Exactly how much better is rolling with advantage? What are the chances of rolling a natural 20 and scoring a critical hit? Let’s find out!
Chapter I: A Hero’s (Re)quest, Or, Pippa Meets Spittlefleck
As you are undoubtably aware, the 5th Edition of Dungeons & Dragons makes extensive use of the 20-sided die. Otherwise known as the ubiquitous D20, this die is used to determined the success, or failure, of almost everything a D&D character does.
For example, if a player’s character (let’s call her Pippa Crocker) attempts to bluff her way into a goblins’ lair, her player will roll a single D20 to determine the result. If the roll (plus any applicable modifiers1) meets or exceeds a predetermined target number,2 then Pippa succeeds. She dupes the credulous goblin guards. Otherwise, she faces the consequences of her failure (and a horde of angry goblins)!
Sometimes however, circumstances conspire to give Pippa an advantage (or disadvantage). Perhaps she forged a letter of recommendation bearing the seal of a neighbouring goblin clan? Maybe she accidentally insulted the goblin’s king, Spittlefleck the Gnarled? In either case, her player rolls a second D20. If she has an advantage, she takes the higher result. With a disadvantage, she takes the lower number.
Intuitively, rolling with advantage gives our fictional hero, Ms. Crocker, a better chance of success. But, just how much better? We can use a little math to take a look…
What is the Average Roll?
Let’s start prying around the edges of this question by considering the average roll on a D20. As you may remember from grade-school, the three most common measures of central tendency are the mean, the median and the mode. Since we are dealing with discrete data3, the median and mode are probably the most useful measures. Nevertheless, for completeness sake, we will take a look at all three.
If we let be the set of all possible outcomes when rolling 1D20. Then, we find that:
In this case, the mean and median aren’t particularly illuminating. Although, the fact that they are equal suggests that the set of possible outcomes does not skew high or low. The lack of a mode is a little more telling, however. It indicates that no outcome is more common than any other. To reiterate, you are no more likely to roll a quote/unquote “average” 10 or 11 than any other number! Once again, at the risk of beating a dead riding lizard, average is not synonymous with typical.
Now, lets see what we can discover about rolling with advantage…
What is the Average Roll with Advantage?
Let be the set of possible outcomes when rolling 2D20. Also, let and . Finally, let be the set of possible outcomes when rolling 2D20 and choosing the higher result. We can determine that:
The mean and median indicate that the average roll (with advantage) is between 2 and 5 points higher than the average for 1D20. This suggests that the set of possible outcomes skews high. This is good news for dear Pippa!
The existence of a mode tells us that, unlike 1D20, some numbers are statistically more common than others. Even better for our fictional hero, we are most likely to roll a natural 20. In fact, if we peek under the hood a little, we can see that (when we roll with advantage) we are 39 times more likely to role a 20 than a 1 — critical hits4 here we come!
But, what happens if we roll with disadvantage…
What is the Average Roll with Disadvantage?
If we let and , we can let be the set of possible outcomes when rolling 2D20 and choosing the lower result. It follows that:
The mean and median suggest that the average (with disadvantage) is between 3 and 5 points lower than rolling 1D20. This does not bode well for our hero, Miss Crocker. Finally, the mode indicates that a 1 is the most likely outcome — very bad news indeed!
Some Tentative Conclusions
Our look at the average roll has suggested two possible advantages of rolling with advantage.
- Over time, rolling with advantage is — in the parlance of D&D — almost like adding a modifier (whose value is between +2 and +5).5
- Over time, we will roll more natural 20s.6
However, it would be nice to know a little more about that virtual modifier. It might also be nice to compare likelihood of rolling a critical hit with and without advantage. Luckily, averages aren’t the only way to think about the advantage of having advantage.
Forget the Average! What’s the Probability?
In fact, we have a much better tool in our toolbox — probability theory. Now as we get started, we might be tempted to find the probability of rolling a specific number — such as the elusive natural 20. But, in our example above, Miss Pippa Crocker will successfully bluff her way past the goblin guards if her player’s roll meets or exceeds the target number. Therefore, it would be better to know the probability of rolling a given number or higher.
Happily, the formula for calculating probabilities like this is rather straightforward.
