[RPG] How does rolling two d20 and taking the higher affect the average outcome

advantage-and-disadvantagedicednd-5estatistics

D&D 5E has an "advantage" concept where instead of rolling 1d20, you roll 2d20 and take the higher. Likewise, disadvantage means rolling 2d20 and taking the lower.

How does this affect the expected average outcome of the roll?

Best Answer

All this does is linearly adjust the normally-flat 5% probability for each number to occur. What results is a increased or decreased probability of any number above or below average to occur, positively for advantage and negatively for disadvantage. See this AnyDice function set, which yields the following:

^{Black is d20, orange is highest of 2d20, blue is lowest of 2d20.}

Since the probability of achieving any given number is a linear function, we can use linear regression (via Wolfram Alpha and our sample data from AnyDice to eventually solve for probability of x = 0.5x - 0.25 - multiply by 100, and there's your percent chance that you'll roll any particular number.

Additionally, what you're likely looking for is the probability that at least a particular number will be rolled, using either advantage or disadvantage. AnyDice, again, is king:

^{Black is d20, orange is highest of 2d20, blue is lowest of 2d20.}

Data:

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

Assumptions

I'm going to be making the following assumptions, based on what you've already provided:

3rd level (since you get a crit on a 19 or 20)
16 Strength (no ASI to bump up to 18)
Fighting a CR 3 creature (for base math)
Average damage is 7.50 (4.50 from the die +3 Str mod)
Average crit damage is 12.00 (4.50 per die +3 Str mod)
Attack bonus is +5 (+2 prof, +3 Str mod)
The enemy has AC 13 (per the DMG guidelines on page 274)
DPR calculations are:
- Crit damage = Crit % x average crit hit damage
- Normal damage = Hit % x average hit damage
- hit % = 100 – [miss %] – [crit %]

Note that any flat modifier to the damage total won't change with a crit, since you only double the dice rolled, not the modifiers added.

Champion Fighter

Per the DMG page 274, a CR3 creature has an average AC 13, meaning you need to roll an 8 or higher.

With Advantage

AnyDice can tell us our miss chance and our crit chance. From there, we know our hit chance.

A normal roll of 1d20 will have a 35% miss, 10.00% crit (1.20 DPR), and 55.00% normal hit (4.13 DPR) for a total DPR of 5.33
A roll with advantage will have a 12.25% miss, 19.00% crit (2.28 DPR), and 68.75% normal hit (5.16 DPR) for a total DPR of 7.44
A roll with your DMs +2 rule will have a 25.00% miss, 10.00% crit (1.20 DPR), and a 65.00% normal hit (4.88 DPR) for a total DPR of 6.08

With Disadvantage

AnyDice can tell us our miss chance and our crit chance. From there, we know our hit chance.

A normal roll of 1d20 will have a 35% miss, 10.00% crit (1.20 DPR), and 55.00% normal hit (4.13 DPR) for a total DPR of 5.33 (unchanged)
A roll with disadvantage will have a 42.25% miss, 1.00% crit (0.12 DPR), and 56.75% normal hit (4.26 DPR) for a total DPR of 4.38
A roll with your DMs -2 rule will have a 45.00% miss, 10.00% crit (1.20 DPR), and a 45.00% normal hit (3.38 DPR) for a total DPR of 4.58

Rogue

How does this change affect other classes, specifically those who rely on bonus damage dice? I'm using a rogue for this example since sneak attack is easy enough to calculate, but a paladin falls under the same heading with their smite spells and the like.

We use the same percentages and base damage (assume Dex and a rapier) for our fighter, but we add sneak attack damage. That's 2d6 at level 3, so with advantage we add +2d6 (7) on a hit and +4d6 (14) on a crit. Attacks without advantage don't get sneak attack damage added in, so the disadvantage numbers from above carry over (I know you can get sneak attack damage without advantage, but we'll ignore that for simplicity).

