Probability question: Advantage and Disadvantage

OnlineDM · May 24, 2012

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.

Radiating Gnome · May 24, 2012

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

Kinak · May 24, 2012

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

Cheers!
Kinak

OnlineDM · May 24, 2012

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.

jasimon951 · May 24, 2012

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?

mkill · May 24, 2012

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.

Tovec · May 24, 2012

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.

OnlineDM · May 24, 2012

[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.

Ellington · May 24, 2012

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.

OnlineDM · May 24, 2012

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.