Question

Two fair dice are thrown what is the probability of the sum is less than 10

Answer 1

I’ll assume the two dice are both fair 6 sided dice marked 1 through 6, although I personally sell dice ranging from D4 through D100 and find it just a little… pedestrian… to hear people talk of dice as though there was only one possibility. Still, I suppose the D6 is the default.

The possible results range from 2 to 12.

Of those, 2, 3, 5, 7, and 11 are prime.

Rather than figure the number of ways to get each result out one at a time, I’ll make you a little chart:

6 7 8 9 10 11 12

5 6 7 8 9 10 11

4 5 6 7 8 9 10

3 4 5 6 7 8 9

2 3 4 5 6 7 8

1 2 3 4 5 6 7

1 2 3 4 5 6

Now, stack up the times each total appears

7

6 7 8

5 6 7 8 9

4 5 6 7 8 9 10

3 4 5 6 7 8 9 10 11

2 3 4 5 6 7 8 9 10 11 12

So here we go, below each total, the number of times it appears in 2d6

2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 5 4 3 2 1

We’re looking for the number of times prime numbers are rolled, adding them together:

1+2+4+6+2=15 total times out of 36

15 is 3*5, 36 is 2*2*3*3, so we can cancel a set of 3s to get 5/12 or 41.666…%