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D&D Next Advantage/Disadvantage Math

I have the public beta package for D&D Next that has been released so far. Just the first batch. I haven’t play tested it, and I don’t intend to write a review, now. This is just a quick post for people looking for a bit of math regarding one of the new mechanics they’ve revealed, Advantage/Disadvantage.

A brief description: the guide states that if the DM declares you, the player, to have Advantage in a situation any d20 roll you would normally make you instead roll 2 d20’s and use the highest roll. If you have Disadvantage you use the lowest. The DM can determine Advantage or Disadvantage for any number of circumstances, but a common one would be during a surprise round, or if the target of an attack was unconscious or something.

So, I was curious so I did some calculations and I’ll share my results. The average d20 roll is 10.5. The average result from an Advantaged roll, basically the max of 2d20, is 13.825. The reciprocal average Disadvantage roll is 7.175. That’s a change of 3.325 on average. If you have advantage against an opponent with disadvantage you have on average a 6.65 bonus.

Still, the nature of the system is pretty volatile. The standard deviation of a normal single d20 roll is 5.916. The standard deviation of an Advantage/Disadvantage roll is 4.717, so you don’t lose much of the luck factor. I don’t have all the numbers, and don’t want to calculate them now, but I would be remiss if I didn’t point out that this system introduces and uneven distribution. Basically, it’s not an even benefit for each number. With 1d20 you have a 5% chance of getting any result, so if you have to roll more than 1 you have a 19/20 chance, higher than 2 a 18/20 chance, and so on. If you have Advantage it’s not so simple. You have a 399/400 chance to get higher than 1 and a 396/400 chance to get higher than a 2, and a 391/400 chance to get higher than a 3. See how the differences in odds aren’t increasing evenly? This means that depending on the DC the boon/bane of Advantage/Disadvantage varies. But that’s as deep as I want to get into that subject.

Anyway, there it is. Hope you find it interesting/useful.

Edit: Well, this was bugging me and I can’t sleep so I ran some more numbers. I have created a public doc here. Please check it out if you want all the numbers related to the asymmetrical distribution mentioned at the end of the original post. I have graphed the benefit of Advantage compared to a normal roll for each possible required roll to make it as clear as possible. All the raw numbers are there as well, so you can see that Disadvantage’s curve¬†would be identical, just upside down.

One additional thing that comes to mind is critical hits. So far the rules for criticals are that if you roll a 20 you do maximum possible damage. Pretty simple. But the Advantage system tied into the critical system though. If you have Advantage you have almost 2x the chance to crit, and if you have Disadvantage you have almost no chance, 1 in 400. Still, the effect of Advantage/Disadvantage is minimized at the extremes of rolling, so Advantage is much more likely to make you hit when you would have missed than to make you crit when you would have hit. Of course, if you have a 15-20 crit range, like is possible is some systems, then this is dramatically no longer true. I suspect they don’t intend to allow this, however.

I think this pretty much covers all the math related to the mechanic.