FOB

So as I said, we're rolling six stats for D&D, where each stat is determined by rolling four six-sided dice and taking the top three…

The probability of getting at least one 18 in six rolls is computed in the usual way. Call the probability of getting 18 in a single roll p. The probability that a given roll is not an 18 is (1 - p). To not roll an 18, all six rolls must independently be not 18s, so we can multiply their probabilities to get the probability (1 - p)**6 that no 18 was rolled. We want the probability that 18 was rolled, so we take the complement 1 - (1 - p)**6 .

Computing p is also straightforward. Consider an ordered roll of four six-sided dice. There are thus 6**4 = 1296 possible rolls. Of these, there is exactly one in which all four dice are sixes, which will give a roll where the top three are 18. The only other way this can happen is if three of the four dice are sixes, and the fourth is less than six. There are five ways a die can be less than six, and there are four different dice this could happen to, so there are 4×5 = 20 possible rolls with this property. Thus, the overall probability p that a given roll is an 18 is (1 + 20)/1296 = 7/432, which is very close to 1/62.

We thus have an overall probability of about 1 - (1 - 1/62)**6 = 1 -(61/62)**6 that a given character will have an 18 stat. But this isn't too convenient to calculate with pencil, paper, and four-function calculator. So we back up a step and note that (61/62)**6 can be approximated by (1 - 6/62 - 15/62**2-…) using the binomial theorem. The displayed terms of this expression are straightforward, and could be done pencil-and-paper: a calculator gives 0.8993, very close to the value of 0.9070 given by the explicit exponential.

There's our answer: about 10% of D&D characters will have an 18 stat. To check this, I hand-rolled 15 characters, and got two with 18 stats. Then I computer-rolled 100000 characters, and got 9425/100000 with about two significant figures of repeatability. The true answer using Nickle as an arbitrary-precision full-function calculator without the approximations done above is (1-(1-21/1296)**6) = 606876045137999/6499837226778624, or about 9.337%. Thus, experiment and theory seem to agree, and we go away happy.