What's the Math on Advantage?

[Sparky McDibben]

OK, so what exactly is the math on advantage? I mean the actual calculations here.

Also, if this is in the wrong forum, let me know.

[Man_Over_Game]

OK, so what exactly is the math on advantage? I mean the actual calculations here.

Also, if this is in the wrong forum, let me know.

Depends on how you look at it.

If you compare the 50% mark of both the standard 1d20 and 2d20k1 (AKA Advantage), then they hit 50% at 10.5 and 15, respectively. That is, Advantage hits 15 50% of the time, and a standard 1d20 hits 10.5 50% of the time. So that comes to a +4.5 difference, which a lot of people just round up to +5.0 (which is likely where the +/-5 to Passive checks comes from). But that's not very accurate of what Advantage ACTUALLY does. Because you don't always roll 15's. Sometimes, your Advantage roll is a 9. If you account for Advantage when it's strongest, it makes a +4.5 difference, but when you consider all of the other numbers (as in, 1 through 20), the gain from Advantage is quite a bit smaller. For example, while you are 44.5% more likely to roll a 10 or higher on Advantage, you are only 4.75% more likely to roll higher than a 2 (which is less than a +1 bonus).


If you count the numbers furthest from the rolls of 10.5-15, Advantage actually increases your average roll by +3.33 (or +16.65%). That is, a roll with Advantage will, on average, increase your roll by +3.33 points. However, there's another catch to this. Because of the fact that 5e uses bounded accuracy, it's unlikely that a loss/gain on your roll of 17, or a loss/gain on your roll of 3, is going to matter. The fact is, the only numbers that really matter on the d20 are roughly between 7 and 16. If you roll higher than 16, it doesn't change anything, and if you roll under a 7, it also doesn't change anything. So it's best just to exclude those numbers for the sake of calculating the gain on Advantage.

So just counting the average numbers on Advantage gets us a +4.5 gain, which is incorrect and biased. We just use that because it's easy, and we like rounding to 5. +3.33 is also incorrect, because those 19's and 2's don't actually contribute anything, which is kinda like counting wild dogs when determining how many dog licenses a state needs.

If you calculate only the range that matters, you end up with basically a +4.0 gain on Advantage, or a -4.0 loss on Disadvantage. This is probably the most realistic shorthand of Dis/Advantage.

If you want to check my work and run your own calculations, here's a link to Anydice that has the probabilities of all the possible results of Dis/Advantage: https://anydice.com/program/167da You'll notice that the Summary page has Advantage with a mean of +3.33 higher than 1d20, and Disadvantage is -3.33 lower than 1d20.

[Steel Mirror]

Depends on how you look at it.Good post. I was working up a reply of my own that delved into the numbers a bit, but this covers it all well and with a couple of good perspectives on it as well.

The only thing I will add is that advantage also increases your chance of scoring a crit, from 5% to 9.75%, slightly less than double the chance. So although the bonus it gives you on likelihood to hit is in the +4ish range, there is also the added chance of scoring a crit which can be a big advantage, especially when attacking with big or magical weapons that let you roll lots of big damage dice. So getting advantage also makes your attacks potentially more explosive, not to mention the fun factor of scoring crits roughly twice as often.

[Theron_the_slim]

Chance to hit = 1-(1-z)(1-z), where z is your Chance to hit without advantage

For example: You have a +5 to hit, you are attacking someone with an AC of 10

so all numbers except 1-4 are enough to hit, your chance z is therefore 0.8 (80%)

So your new chance to hit = 1-(0.2*0.2)= 0.96 (96%)

The same formular works also for your crit chance with advantage, you only have to use 0.05 for z (assuming the usuall crit range)


In case you are interested when you get the most benefit from your advantage (in terms of a + chance to hit), its when you have z=0.5 (so a 50% chance to hit)
In that case your chance to hit goes from 50% to 75% (so basiclly a +5 to hit) ... compared from the first example where there is only a 16% difference (so basiclly a +3 to hit)
The peak of mathematical usefullness of advantage is at z=0.5 and declines in both directions (in an equal speed), first slowly, then faster

[LudicSavant]

Advantage changes your hit rate to 1-(1-P)^2, where P is the probability of hitting without Advantage. For example, if your hit rate was 60%, your Advantage hit rate would be 1-(1-.6)^2 = 84%.

