Question

For the event of rolling three standard fair 6-sided dice, find the following probabilities. Give all answers as a simplified fraction. The probability of rolling a sum of 8 or a sum of 12. P(Sum is 8 È Sum is 12) =

Answer 1

One way we could figure this out is by finding all the combinations of 3 dice rolls that add up to 8 and dividing that by the total number of possible outcomes
Likewise, we can find all the combinations of 3 rolls that add up to 12 and divide that number by the total number of possible outcomes

Answer 2

First of all, the number of possible outcomes for 3 rolls of 6-sided dice is 6 * 6 * 6 = 196
This is true because the first die can be any of 6 possibilities. Then, for each of those 6 possibilities, the second die can be 6 possibilities. That gives us 6*6 or 36 possibilities for the first 2 dice. Then, for each of those 36 possibilities, the third die can be 6 possibilities, giving us a total of 36*6 or 196 possibilities

Answer 3

So first 8: the combinations of 3 rolls that could add up to 8 are:
1 1 6
1 2 5
1 3 4
1 4 3
1 5 2
1 6 1
2 1 5
2 2 4
2 3 3
2 4 2
2 5 1
3 1 4
3 2 3
3 3 2
3 4 1
4 1 3
4 2 2
4 3 1
5 1 2
5 2 1
6 1 1

Answer 4

That's 21 ways to get 8 from 3 dice rolls (I hope I didn't miss any)
So, the chances of rolling 8 should be 21/196

Answer 5

I'm starting to think this probably isn't the easiest way to do this problem, but oh well, here goes for 12:

Answer 6

1 5 6
1 6 5
2 4 6
2 5 5
2 6 4
3 3 6
3 4 5
3 5 4
3 6 3
4 2 6
4 3 5
4 4 4
4 5 3
4 6 2
5 1 6
5 2 5
5 3 4
5 4 3
5 5 2
5 6 1
6 1 5
6 2 4
6 3 3
6 4 2
6 5 1
That's 25 ways to roll 12 with 3 dice rolls, so it should be 25/196 chance to roll 12 (assuming I didn't mess up)

Answer 7

Again, my intuition tells me there's definitely easier ways to solve these kinds of problems. It might be good to do some research about that if you run into similar problems in the future. Anyway, I hope that's good enough for now!