D&D 5E Math Question

[trentonjoe]

Short Question: What's the average of the better die roll on a d20 made with advantage?

 

[MerricB]

The average is 13.825. The median is 15.

Cheers!

 

[trentonjoe]

Is there a mode? Just kinda wondering.

 

[Uller]

I'm sure some statistics jedi can come up with the mathematics...but a quick google sheet with 10K rows says it's about 13.8.

 

[Staffan]

13.825.

That's fairly irrelevant though, because what you're really interested in most of the time is how it affects your chance of success, which is not as straight-forward as "equal to +4 on the roll". This page has a table for that.

 

[Staffan]

Is there a mode? Just kinda wondering.

The mode is the most common result, which would be 20 (a 39/400, or 9.75% probability).

 

[CapnZapp]

13.825.

That's fairly irrelevant though, because what you're really interested in most of the time is how it affects your chance of success, which is not as straight-forward as "equal to +4 on the roll". This page has a table for that.

This is the answer you seek.

In other words, advantage is about the same as a +1 to a +5 bonus, depending.

When your task is very hard, advantage does not help very much. When your task is average, it provides a major buff.

 

[Staffan]

If you want it expressed in mathematical terms, advantage squares your chance of failure, and your chance of success is what's left over. Disadvantage similarly squares your chance of success.

 

[EzekielRaiden]

If you want it expressed in mathematical terms, advantage squares your chance of failure, and your chance of success is what's left over. Disadvantage similarly squares your chance of success.

To expound on this just a little bit more:
All probabilities are values between 0 and 1, inclusive. Thus, if you square them, at best they will remain exactly the same (for the degenerate case of 0, and the best case of 1). All other values will get decreased by squaring.

The benefit of Advantage (and the detriment of Disadvantage) changes depending on what you needed, and how you view it. First, consider the raw percents. If you had to get a crit before, your chance to succeed nearly doubles (from 5% to 9.75%). If you had to roll 15 before, your chance to succeed is now 51% instead of 30%. The gap remains at least (roughly) 20% until you get down to the mid-single-digits for the target roll value.

But if we instead consider it as the "effective expected bonus," for very high TN it is equivalent to only a +1, while for the bulk of possible target rolls, it's more like a +4 or +5. This captures a different perspective on things. That is, you'll hardly notice the benefit of Advantage on a bunch of rolls that were almost guaranteed (whether for success or failure). It's only for checks that could relatively easily go either way (e.g. at least 25% chance of success, or failure, without Advantage) that you'll really notice it over the long term, and for truly "it could go either way" checks (e.g. target roll of 11, 50% chance of success without Adv) you'll see the biggest impact. Ironically, Advantage is far more useful on passive checks, because it's a flat +5 rather than varying between exactly +5 (rare but happens) and just-barely-under +1 (for needing 2 or 20).

Disadvantage, of course, will be exactly inverse of all these benefits.

 

[Mercule]

Running the actual math you get these effective bonus/penalty values. These are rounded to the nearest whole number and happen to be the same, whether a positive or negative adjustment.

[table="width: 100, class: grid"]
[tr]
[td]D20 Target[/td]
[td]Modifier[/td]
[/tr]
[tr]
[td]20[/td]
[td]+/-1[/td]
[/tr]
[tr]
[td]19[/td]
[td]+/-2[/td]
[/tr]
[tr]
[td]18[/td]
[td]+/-3[/td]
[/tr]
[tr]
[td]17[/td]
[td]+/-3[/td]
[/tr]
[tr]
[td]16[/td]
[td]+/-4[/td]
[/tr]
[tr]
[td]15[/td]
[td]+/-4[/td]
[/tr]
[tr]
[td]14[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]13[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]12[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]11[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]10[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]9[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]8[/td]
[td]+/-5[/td]
[/tr]
[tr]
[td]7[/td]
[td]+/-4[/td]
[/tr]
[tr]
[td]6[/td]
[td]+/-4[/td]
[/tr]
[tr]
[td]5[/td]
[td]+/-3[/td]
[/tr]
[tr]
[td]4[/td]
[td]+/-3[/td]
[/tr]
[tr]
[td]3[/td]
[td]+/-2[/td]
[/tr]
[tr]
[td]2[/td]
[td]+/-1[/td]
[/tr]
[tr]
[td]1[/td]
[td]+/-0[/td]
[/tr]
[/table]