Probability sum of two dice is less than 10?

[I can't remember]

Two standard dice are rolled. What is the probability of the sum on the two dice being less than 10?

I thought since there are 6 ways each dice could potentially end up, that I should multiply them to get 36 possible outcomes. Then I said well it would be easier to find the probability of rolling a 10 or higher. so I wrote out all the possabilities of combinations of 10 or higher.

4,6
6,4
5,5
5,5
5,6
6,5
6,6
6,6

That totals 8 combination out of 36 that could be ten or higher, so 8/36= 2/9. since I wanted less than ten 1-(2/9) = 7/9 probability of getting less than 10.

my question is did I do this right, and if so, was there an easier way?

 

[Loren]

Your idea is okay. The problem is that you list 5,5 twice and 6,6 twice. They should only be listed once. One way to see this is to draw a 6 by 6 grid with 1,2,3,4,5,6 down one side and 1,2,3,4,5,6 across the top. Then in the body of the grid in each cell enter the sum of the numbers that appear one at the top, and the other at the side. You will see that there are two cells containing a 10 that are produced, one by 4 and 6, and the other by 6 and 4. But there is only one cell containing a 10 that is produced by 5 and 5. Likewise, there is only one way to get a sum of 12.