# D&D Probability question: why am I wrong?

In Dungeons and Dragons, the primary die used for success or failure is the D20, or 20 sided die. A 1 is a critical failure, and a 20 is a critical success. When one has advantage, one rolls 2 dice and takes the higher of the 2 rolls. Recently, we had a game where a character rolled 6 critical successes over the course of 18 rolls, some with advantage and some without. The odds of a critical success by itself are 1 in 20 (p1), the odds with advantage are 39 in 400 (p2). What are the odds of 6 over the course of 18 rolls?

I modelled it as 18 choose 6 / pn^6. This yielded 1 in 3500 for p1 and 1 in 62 for p2. I also wrote a simulation, and got significantly different results, but in the ballpark. I rolled a billion simulations and the results converged to about 1/6300 for p1 and 1/220 for p2. I repeated this several times with essentially no change.

Where did I go wrong? I would really like to know why these 2 methods don't yield similar results.

• The formula should be not $^{18}C_6{p_n}^6$, but $^{18}C_6{p_n}^6(1-p_n)^{12}$ Nov 3, 2021 at 21:19
• @Dave: I agree about disagreeing with the close vote. I would say questions like these are squarely on-topic in the probability and dice tags, though they might once in a while cross over into self-study territory. Nov 3, 2021 at 21:21
• 1. You seem to be using the word odds as if it meant the same as probability. They are related but are not synonyms. See the Wikipedia page on odds. 2. It wouldn't be hard to specify that you're rolling a 20 sided die , that rolling a 20 is a critical, and that advantage means taking the higher of two such rolls - then you would not cut out people who did not already know these facts from the pool of potential answerers. Nov 3, 2021 at 21:43
• @Glen_b IMO "the odds are 1 in 20" is acceptable colloquial English. It wouldn't be appropriate in academic writing but is fine and unambiguous for this kind of informal discussion. Nov 3, 2021 at 23:33
• It would leave an unnecessary ambiguity or confusion in many situations. By analogy, while people commonly conflate some combination of force, momentum and kinetic energy in ordinary language, if I did so posting to physics.SE I would expect (and indeed want) to be corrected, since -among other things- that helps for finding answers in the future. Nov 4, 2021 at 0:35