system
Given the following post, respond as if you're commenting on Reddit:
human
What is the probability that a rolled d10 has a higher role than two d20's?

 I'm in need of a good statistics mathematician.

I was rolling some dice playing DnD and was wondering how to figure out this question about probabilities.

If I roll 2 D20's and 1 D10 what is the likely hood that the D10 roll is a higher number than both the D20's (indivudally, not combined).

For example, one d20 is a 5, the other d20 is a 6 and the d10 is a 7.
gpt
Everyone is doing coding simulations that obfuscate the reasoning when this is really just a pen and paper probability/math problem.

Imagine you roll a 1 on D10, what are the odds that's higher than both D20?  It's 0.  What about a 2?  That's only higher if you roll two 1s D20, so 1/20 x 1/20, or 1/400.  If you roll a 3 D10 then the D20s can be 1 or 2, so the probability is 2/20 x 2/20, or 4/400.  Continue this way and you find that if you roll an n, the odds it beats both D20s is (n-1)^2 /400.  So we just sum this from 1 to 10, and divide the total answer by 10 because each D10 roll has a 1/10 chance.

Re-indexing, this is the same as summing n^2 /4000 from 0 to 9.  The sum of n^2 to k equals k(k+1)(2k+1)/6, so for k=9 we get 285/4000, or 7.125%.  This procedure generalised well, the 4000 is from 20x20x10 and the 285 is the k(k+1)(2k+1)/6 formula for k=10-1 (because you didn't count ties we didn't sum to 10).  If you use a D8 and 2 D12s, the probability is 140/1152 or ~13%