# Probability question: Advantage and Disadvantage



## OnlineDM

So, I'm still working my way through the playtest documents (loving them so far), but I had a question about the advantage/disadvantage mechanic. For those who haven't seen it yet, advantage means you roll two d20s and take the better roll, and disadvantage means you roll twice and take the worse roll.

Then, in reading the conditions, I saw that Prone still applies a -2 penalty to attacks, just as in 4th Edition, rather than giving you disadvantage. Blinded, on the other hand, gives you disadvantage on attacks.

My question is, what is the numerical impact of having advantage or disadvantage, on average?

The average result of a d20 is 10.5. What's the average result of MAX(2d20) and MIN(2d20)? I'm not enough of a dice probability guy to know off the top of my head.


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## Radiating Gnome

Someone much better with probability will do better than me, but a few years back looking at avengers I did some poking around with the double roll mechanic.  

The trick is to understand that this takes a flat probability and turns it into a bell curve of sorts -- so the probability improvement  depends upon the number you need to roll.  

So, if you need an 11 or better to succeed, for example, you succeed on one die 50% of the time.  And of the 50% of the time you fail, a second die would succeed, so your chance of success becomes 75%.  

If, on the other hand, you need a 16 or better, you succeed 25% of the time on the first die, and of the 75% of the time you fail, you succeed 25% of the time on the other die, so with two rolls your chance of success is 31.25% -- so a little better than 25% with one die, but it's not as big a difference as in the easier tests.  

This sort of advantage mechanic does a whole lot more to protect against bad rolls when the odds of success are high than it does to improve the chances of a high roll when the odds of success are low. 

But, that's what you get when you trust a guy with a few degrees in english to talk probability. I'm probably dead wrong. 

-rg


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## Kinak

Looks like advantage gives an average of 13.825 and disadvantage gives an average of 7.175

Cheers!
Kinak


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## OnlineDM

I just did a quick numerical approximation in Excel (65,000 pairs of d20 rolls), and here's what I get:

Average for a single d20: 10.5
Average with advantage: 13.8
Average with disadvantage: 7.2

So, advantage is approximately equivalent to getting +3.3, and disadvantage is about -3.3.

I'm sure the folks who are actual probability whizzes can figure this more precisely.


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## jasimon951

I'm no expert and someone much smarter than I will probably come with a better explanation, but I believe your probability of success when given multiple chances to make a success can be found by:

1-(1-m)^n, where m is your "miss" chance and n is the number of attempts you get.  

So if you had a 50/50 chance on a roll, you'd have roughly a 75% chance to succeed if given two rolls.

I'm not sure how it works for disadvantage.  Probably very close but you use success chance instead of miss chance?


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## mkill

On a flat average, roll twice and take the better is equal to a +3. 

However, PC typically have rolls where an 11 or better is a success (50%). In that case, rolling twice is a massive boost to 75%, or +5 to the roll. The better or worse your success chance gets from 50%, the less effective a reroll is compared to a flat bonus.

If 20 is a crit, it also (nearly) doubles your crit chance, and if 1 is a fumble, it almost fully avoids them (0.25% instead of 5%)

At the table a reroll is therefore roughly worth a +4, or even +5.


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## Tovec

Kinak said:


> Looks like advantage gives an average of 13.825 and disadvantage gives an average of 7.175
> 
> Cheers!
> Kinak




I did a simple calculation so I wouldn't have to figure it out mathematically, over the course of 20 rolls I got _About _what you stated here. Good to know my results were average.
Average Maxs: 13.8
Average Mins: 6.75
Difference between: 7.05
YMMV.


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## OnlineDM

[MENTION=55985]mkill[/MENTION] I see what you mean. I used my set of 65,000 rolls in Excel to see what the probability of meeting or beating a target number is with a single d20, advantage, and disadvantage:



		Code:
	

Target	1d20	Adv	Disadv
1	100%	100%	100%
2	95%	100%	90%
3	90%	99%	81%
4	85%	98%	72%
5	80%	96%	64%
6	75%	94%	56%
7	70%	91%	49%
8	65%	88%	43%
9	60%	84%	36%
10	55%	80%	30%
11	50%	75%	25%
12	45%	70%	20%
13	40%	64%	16%
14	35%	58%	12%
15	30%	51%	9%
16	25%	44%	6%
17	20%	36%	4%
18	15%	28%	2%
19	10%	19%	1%
20	5%	10%	0%


So, if you need to roll a 13 on the die to hit, for example, you have a 40% change of success normally, 64% with advantage and 16% with disadvantage. To see what a straight +2 or +3 to hit would do for you, simply look two or three rows up on the chart (+2 to hit would give you a 50% chance normally; +3 would be 55%). Advantage when you need a 13 on the die is equivalent to a +5 (the chance of an 8 or better on one die is close to the chance of a 13 or better on either of two dice - around 65%).

