D&D 5E Advantage vs Disadvantage : What's the Math?

TwoSix said:

You're absolutely right. In a game with a 2-3 combats per session, rolling attacks normally will probably give you a 1 about once a session. Rolling attacks with advantage gives you a 1 about once a campaign.

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guachi said:

This highlights the biggest impact of advantage - the massive decrease in bad rolls. Yes, the average on advantage is only 3.25 higher than the average on a normal roll, but most of the difference is at the bottom end.

That is, the chance of rolling a 1-5 without advantage is 25% but it's only 6.25% with advantage. It's an 18.25% difference (the same difference between a 16-20 with and without advantage) but in relative terms you'd roll a 1-5 four times more often without advantage. Depending on the final roll you need and your bonuses it's possible for advantage to make it highly unlikely you'd ever fail.

It's this highly variable benefit of advantagethat makes me like it so much more than a flat bonus of +3, for example. if your first roll is a 20 then advantage gives you +0. If you're first roll is a 1 then advantage gives you an average bonus of roughly +10.

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Dausuul said:

A 19 to succeed is unheard-of in the MM, but a 2 to succeed is not. There are monsters in the DMG that even low-level PCs can hit on a 2; zombies, for example.

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nexalis said:

Here is a mathematically precise and succinct way to compute your chance to hit with advantage and disadvantage. You can turn this into a set of simple functions in a programming language or Excel without the need to perform simulations or permutations of any sort.

// First, compute the basic probability of a hit
double probabilityOfHit = (21 + attackModifier - targetArmorClass) / 20;
probabilityOfHit = Math.min(0.95, probabilityOfHit); // if you roll a 1, you always miss
probabilityOfHit = Math.max(0.05 * (21 - criticalHitOn), probabilityOfHit); // if you equal or exceed your critical hit roll, you always hit

// From the probability of a hit, derive the probability of a miss
double probabilityOfMiss = 1 - probabilityOfHit;

// Compute the probability of hit with advantage
probabilityOfHit = probabilityOfHit + (probabilityOfMiss * probabilityOfHit);

// Compute the probability of hit with disadvantage
probabilityOfHit = 1 - (probabilityOfMiss + (probabilityOfHit * probabilityOfMiss);

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nexalis said:

This is precisely the way to look at it. If you focus only on the difference as it applies to a single roll, a 0.25% chance of failure doesn't look much different from a 5% chance failure. If you focus on all of the rolls you will make for an entire year, however, it is the difference between failing once per week as opposed to failing once per year. The graph is deceptive in that it makes advantage/disadvantage appear inconsequential at the extremes of the DC, when this is not the case at all, as you rightly point out. The relative difference is just as important as the absolute difference when you have a reliable source of advantage. In other words, you are seeing the forest just fine!

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Hemlock said:

I've actually got a web app that implements this: http://maxwilson.github.io/RollWeb/Roll/

You can ask it questions like "avg.3a?3d8+7" (if I hit on a 3+ with advantage and do 3d8+7, what is my average damage including crits?). If you take off the "avg" part it gives you a random roll. The "Help" link has further examples.

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nexalis said:

Nice! I've written similar programs myself. I notice yours is hosted on GitHub. Is the source code available in a public GitHub repo?

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Hemlock said:

Yep, it's at the usual relative URL for the github project: https://github.com/MaxWilson/RollWeb

There's also an .exe there with slightly better capabilities, e.g. only the exe can parse "avg.(d20a+7)?15". I wrote those tools primarily as an exercise in packrat parsing (an easy way to parse recursive grammars). The transpilation to JavaScript (using WebSharper) ended up losing some capabilities, and I never hunted down the bug because I switched to using Fable instead of Websharper.

-Max

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jrowland said:

working as intended

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