D20 with Advantage

Dungeons & Dragons players spend a lot of time rolling 20-sided polyhedral dice, known as D20s.

In general, they’re looking to roll as high as possible to successfully stab a wyvern, jump a chasm, pick a lock, charm a Duke¹, or whatever.

Roll with advantage

Sometimes, a player gets to roll with advantage. In this case, the player rolls two dice, and takes the higher roll. This really boosts their chances of not-getting a low roll. Do you know by how much?

I dreamed about this very question last night. And then, still in my dream, I came up with the answer². I woke up thinking about it³ and checked my working.

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	2	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	3	3	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
4	4	4	4	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
5	5	5	5	5	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
6	6	6	6	6	6	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
7	7	7	7	7	7	7	7	8	9	10	11	12	13	14	15	16	17	18	19	20
8	8	8	8	8	8	8	8	8	9	10	11	12	13	14	15	16	17	18	19	20
9	9	9	9	9	9	9	9	9	9	10	11	12	13	14	15	16	17	18	19	20
10	10	10	10	10	10	10	10	10	10	10	11	12	13	14	15	16	17	18	19	20
11	11	11	11	11	11	11	11	11	11	11	11	12	13	14	15	16	17	18	19	20
12	12	12	12	12	12	12	12	12	12	12	12	12	13	14	15	16	17	18	19	20
13	13	13	13	13	13	13	13	13	13	13	13	13	13	14	15	16	17	18	19	20
14	14	14	14	14	14	14	14	14	14	14	14	14	14	14	15	16	17	18	19	20
15	15	15	15	15	15	15	15	15	15	15	15	15	15	15	15	16	17	18	19	20
16	16	16	16	16	16	16	16	16	16	16	16	16	16	16	16	16	17	18	19	20
17	17	17	17	17	17	17	17	17	17	17	17	17	17	17	17	17	17	18	19	20
18	18	18	18	18	18	18	18	18	18	18	18	18	18	18	18	18	18	18	19	20
19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	19	20
20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20

The chance of getting a “natural 1” result on a D20 is 1 in 20… but when you roll with advantage, that goes down to 1 in 400: a huge improvement! The chance of rolling a 10 or 11 (2 in 20 chance of one or the other) remains the same. And the chance of a “crit” –  20 – goes up from 1 in 20 when rolling a single D20 to 39 in 400 – almost 10% – when rolling with advantage.

You can see that in the table above: the headers along the top and left are the natural rolls, the intersections are the resulting values – the higher of the two.

The nice thing about the table above (which again: was how I visualised the question in my dream!) is it really helps to visualise why these numbers are what they are. The general formula for calculating the chance of a given number when rolling D20 with advantage is ( n² – (n-1)² ) / 400. That is, the square of the number you’re looking for, minus the square of the number one less than that, over 400 (the total number of permutations)⁴.

Why roll two dice when one massive one will do?

Knowing the probability matrix, it’s theoretically possible to construct a “D20 with Advantage” die⁵. Such a tool would have 400 sides (one 1, three 2s, five 3s… and thirty-nine 20s). Rolling-with-advantage would be a single roll.

This is probably a totally academic exercise. The only conceivable reason I can think of would be if you were implementing a computer system on which generating random numbers was computationally-expensive, but memory was cheap: under this circumstance, you could pre-generate a 400-item array of possible results and randomly select from it.

But if anybody’s got a 3D printer capable of making a large tetrahectogon (yes, that’s what you call a 400-sided polygon – you learned something today!), I’d love to see an “Advantage D20” in the flesh. Or if you’d just like to implement a 3D model for Dice Box that’d be fine too!

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