Note that I've rounded these to the nearest percentage point, which is most noteworthy for the extremes. You don't actually have 0% chance of a crit with disadvantage - it's 0.25% (1 in 400). Same goes for avoiding a natural 1 with advantage - you have a 0.25% chance on rolling a 1 on both dice.


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## Ellington

Don't forget to take into account natural 1s and natural 20s. You normally have a 1 in 20 to get a critical hit or a natural miss. With advantage and disadvantage respectively, that changes to 1 in 400.


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## OnlineDM

Ellington said:


> Don't forget to take into account natural 1s and natural 20s. You normally have a 1 in 20 to get a critical hit or a natural miss. With advantage and disadvantage respectively, that changes to 1 in 400.




Very true, but I don't care about "damage per round" or anything like that (for which crit chance matters), but just the chance of success or failure on a given roll.

If I've done my math right, the chance of a crit with advantage goes from 5% up to 9.75% [ 1 - (19/20)*(19/20) ]. I don't think the chance of a natural 1 going from 5% to 9.75% with disadvantage really matters (a miss is a miss). So, DPR calculators will care about the increased crit chance; I don't (at least not at the moment). But it's a fair point.


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## Kinak

OnlineDM said:
			
		

> @mkill   I see what you mean. I used my set of 65,000 rolls in Excel to see  what the probability of meeting or beating a target number is with a  single d20, advantage, and disadvantage:





I was poking at the same thing, although trying to figure out the effective bonus in each case. Based on what you'd have to roll naturally, here's how advantage/disadvantage effects you're chances (it's symmetrical, so advantage gives + the listed amount and disadvantage gives -).



0% (+0)
4.75% (+1)
9% (+2)
12.75% (+3)
16% (+3)
18.75% (+4)
21% (+4)
22.75% (+5)
24% (+5)
24.75% (+5)
25% (+5)
24.75% (+5)
24% (+5)
22.75% (+5)
21% (+4)
18.75% (+4)
16% (+3)
12.75% (+3)
9% (+2)
4.75% (+1)
What I think is pretty interesting about this is that the target number in actual play tends to cluster in the 5 to 15 zone. Which gives you an average of 22.25% difference (over +/- 4).


Anyway, the morale of the story is that advantage is awesome and I think it'll be a lot of fun.


Cheers!
Kinak


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## TerraDave

Radiating Gnome said:


> ....
> 
> The trick is to understand that this takes a flat probability and turns it into a bell curve of sorts -- so the probability improvement  depends upon the number you need to roll....
> 
> ...This sort of advantage mechanic does a whole lot more to protect against bad rolls when the odds of success are high than it does to improve the chances of a high roll when the odds of success are low.
> 
> ...




I think that's key. Advantage may help your wizard a little bit to konk someone with your staff, but its deadly for the rapier wielding rogue.

And if you have high charisma and a charmed target, you own them. But I guess that is the point.


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## Ainamacar

Since this topic will probably come up often, it can't hurt to have pictures. I've already been beaten to the punch on tables. 

It is easiest to start with disadvantage.  If a character has disadvantage the only way to succeed is if both dice meet or exceed the DC.  If the probability of success when rolling a single die is p, then both dice succeed with probability p^2.  Therefore the probability of success with disadvantage is just p^2.

In the case of advantage, at least one die must beat the DC.  That is equivalent, however, to saying that the check only fails if both dice fail.  Well, the probability of failure with a single die is 1-p, so the probability of failing with both dice is (1-p)^2.  Therefore the probability of success with advantage is 1-(1-p)^2.

These results are shown below.







How does that translate into equivalent bonuses?  It does not do so directly, because the benefit or penalty depends on the original probability.  That is shown on the graph below, where we can clearly see that advantage and disadvantage are mirror images.




The peaks are clearly when the probability of success on the single roll is 0.5, in which case the increase or decrease in the probability of success is exactly equal to 0.25, equivalent to +/- 5 on a d20.  At the edges this tapers, so advantage when one needs a 20 is still useful, but increases the probability of success by just 19/400=.0475, which is just less than the benefit of a +1 on a single roll.