Unhappily, actually calculating probabilities is thankless drudgery. Nevertheless, with this formula in mind, here are the probabilities for (1D20), (Advantage) and (Disadvantage).
Rolling with (Dis)Advantage vs Adding Modifiers
Lighting the mystical Torches of Central Tendency left us with the suspicion that rolling with advantage can be compared to adding a modifier to a single D20. However, we were left with an unhelpfully large range of modifiers. Hopefully, a quick look into the sibylline Pool of Probability will add some clarity.
Studying the previous graph, we see that rolling a 2 or higher (with advantage) has nearly the same probability as rolling a 1 or higher on a single D20. This suggests, the chance of scoring 2 or higher with advantage is similar to rolling 1D20 and adding +1. Say goodby to critical failures!
Additionally, the probability of rolling 10 or higher (with advantage) is almost the same as rolling a 5 or higher on a single D20. Thus, if you need a 10 or higher to succeed, having advantage is like rolling 1D20 and adding +5. (The same goes for rolling 15 or higher.) Very good indeed!
So, having advantage can be compared to rolling a D20 and adding a modifier, but the modifier’s value varies depending on the target number! In fact, it ranges from +0 to +5. And, as you can see from the graph below, having advantage is most advantageous if you need to roll an 8-14 to succeed. It is less valuable if your target is higher or lower.
A Few Provisos and a Couple of Quid Pro Quos
I should note that, with the exception of the modifiers associated with 1 and 11, these are approximate values. Also, you will have noticed that disadvantage mirrors advantage, however, disadvantage is missing a few modifiers. I’m not sure it’s helpful to compare rolling an 18, 19 or 20 with disadvantage to adding a modifier — their probabilities are closer to 0 than to rolling a 20 without disadvantage. You could argue that they do mirror advantage, but the odds become so abysmal I think the comparison starts to break down.
Rolling a Natural 20
By studying the average roll, we discovered that we were more likely to roll a natural 20 with advantage. But, we weren’t able to compare that likelihood to the likelihood of rolling a critical hit with 1D20. Now, we have the tools to dig a little deeper. The graph above shows that there is a 5% chance of rolling a 20 with a single D20. It also reveals that the probability of rolling a 20 nearly doubles with advantage! (In fact, it is nearly the same as rolling a 19 or higher without advantage.) In other words, the chance of scoring 20 with advantage is similar to rolling 1D20 and adding +1 to the result. Not too bad.
On the other hand, with disadvantage, the odds of rolling a 17 or higher are less than rolling a 20 with a single D20. Not too good!
Happily, our dip into the Pool of Probability has added some depth and clarify to the two advantages of rolling with advantage we posited earlier.
- We can liken having advantage to adding a virtual modifier. However, the modifier’s value varies depending on the target number.
- Our chance of rolling a natural 20 doubles with advantage!
Clearly, our imaginary hero Miss Pippa Crocker, will be well served if she takes the time to engage in a little forgery. We should remember, though, that her letter of introduction won’t do her quite as much good if the goblin guards are abnormally difficult to dupe. Regardless, she should definitely take pains to avoid insulting his excellency, the goblin king!
Check out these pages for more on Dungeons & Dragons 5E‘s advantage mechanic:
- Just How Lucky are Halflings? — A quick peak at the statistical benefits of re-rolling 1’s.
- The Truth About Randomness, Rolling at Advantage, and How to Equip Your Barbarian — A look at randomness, superstition and rolling a D20, by Sax Carr (with input from Glen Whitman).
- Probability for Dumm… for Gamers — A thorough analysis of probability in the context of D&D, by Scott Rehm (a self-proclaimed misanthrope).
- In this case, her player would add Pippa’s Charisma (Deception) modifier.
- In this case, the target number is called the Difficulty Class.
- Anything you can count is discrete data. The integers from 1 to 20 (on a D20) are discrete.
- Officially, the critical hit rule doesn’t apply to Pippa’s (deception) ability check. It is however, a common house rule.
- This virtual modifier does minimizes, but does not negate, the possibility of rolling a 1.
- This does not mean that if you haven’t rolled a critical in a while you are more likely to roll a 20. Every die roll is independent; they cannot effect each other in any way. Suggesting otherwise courts superstition and madness!