A mathematician that I follow, Matt Parker, made a video exploring the statistics and average value of rolling with advantage.  I found this video very interesting...and thought you guys would like it too.

The unexpected logic behind rolling multiple dice and picking the highest.

Thanks for posting. That was really interesting. 

Also, it seems to challenge the conventional wisdom that rolling with advantage on a d20 is like adding 5. It’s more like adding 3.3. Since the average of a straight roll is 10.5, but the average of rolling with advantage is 13.8. Rolling super advantage does get you to that +5, though. 

At least that’s how I, not a math person, understands it. I’m happy to be corrected. 

Quote from Xalthu >>

Thanks for posting. That was really interesting. 

Also, it seems to challenge the conventional wisdom that rolling with advantage on a d20 is like adding 5. It’s more like adding 3.3. Since the average of a straight roll is 10.5, but the average of rolling with advantage is 13.8. Rolling super advantage does get you to that +5, though. 

At least that’s how I, not a math person, understands it. I’m happy to be corrected. 

 I haven't watched the video or anything like that, but I just checked this by simulation, and that's correct. (Well, the advantage part is; I didn't bother with checking super advantage.)

>>> for x in range(1,1000000):
...     a = random.randint(1,20)
...     b = random.randint(1,20)
...     if (a>b):
...             total += a
...     else:
...             total += b
... 
>>> print(total/1000000)
13.827884

Quote from Xalthu >>

Thanks for posting. That was really interesting. 

Also, it seems to challenge the conventional wisdom that rolling with advantage on a d20 is like adding 5. It’s more like adding 3.3. Since the average of a straight roll is 10.5, but the average of rolling with advantage is 13.8. Rolling super advantage does get you to that +5, though. 

At least that’s how I, not a math person, understands it. I’m happy to be corrected. 

 You're correct. The game seems to assume that it's the equivalent of +5, but yeah, it's actually a tad over +3 (for a d20, smaller dice give smaller margins). Guess you should take the +5 it offers as an alternative, if the DM allows it.

What I found useful is that it gives you a good rule of thumb for working out the averages, so you can apply it to other rolls. Not that it applies to 5e really, but helpful for other games.

I've done the math personally.

To the people saying advantage is more like a 3.3 than a 5, that's only partially correct. This is true over the entire range of rolls, but D&D isn't played against an even distribution of ACs. You're far more likely to fight something requiring a roll of 10 to hit than a 2 or a 19. It's around that 10 roll where advantage offers the best gain and can be worth up to a bonus of 5.5. For martials this stays relevant even as their hit bonus outscales AC due to being able to take a -5 hit for +10 damage.

Lastly, there aren't many situations in 5e that offer the choice of either advantage or +5 hit, so advantage is often a straight benefit.

I've not seen many people equate advantage as +5 for a d20 in general, but there is a written rule where advantage to a skill check bumps the passive skill value by 5, most notably, for passive perception. There might actually be surprising wisdom in that rule, using +5 instead of +3.3.

The calculation for a passive skill value is 10 + Ability Modifier + Skill Proficiency, in other words, equivalent to a skill check with the result of 10. Incidentally, 10 is the highest value you can expect to meet more than half the time (55%) from a normal d20 roll - an 11 or higher is expected to happen exactly half the time (50%). While rolling d20 with advantage, there is a 57.75% chance of getting 14 or higher, and a 51.00% chance of getting 15 or higher - now, 15 is the highest value that we can expect to meet more than half the time, and advantage here was worth at least more than +4. 

What is “super-advantage”?

When three dice are rolled for one result, Elvin Accuracy being the most obvious example.

Thank you!

It's fairly uncommon; I think Elven Accuracy is the only thing that does it naturally, though Silvery Barbs can essentially create super disadvantage if used on a roll that was already under advantage or disadvantage.

Here's the correct calculation:

For each number $n$ from 1 to 20:

• The probability of rolling $n$ on the first die and $n$ or higher on the second die is �20×1=�2020n​×1=20n​.
• The probability of rolling $n$ or higher on the first die and $n$ on the second die is (1−�−120)×120=21−�202(1−20n−1​)×201​=20221−n​.