If in play DCs were presented such that all probability of success were equally common then the average increase or decrease in probability would be 19/120=.1583bar, which is better than a +3 and less than a +4. (These values are just the mean of p^2-p and 1-(1-p)^2-p, which turn out to be the same except for the sign, which isn't surprising considering the graph above.)

In actual play, of course, certain probabilities of success are more likely to occur.  I suppose one could attempt to model this, but it will vary from table to table anyhow.  At most tables, however, the PCs probably tend to avoid rolling on challenges they almost certainly succeed at as well as challenges where they are very certain to fail.  Therefore, the actual bonus at the table will be somewhat higher than .1583bar, but cannot be larger than .25 (which would mean every single roll always had p=0.5).  It's probably slightly stronger than a +/- 4 in typical play.

It is worth remember some cases not covered above.  First of all, a bonus to a d20 roll itself can change whether certain tasks are possible or impossible, and advantage/disadvantage never does this.  Secondly, in opposed rolls this math only applies after one party has resolved their roll (effectively setting the DC for the other person).  Thirdly, not all cases are about meeting a DC, but involve exceeding an interval and getting different outcomes based on that.  (In earlier editions the jump check is the classic example, where one simply jumps as far as one can roll.)  In that case the probabilities above don't capture any notion about "how well" or "how poorly" the roll succeeds.

Edit:  I've added a table anyway.  These are the exact probabilities of success in decimal form.


		Code:
	

p     Adv      Dis     Difference
0.    0.       0.      +/-0.
0.05  0.0975   0.0025  +/-0.0475
0.1   0.19     0.01    +/-0.09
0.15  0.2775   0.0225  +/-0.1275
0.2   0.36     0.04    +/-0.16
0.25  0.4375   0.0625  +/-0.1875
0.3   0.51     0.09    +/-0.21
0.35  0.5775   0.1225  +/-0.2275
0.4   0.64     0.16    +/-0.24
0.45  0.6975   0.2025  +/-0.2475
0.5   0.75     0.25    +/-0.25
0.55  0.7975   0.3025  +/-0.2475
0.6   0.84     0.36    +/-0.24
0.65  0.8775   0.4225  +/-0.2275
0.7   0.91     0.49    +/-0.21
0.75  0.9375   0.5625  +/-0.1875
0.8   0.96     0.64    +/-0.16
0.85  0.9775   0.7225  +/-0.1275
0.9   0.99     0.81    +/-0.09
0.95  0.9975   0.9025  +/-0.0475
1.    1.       1.      +/-0.


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## dammitbiscuit

So, OnlineDM, if you park someone with the Guardian theme behind whichever character is currently taking a beating, _you push crits off of the combat table_. Now you don't have to worry about sudden, horrific spike damage removing your friend from the waking world - just try to manage the incoming normal hits.


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## OnlineDM

dammitbiscuit said:


> So, OnlineDM, if you park someone with the Guardian theme behind whichever character is currently taking a beating, _you push crits off of the combat table_. Now you don't have to worry about sudden, horrific spike damage removing your friend from the waking world - just try to manage the incoming normal hits.




First, I haven't gotten to themes yet, but I gather that the Guardian makes it so that enemies have disadvantage against allies adjacent to the Guardian. Cool.

Second, as I said, I rounded the percentages in my table to the nearest point. Having disadvantage doesn't make a crit impossible, but it makes it a 0.25% chance (1 in 400).

Third, are there massive spikes in crit damage for monsters? I haven't gotten to the bestiary yet, but for PCs it seems to be the case that you just maximize the damage roll. Not fun to be hit by it, sure, but not super-devastating either.


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## B.T.

Advantage/disadvantage is a brilliantly elegant mechanic that much simplifies unnecessary modifiers.  Math done on the fly is cumbersome and clunky.  I've toyed with the mechanic before but never thought to codify it in the way that the 5e playtest done.  Along with the HD mechanic, it's something I'll be stealing for my own work.


