# D&D GeneralVideo on the math behind advantage (and Elven Accuracy)

#### Charlaquin

##### Goblin Queen (She/Her/Hers)
Just had this video pop up in my recommended feed and thought some folks here might appreciate it.

Summary: Mathematician works out the average result of an unmodified d20 roll with advantage (turns out its 13.833), then extrapolates to create a formula for doing the same for a die with any number of sides. Then does the same for roll 3 and take the best - functionally elven accuracy (for d20s it’s about 15.5). Then further extrapolates for taking the best of any number of rolls.

Interesting stuff, but I think we can all agree that the most valuable takeaway here is that the average result of rolling an infinity sided die with advantage is 2/3 (roughly 0.667).

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#### CharlesWallace

##### enworld.com is a reminder of my hubris
Just had this video pop up in my recommended feed and thought some folks here might appreciate it.

Summary: Mathematician works out the average result of an unmodified d20 roll with advantage (turns out its 13.833), then extrapolates to create a formula for doing the same for a die with any number of sides. Then does the same for roll 3 and take the best - functionally elven accuracy (for d20s it’s about 15.5). Then further extrapolates for taking the best of any number of rolls.

Interesting stuff, but I think we can all agree that the most valuable takeaway here is that the average result of rolling an infinity sided die with advantage is 2/3 (roughly 0.667).
Wow- he’s really charismatic! That was a great video. I lasted until hypercubes and then had to bail. I’m barely functional in a 3D world. Try as I might, I can’t grok 4D!

#### Olaf the Stout

##### Hero
I love Matt Parker. He's awesome.

#### Rune

##### Once A Fool
I love Matt Parker. He's awesome.
His stand-up is pretty good, too. If you’re into spreadsheets.

#### Blue

##### Ravenous Bugblatter Beast of Traal
I need to watch the video (but doubt I will clear half an hour to do so tonight), but don't get confused about what average means with a non-normalized distribution. Especially when your end-goal is just a boolean pass/fail for a threshold.

There's a 51% of rolling a 15 or higher. There's a 75% chance to roll a 10 or higher. Considering the rolls needed to hit, those are pretty impressive.

So if you normally have a 65% chance to hit (8+), you now have an 84% chance to hit. But if you applied GWM and now had a 40% chance to hit (13+), with advantage you have a 64% chance to hit. Basically offset the entire -5 penalty.

Chance to roll the target number or higher on a die.

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#### Charlaquin

##### Goblin Queen (She/Her/Hers)
Well, that’s just the mean, which is far from the whole story. For example, this also demonstrates that 20 is the mode! (or for folks who don’t speak math, the most likely result). The overall impact of advantage depends on what number you need on the d20 to succeed, and it’s at its most impactful when you need a 10 or 11. That’s where it’s actually worth the “about +5” people often shorthand it to. The further your target number is above or below that, the less Advantage matters.

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#### Blue

##### Ravenous Bugblatter Beast of Traal
Well, that’s just the average result of an unmodified advantage d20 roll. How much of an impact advantage has depends on what number you need on the die to succeed.
This is very true, but with bounded accuracy the threshold of what you need to roll usually is closer to the middle that 1 or 20 - and that's where you have the biggest gains. 6-14 as your target numbers have bigger improvements than 1-5 or 15-20.

#### Charlaquin

##### Goblin Queen (She/Her/Hers)
I need to watch the video (but doubt I will clear half an hour to do so tonight), but don't get confused about what average means with a non-normalized distribution. Especially when your end-goal is just a boolean pass/fail for a threshold.

There's a 51% of rolling a 15 or higher. There's a 75% chance to roll a 10 or higher. Considering the rolls needed to hit, those are pretty impressive.

So if you normally have a 65% chance to hit (8+), you now have an 84% chance to hit. But if you applied GWM and now had a 40% chance to hit (13+), with advantage you have a 64% chance to hit. Basically offset the entire -5 penalty.

Chance to roll the target number or higher on a die.
View attachment 252477

Oh, yeah, absolutely. The mean is actually not all that useful to know in the context of D&D. Which is why the person who asked him the question did so in the first place - all the data they were actually able to find online was specifically about D&D, so it was all spreadsheets and graphs that show how various bonuses and target numbers are affected. But they just wanted to know what the mean was and couldn’t find it anywhere.

#### Charlaquin

##### Goblin Queen (She/Her/Hers)
This is very true, but with bounded accuracy the threshold of what you need to roll usually is closer to the middle that 1 or 20 - and that's where you have the biggest gains. 6-14 as your target numbers have bigger improvements than 1-5 or 15-20.
Yeah, I edited my post to clarify that.

#### EzekielRaiden

##### Follower of the Way
As noted, this is an issue of considering mean difference, rather than typical difference. If we think of advantage from the perspective of the lowest die, then if you rolled a 1 advantage will (on average) give you +9.5. If the "lowest" die is 20 (meaning you double-crit), your average gain is zero. Over the range of values most likely to occur, however, the net change in probability is about 20-30 percentage points, hence, most people gloss the benefit as roughly +5.

Wow- he’s really charismatic! That was a great video. I lasted until hypercubes and then had to bail. I’m barely functional in a 3D world. Try as I might, I can’t grok 4D!
Oh yeah, Matt's great. I'm a big fan of both of the presentations he's given at the Royal Institution, on his two books, Things To Make And Do In The Fourth Dimension and Humble Pi: A Comedy of Maths Errors. Sadly I haven't read either book yet but someday, if I have actual money, I might snag a copy of each.

I love Matt Parker. He's awesome.
Indeed.

#### Olaf the Stout

##### Hero
His stand-up is pretty good, too. If you’re into spreadsheets.
I have spreadsheets for so much of my stuff!

#### overgeeked

##### B/X Known World
Interesting stuff, but I think we can all agree that the most valuable takeaway here is that the average result of rolling an infinity sided die with advantage is 2/3 (roughly 0.667).
I watched that earlier today, at least up until he started talking about roll 3, keep 1.

It is really interesting that the average result of roll 2, keep the highest 1 will always be 2/3 of the max result of that die type.

#### Charlaquin

##### Goblin Queen (She/Her/Hers)
I watched that earlier today, at least up until he started talking about roll 3, keep 1.

It is really interesting that the average result of roll 2, keep the highest 1 will always be 2/3 of the max result of that die type.
Well, 2/3 if the max result plus 0.5. And If you’d kept watching, it turns out the average result of roll 3, keep the highest is always 3/4 of the max result plus 0.5. And the average of roll 4, keep the highest is always 5/6 of the max result plus 0.5. And we know that the average result of a single die roll is always 1/2 of the max result plus 0.5. So, he conjectures (though doesn’t prove) that the average result of rolling an n sided die m times and keeping the highest is always n*(m/(m+1)+0.5)

#### Paul Farquhar

##### Legend
That's assuming the dice is numbered with the complete set of integers from 1 to n.

But it is provable, it was probably just too long for video.

#### Charlaquin

##### Goblin Queen (She/Her/Hers)
That's assuming the dice is numbered with the complete set of integers from 1 to n.
Yes.
But it is provable, it was probably just too long for video.
Oh, cool!

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