Mathematically Confirmed: Advantage is NOT +5

 Last year I wrote a blog post pointing out that the D&D rulebooks are WRONG when they say that "Advantage" is equivalent to +5. It's actually equivalent to just +3.25. 

A couple days ago, the excellent youtube channel "Stand-Up Maths" did a video on exactly this topic, and came to the same conclusion I had (though he proceeded to push the math much further because, y'know, he's a mathematician, and I'm not). 

 The average roll of 2d20k1, aka "d20 with Advantage", is just 13.825.

https://www.youtube.com/watch?v=X_DdGRjtwAo 

This is most important for DMs when planning adventures, but can also be really critical for PCs who have some reliable way of achieving advantage on a common die roll. If you use the recommended short-hand equivalency that the rulebook suggests and think of advantage as if it were +5, you will over-estimate the power of the PCs and generally make things much harder for the players than you intended.

The video also helpfully provides formulas for similar rolls of various dice sizes.

Average roll of n-sided die "with advantage" is  ((1/6n) x (n+1) x (4n-1))
So average of 2d6k1 is 4.472  (1/36  x 7 x 23)
So average of 2d20k1 is 13.825 (1/120 x 21 x 79)
The average of any 2k1 roll is roughly 2/3 the number of sides. The larger the die, the closer they get to that 2/3rd limit.

Average of 3dnk1 is ((1/4n) x (n+1) x (3n-1))
So 3d6k1 is 4.9583 ((1/24) x (7) x (17))
So 3d20k1 is 15.4875 ((1/80) x (21) x (59))
The average of any 3k1 roll is roughly 3/4 the number of sides.

Average of any 4k1 is roughly 4/5 the number of sides.

Average of xdnk1 is  ((x/(x+1))n +1/2) where x is number of dice rolled, and n is number of sides on each die. So 5dNk1 is 5/6 N, 6dNk1 is 6/7 N, etc. It's a really cool formula that makes it super easy to figure out the average roll on some seemingly complicated dice pools.

Advantage Math (It's not +5)

 I'm sure this is posted elsewhere on the internet, but I'm putting it here so I can remember so I don't have to do the math again if I forget it. In D&D 5th, Advantage is roughly equal to +3.325 on your roll on average, and Disadvantage works out to -3.325 on your roll on average.

That said, for typical difficulties of your character level in a thing your PC is built to be good at, the effect of Advantage is probably functionally better than that +3.325 suggests. Example: if you need to roll a natural 11+ (just counting the die, not considering modifiers) to hit, you'd normally hit 50% of the time, but with Advantage you'd be hitting 75% of the time. Which feels like it's a +5 bonus (a +25% success chance, since each +1 on a linear d20 = +5% success rate) in that situation, and if you look online you'll see people talk about it as if +5 were accurate.

I'd argue that +5 equivalency is really more of an illusion or an ideal, than an actual mechanical reality. You can't count on it reliably. Those people are just wrong.

I mean, the "it's practically +5" notion is predicated on a very middle-of-the-bell-curve target number. Aside from the fact that a roll of 1d20 isn't an actual bell curve, you also can't really expect to have the ideal mathematically balanced scenario show up at the table with any consistency. Yes, if you built your character really well, and your GM is very careful about the Difficulties they set and Monsters you encounter, then, sure, pretty often you'll find Advantage pays off roughly like it were a +5 boost to your average. But not every time. The moment the DM picks an above-average challenge for your climactic final encounter (or, for that matter, a real low-ball DC because they want you to succeed and some minor hurdle), or rolls on a Random Encounter Table, or the situation causes you to try something "outside the box" rather than what your PC is actually good at, this whole notion of it being equal to +5 is out the window. 

Lets say the GM puts a high DC on something, and you need a natural 18+ on the die to succeed. Your base chance of success is 15%. If you have Advantage, your chance of success goes to 27.75%. So that's just a hair better than a flat +2 boost. On the rolls where it really matters because you're trying to do the impossible, thinking of Advantage as being +5 will make you badly overestimate your odds of success. It's not only more accurate, but also safer, to think of it as if it were +3.

This fact is of greatest importance for GMs. If you plan out your scenarios thinking Advantage is +5, you'll end up making things impossibly hard for the PCs by mistake, and do so fairly often.

 

UPDATE: The excellent youtube channel "Stand-Up Maths" has done an a video on exactly this topic. So if you're reading this and you think "rbbergstrom's out of his gourd, as the PHB says advantage is the equivalent of +5 and the PHB wouldn't lie to me", you should watch this video:  https://youtu.be/X_DdGRjtwAo  As he points out and explains, the average roll of a d20 with advantage is 13.825. In other words, just +3.25 more than the normal 10.5 average of a single d20 roll.