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## Neonchameleon

Target 			Advantage 			Normal 			Disadvantage 		 		 			>=2 			99.75% 			95% 			90.25% 		 		 			>=3 			99.00% 			90% 			81.00% 		 		 			>=4 			97.75% 			85% 			72.25% 		 		 			>=5 			96.00% 			80% 			64.00% 		 		 			>=6 			93.75% 			75% 			56.25% 		 		 			>=7 			91.00% 			70% 			49.00% 		 		 			>=8 			87.75% 			65% 			42.25% 		 		 			>=9 			84.00% 			60% 			36.00% 		 		 			>=10 			79.75% 			55% 			30.25% 		 		 			>=11 			75.00% 			50% 			25.00% 		 		 			>=12 			69.75% 			45% 			20.25% 		 		 			>=13 			64.00% 			40% 			16.00% 		 		 			>=14 			57.75% 			35% 			12.25% 		 		 			>=15 			51.00% 			30% 			9.00% 		 		 			>=16 			43.75% 			25% 			6.25% 		 		 			>=17 			36.00% 			20% 			4.00% 		 		 			>=18 			27.75% 			15% 			2.25% 		 		 			>=19 			19.00% 			10% 			1.00% 		 		 			>=20 			9.75% 			5% 			0.25% 		 	  
Edit:


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## Fazza

Kinak said:


> Looks like advantage gives an average of 13.825 and disadvantage gives an average of 7.175
> 
> Cheers!
> Kinak




This man is correct, I checked this in regards to Avengers in 4e and it's 13.825. Just did a check and disadvantage would give 7.125.

The way I did this was in excel made a grid with 1 to 20 as headers(to simulate possible first roll outcomes) and 1 to 20 at the sides(for second roll) then use MAX() and MIN() functions to decide which of the 2 rolls to take and then averaged these 400 possible results.

EDIT: Chances of Nat 20 and Nat 1 with advantage is 9.75% and .25% respectively(reverse with disadvatage)


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## Yora

B.T. said:


> Advantage/disadvantage is a brilliantly elegant mechanic that much simplifies unnecessary modifiers.  Math done on the fly is cumbersome and clunky.  I've toyed with the mechanic before but never thought to codify it in the way that the 5e playtest done.  Along with the HD mechanic, it's something I'll be stealing for my own work.



However, for no objective reason, I absolutely hate variable numbers of dice. I won't touch any dice pool game unless it's a one-shot with friends in which I am not GM.

Roll two dice, use only one goes in that direction. The moment I read I immediately went to find out what that means for the avarage and how to translate that into a fixed bonus or penalty to a single dice roll.

I think Advantage and Disadvantage will probably come up a lot, so the only factor that should matter in the long run is average. +/-3 is good enough for me.


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## Rampaging Lawn Gnome

If I'm understanding this correctly, in a battle with 2 goblins: a guard (AC12) and his chief (AC19), advantage against the guard gives me a +5 or so, whereas advantage against the chief only gives me a +2 effectively.  It makes sense that the tougher opponent would be more difficult to hit, but there is a bit of a law of diminishing returns isn't there.  What's the incentive to try to gain advantage against the tougher opponent?

I think I might think its a good mechanic, but maybe I'm missing it.


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## OnlineDM

Rampaging Lawn Gnome said:


> If I'm understanding this correctly, in a battle with 2 goblins: a guard (AC12) and his chief (AC19), advantage against the guard gives me a +5 or so, whereas advantage against the chief only gives me a +2 effectively.  It makes sense that the tougher opponent would be more difficult to hit, but there is a bit of a law of diminishing returns isn't there.  What's the incentive to try to gain advantage against the tougher opponent?
> 
> I think I might think its a good mechanic, but maybe I'm missing it.




This is only true if your attack has +0 to hit. If you take the dwarf fighter from the playtest, he has +6 to hit with his Greataxe. This means he only needs to roll a 6 to hit the guard or a 13 to hit the chief. Having advantage when you need a 6 on the die is equivalent to just shy of a +4, and having advantage when you need a 13 on the die is equivalent to almost a +5.

It only has diminishing returns at the extremes - when you'll be hitting on a 2 or 3 on the die, or when you need an 18 or 19 on the die to hit, for instance. D&D rarely puts players in that situation.


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## Rampaging Lawn Gnome

Double post.


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## mkill

Rampaging Lawn Gnome said:


> What's the incentive to try to gain advantage against the tougher opponent?




Because even if you need a 20 to hit, it's still better if you get two tries to do it.

However, if you need a 20 to hit an opponent, it's probably better to run.


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## DNH

Rampaging Lawn Gnome said:


> What's the incentive to try to gain advantage against the tougher opponent?






mkill said:


> Because even if you need a 20 to hit, it's still better if you get two tries to do it.



Precisely. All this talk about probabilities and averages but those averages are only valid over time; there is no average result of a single die roll - each result , 1 through 20, is equally probable. So having two throws of the dice is clearly desirable.

[And I really need to change my sig line.]


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