Probability question: Advantage and Disadvantage

[Kinak]

@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 -).

1. 0% (+0)
2. 4.75% (+1)
3. 9% (+2)
4. 12.75% (+3)
5. 16% (+3)
6. 18.75% (+4)
7. 21% (+4)
8. 22.75% (+5)
9. 24% (+5)
10. 24.75% (+5)
11. 25% (+5)
12. 24.75% (+5)
13. 24% (+5)
14. 22.75% (+5)
15. 21% (+4)
16. 18.75% (+4)
17. 16% (+3)
18. 12.75% (+3)
19. 9% (+2)
20. 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

 

[TerraDave]

....

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.

 

[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.

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.

 

[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.

 

[OnlineDM]

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.

 

[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.

 

[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:

 

[Fazza]

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)

 

[Yora]

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.

 

[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.