It also changes your crit rate in the same fashion. For example, if your previous crit rate was 5%, your new crit rate would be 1-(1-0.05)^2, or 9.75%.

If you'd instead like the formula for Elven Accuracy, it's just 1-(1-P)^3 instead of ^2.

[EnnPeeCee]

In case you are interested when you get the most benefit from your advantage (in terms of a + chance to hit), its when you have z=0.5 (so a 50% chance to hit)
In that case your chance to hit goes from 50% to 75% (so basiclly a +5 to hit) ... compared from the first example where there is only a 16% difference (so basiclly a +3 to hit)
The peak of mathematical usefullness of advantage is at z=0.5 and declines in both directions (in an equal speed), first slowly, then faster

This is also somewhat subject to how you look at the numbers.
Take a look at the 3 extreme cases, where your base chances are 1/20, 10/20, and 19/20:



Base %
Advant. %
Flat % Increase
Mult. % Increase


1/20 5%
9.75%
4.75%
195%


10/20 50%
75%
25%
150%


19/20 95%
99.75%
4.75%
105%



Where the flat% increase in your (advantage% - base%), and the mult% increase is (advantage% / base%).

At the base 50% mark, you get the greatest flat increase of chance to hit, 25%, which is most cases will be what most people will care about.
But at the base 5% mark, looking at the mult % increases, you can see that your chance of success nearly doubles, compared to only one and a half at the 50% mark. So in that sense, you get the most benefit from advantage when you have only a 5% base chance of success.
Another way to look at is is how much your chance of failure goes down with advantage, which I did't include in the table. But those numbers work out similar to above, where at 50% base you have the greatest flat decrease in your failure chance (-25%), but at 95% base you have the greatest % decrease in your failure chance (2000%!).

Anyway, just some other ways to look at what "best" actually means.

[Chronos]

Quoth Man_Over_Game:

If you compare the 50% mark of both the standard 1d20 and 2d20k1 (AKA Advantage), then they hit 50% at 10.5 and 15, respectively. That is, Advantage hits 15 50% of the time, and a standard 1d20 hits 10.5 50% of the time.
The number for advantage is actually closer to 14, not 15. What you might be thinking of is the chance of success if a normal roll has a 50% chance: In that case, the advantaged roll has a 75% chance of success (i.e., the same chance you'd get if you had +5 on your roll instead of advantage).

[Sorlock Master]

OK, so what exactly is the math on advantage? I mean the actual calculations here.

Also, if this is in the wrong forum, let me know.

That is such a vague question. You need to be more specific.

Are you referring to Attacks? Damage? Saves? Skill checks? Initiative?

As a general rule anything that is not an a critical gets its probability increased by 25%. Any critical has its probability doubled.

This is a very short hand way of doing things so if you want exacts then use the already post equations.

[Sparky McDibben]

Chance to hit = 1-(1-z)(1-z), where z is your Chance to hit without advantage

For example: You have a +5 to hit, you are attacking someone with an AC of 10

so all numbers except 1-4 are enough to hit, your chance z is therefore 0.8 (80%)

So your new chance to hit = 1-(0.2*0.2)= 0.96 (96%)

The same formular works also for your crit chance with advantage, you only have to use 0.05 for z (assuming the usuall crit range)


This and LudicSavant's reply were exactly what I needed. Thank you, gentlemen. Man_Over_Game, that is some very helpful context. I appreciate it from all!!!

[Dirclaw]

OK, so what exactly is the math on advantage? I mean the actual calculations here.

Also, if this is in the wrong forum, let me know.

In addition to what the others have said, which answer most of what you wanted, you also said you wanted the actual calculations.

A lot of the calculations for specific rolls needed have been given (crits increasing from 1/20 to 39/400, or from 1/10 to 19/100 if you also crit on a 19).

They have also given regular use numbers for most rolls needed to hit, the 6-16 range. These are easy enough to calculate with some pretty simple math.

If you only fail on a 1-5, you would need to roll in that range (1/4 chance) twice. That is a 1/16 chance so you go from a 75% success rate to a 93.75% success rate, an effective increase of between a 3 and a 4 flat value.