With Advantage

A normal roll of 1d20 will have a 35% miss, 5.00% crit (1.30 DPR), and 60.00% normal hit (8.70 DPR) for a total DPR of 10
A roll with advantage will have a 12.25% miss, 9.75% crit (2.54 DPR), and 78.00% normal hit (11.31 DPR) for a total DPR of 13.85
A roll with your DMs +2 rule will have a 25.00% miss, 5.00% crit (1.30 DPR), and a 70.00% normal hit (10.15 DPR) for a total DPR of 11.45

With Disadvantage

A normal roll of 1d20 will have a 35% miss, 5.00% crit (1.30 DPR), and 60.00% normal hit (8.70 DPR) for a total DPR of 10
A roll with disadvantage will have a 42.25% miss, 1.00% crit (0.12 DPR), and 56.75% normal hit (4.26 DPR) for a total DPR of 4.38 (identical to the fighter, as no advantage means no sneak attack)
A roll with your DMs -2 rule will have a 45.00% miss, 5.00% crit (1.20 DPR), and a 50.00% normal hit (2.25 DPR) for a total DPR of 3.45

Conclusion

With your DMs proposed houserule, the expected DPR for any class is going to be decreased because of the fact that you're still only rolling 1 die, so the chance of a critical hit will not change. The biggest, well, advantage of rolling with advantage is it almost doubles your chance of a crit: 9.75% vs. 5.00% for a normal 20 crit and 19.00% vs. 10% for a champion fighter crit.

Indeed, that simple change reduces the overall expected damage output of the entire party, especially those classes that rely on burst damage in the form of more dice. As you gain in levels and get the extra attack feature, magic items/spells that add damage dice, and class features that change the damage dice done, the gap will only increase.

[RPG] When do you have neither Advantage nor Disadvantage

"Neither advantage nor disadvantage" is the normal ability check.

A normal/typical ability check is rolled with 1d20 (modified for Proficiency and Ability scores). It takes "something extra being involved" for an ability check to accrue advantage or disadvantage. (Spell effects, class features, conditions, etc).

The D20
Every character and monster in the game has capabilities defined by six ability scores. {snip} These ability scores, and the ability modifiers derived from them, are the basis for almost every d20 roll that a player makes on a character’s or monster’s behalf.{snip} Ability checks, attack rolls, and saving throws are the three main kinds of d20 rolls, forming the core of the rules of the game. All three follow these simple steps.

Roll the die and add a modifier.

Apply circumstantial bonuses and penalties.

Compare the total to a target number.

Advantage and Disadvantage
Sometimes an ability check ... is modified by special situations called advantage and disadvantage. Advantage reflects the positive circumstances surrounding a d20 roll, while disadvantage reflects the opposite. When you have either advantage or disadvantage, you roll a second d20 when you make the roll. (Basic Rules, p. 4 / PHB p. 7)

An ability check with advantage is rolled with 2d20, take the highest one.

Example: Ranger. (Class feature Favored Enemy)

You have advantage on Wisdom (Survival) checks to track your favored enemies, as well as on Intelligence checks to recall information about them.
An ability check with disadvantage is rolled with 2d20, take the lowest one.

There are a variety of things which will cause disadvantage on an ability check, such as a level of exhaustion.

Exhaustion (Appendix A, Conditions)
Level...............Effect
1...................Disadvantage on ability checks

Those are the three cases of ability checks: normal, with advantage, with disadvantage.

Trying to create advantage

Whenever the party/players use the "help" action (working together) an ability check can be made with advantage.

Working Together (Sub category of Using Ability Scores)

Sometimes two or more characters team up to attempt a task. The character who’s leading the effort—or the one with the highest ability modifier—can make an ability check with advantage, reflecting the help provided by the other characters. In combat, this requires the Help action. A character can only provide help if the task is one that he or she could attempt alone. For example, trying to open a lock requires proficiency with thieves’ tools, so a character who lacks that proficiency can’t help another character in that task. Moreover, a character can help only when two or more individuals working together would actually be productive. Some tasks, such as threading a needle, are no easier with help.

Where it makes sense

You as DM can, whenever you feel that the circumstances warrant it, apply either advantage or disadvantage based on what the character is trying to do, and what judge makes it easier or harder to achieve based on their situation and their description of the attempt. (from Advantage and Disadvantage)

The GM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.

Best Answer

Related Solutions

[RPG] How does replacing Advantage with a flat +2 affect Champion critical hit damage output

Assumptions

Champion Fighter

With Advantage

With Disadvantage

Rogue

With Advantage

With Disadvantage

Conclusion

[RPG] When do you have neither Advantage nor Disadvantage

"Neither advantage nor disadvantage" is the normal ability check.

Trying to create advantage

Where it makes sense

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