So, the average result would be:

Average=∑�=120�(�20+21−�202)Average=∑n=120​n(20n​+20221−n​)

=120∑�=120�2+1202∑�=120(21−�)�=201​∑n=120​n2+2021​∑n=120​(21−n)n

=120×20×21×416+1400×20×21×202=201​×620×21×41​+4001​×220×21×20​

=287020+2100400=202870​+4002100​

$=71.75+5.25$

$=77$

So, the average result when rolling 2 d20 at advantage in D&D is 77/20, approximately 3.85 or 14.35 when factoring a bassline roll of 10.5

Thank you so much for posting that video! That was mindblowing and revelatory to realise and saves me lots of manual rolls.

I wish he would do another video on calculating saving throw optimization (fe Resilient prof vs War caster adv), taking + bonuses into account. I've been trying to wrap my head around it but it's very complicated to calculate without math (which I'm not good at)

Quote from The_Ace_of_Rogues >>

It's fairly uncommon; I think Elven Accuracy is the only thing that does it naturally, though Silvery Barbs can essentially create super disadvantage if used on a roll that was already under advantage or disadvantage.

 I don't think silvery barbs works this way. Silvery barbs can only be cast as a reaction when a successful roll is made, which the causes a reroll of the success. With advantage, success is determined after two rolls are made and the higher is selected. Silvery barbs would reroll the sucessful roll and pick the lower of the two. The lower roll on advantage would be disregarded entirely since it wasn't the roll which determined success. 

But you can essentially get super advantage by using a luck point on advantage or disadvantage. This works this way because lucky causes a reroll before success is determined. Silvery barbs is cast after it's determined. 

Quote from alexdohm9191 >>

Quote from The_Ace_of_Rogues >>

It's fairly uncommon; I think Elven Accuracy is the only thing that does it naturally, though Silvery Barbs can essentially create super disadvantage if used on a roll that was already under advantage or disadvantage.

 I don't think silvery barbs works this way. Silvery barbs can only be cast as a reaction when a successful roll is made, which the causes a reroll of the success. With advantage, success is determined after two rolls are made and the higher is selected. Silvery barbs would reroll the sucessful roll and pick the lower of the two. The lower roll on advantage would be disregarded entirely since it wasn't the roll which determined success. 

But you can essentially get super advantage by using a luck point on advantage or disadvantage. This works this way because lucky causes a reroll before success is determined. Silvery barbs is cast after it's determined. 

 If they rolled with Disadvantage, then it would be be effectively Super Disadvantage. If they were rolling with Advantage, then it's a weird hybrid where they sort of have to pass two checks, one with Advantage and then one with Disadvantage.

The +5 for advantage is acccurate (roughly).

If you roll a d20, you roll ≥ 10, 55% of the time (11 possible rolls out of 20). If you roll with advantage, you roll ≥ 15, 51% of the time (204 rolls out of 400).  

The +5 for disadvantage is a bit inaccurate (Should be 4, roughly).

If you roll a d20, you roll ≥ 10, 55% of the time  (11 possible rolls out of 20). If you roll with disadvantage, you roll ≥ 5, 64% of the time (256 rolls out of 400), you roll ≥ 6 56.25% of the time (225 rolls out of 400).

For super advantage, the bonus is between +6 (roll ≥ 16, 57.8125% of the time) and +7 (roll ≥ 17, 48.80% of the time)

 The formulas are : 

P(highest ≥ n )= 1− ((n−1​)/20)^numberOfDiceRolled

P(lowest ≥ n) = ((21 - n)/20)^numberOfDiceRolled 

I've found the probability of success with advantage to be (2-p)*p and with disadvantage to just be p^(number of dice rolled, usually 2), where p is the probability of succeeding.

p can be worked out by the formula (21+modifier-DC)/20

I have a code in python which does this, as seen below. it is purposed for attacks, rather than ability checks, but it functions pretty much the same. its part of a larger procedure, so some of the variables aren't relevant

Advantage or Disadvantage is ALWAYS better than a bonus or penalty, because rolling dice is FUN and doing math isn't.  (for most anyway)

On an emotional level you're not wrong, though on a mechanical level it is worth pointing out that advantage can't make an impossible thing possible, nor can disadvantage make a possible thing impossible.

https://anydice.com/

All you answerx there...