If you only fail on a 1-15, then you need to roll that range (3/4 chance) twice. That is a 9/16 chance, so you go from a 25% success rate to a 43.75% success rate, again, an effective increase of between a 3 and a 4 flat value.

You will notice that these increases are the same, that is because advantage is the most helpful (in comparison to being related to a flat bonus) at the middle levels, when you would fail 1/2 the time. In those cases, you need to roll a 1-10 (1/2 chance) twice to fail. You go from a success rate of 25% to 50%, an effective increase of 5. This is why advantage often gets compared to a +5 increase, because that is it's max value, when success and failure happen equally (fairly common in dnd).

The same mirroring will be true at the high and low ends, 1/20 chance and 19/20 chance. Your success rates go from 5% to 9.75% and from 95% to 99.75% respectively, which is of less of a value than even a +1 flat increase.

If you look at all the numbers together (or if you don't know what your needed roll is) then the full effective increase is the same as a 3.325 bonus. The formula for advantage is ((x^2)-1)/(6x), with x being the size of the die, or 399/120 = 3.325 in the case of a 20 sided die.


TLDR: Advantage is the same as a 3.325 flat bonus in total, but depends on the DC needed, with it being higher for middle DCs and lower for very high and very low DCs

Note: This math does not take into account auto fails for auto successes for 1s and 20s

[KorvinStarmast]

OK, so what exactly is the math on advantage? I mean the actual calculations here.

Also, if this is in the wrong forum, let me know. A superb treatment of it here. (https://rpg.stackexchange.com/q/14690/22566)

For a graphical depiction (Orange is advantage, blue is disadvantage)

https://anydice.com/program/1203
Raw numbers.
(That number or better ...)
Advantage
# %
1 100
2 99.75
3 99
4 97.75
5 96
6 93.75
7 91
8 87.75
9 84
10 79.75
11 75
12 69.75
13 64
14 57.75
15 51
16 43.75
17 36
18 27.75
19 19
20 9.75

Disadvantage
# %
1 100
2 90.25
3 81
4 72.25
5 64
6 56.25
7 49
8 42.25
9 36
10 30.25
11 25
12 20.25
13 16
14 12.25
15 9
16 6.25
17 4
18 2.25
19 1
20 0.25

[Keravath]

http://www.enworld.org/forum/showthread.php?654979-The-math-of-Advantage-and-Disadvantage

[Chronos]

One case that I've wondered about, but haven't grinded through all of the math on:

Something's trying to grapple/shove/whatever you, and so you need to make an opposed check with your choice of Str (Athletics) or Dex (Acrobatics). Ordinarily, this is an easy choice: You use whichever one has a higher bonus for you (let's say, without loss of generality, that it's Acrobatics). But now suppose that you've also been hit by a Hex, which puts disadvantage on your Dex checks. Which one is better now?

I know that if your Acrobatics is 5 or more points better than your Athletics, then you should still use Acrobatics, because advantage/disadvantage is never worth more than 5 points. And if it's only 1 point better, then you should switch to Athletics, because advantage/disadvantage is almost always worth more than 1 point. And since it's an opposed check, it probably depends on just how much better/worse the opponent is than you at their Athletics.

[LudicSavant]

One case that I've wondered about, but haven't grinded through all of the math on:

Something's trying to grapple/shove/whatever you, and so you need to make an opposed check with your choice of Str (Athletics) or Dex (Acrobatics). Ordinarily, this is an easy choice: You use whichever one has a higher bonus for you (let's say, without loss of generality, that it's Acrobatics). But now suppose that you've also been hit by a Hex, which puts disadvantage on your Dex checks. Which one is better now?

I know that if your Acrobatics is 5 or more points better than your Athletics, then you should still use Acrobatics, because advantage/disadvantage is never worth more than 5 points. And if it's only 1 point better, then you should switch to Athletics, because advantage/disadvantage is almost always worth more than 1 point. And since it's an opposed check, it probably depends on just how much better/worse the opponent is than you at their Athletics.

You can calculate and compare the success rate of opposed rolls in any given situation like so:

https://anydice.com/program/168c2