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I want to solve this dice problem but I'm not sure if my solution is the most efficient way. If I roll one dice and then one more and then add those 2 numbers together. What is the probability that the sum is $\le 5$ or that the sum is $\ge 11$ ?

I basically wrote out all possible occasions and checked how many of the possibilities were less than or equal to 5 or greater than or equal to 11. 10 possibiliteis has a sum of 5 or less and 3 had a sum of 11 or more. And this is out of $6 \times 6 = 36$ total possibilities. So $\frac {10}{36} + \frac {3}{36} = \frac {13}{36}$

Is there a better way of solving this?

Bioelli
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    If you imagine the $6 \times 6$ grid with the squares in it labeled with the sum, you'll notice a pattern that the diagonals one way all have the same number, and that it comes in "layers" that are increasing. It might help to draw it out to see for yourself.

    Then the places where the sum is $\le 5$ and $\ge 11$ form triangles, which can be quickly computed knowing the formula for triangular numbers, $\frac{54}{2}=10$ and $\frac{32}{2}=3$ for those sections respectively.

     – Merosity Oct 06 '22 at 00:30

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