Can someone help me with this Algebra 2 question?

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 not divisible by 2 and not divisible by 3 . 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 times

 

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 outcomes whose sums are not divisible by 2 or 3. That includes any outcomes where the sum is 5, 7, or 11:

 

4 (sum of 5) + 6 (sum of 7) + 2 (sum of 11) = 12 times

 

So, 12 out of the 36 outcomes are sums that are not divisible by 2 or 3. That means the probability is 12/36 = 1/3.