 11
(6,6) = 12

There are a total of 6×6 = 36 possible outcomes:

2 appears 1 time
3 appears 2 times
4 appears 3 times
5 appears 4 times
6 appears 5 times
7 appears 6 times
8 appears 5 times
9 appears 4 times
10 appears 3 times
11 appears 2 times
12 appears 1 time


Now that we know the sample space, we can answer your questions. How many of those 36 outcomes are greater than 5?

5 (number of outcomes with a sum of 6) + 6 (number of outcomes with a sum of 7) + 5 (sum of 8) + 4 (sum of 9) + 3 (sum of 10) + 2 (sum of 11) + 1 (sum of 12) = 26 outcomes

So, 26 out of the 36 outcomes are sums greater than 5. That means the probability is 26/36 = 13/18.

Now we'll do the same thing for odd sums, which includes sums of 3, 5, 7, 9, and 11:

2 (sum of 3) + 4 (sum of 5) + 6 (sum of 7) + 4 (sum of 9) + 2 (sum of 11) = 18 outcomes

So, 18 out of the 36 outcomes are odd sums. That means the probability is 18/36 = 1/2.