# Math question about bonus/penalty dice

#### overgeeked

##### B/X Known World
So big-brain math types, help me out, please. With D&D 5E, advantage on a d20 roll is worth about a +5. So in d100 games, like Call of Cthulhu, about how much of a benefit are bonus/penalty dice? If you're not sure what bonus/penalty dice are, they're advantage/disadvantage on d100 rolls. You roll 2d10 for the tens place, taking the better/worse one, depending.

I tried to goof around with anydice to answer this myself, but it didn't work out.

So about what kind of bonus that is? Anyone know?

#### Blue Orange

##### Hero
Advantage on a d20 is worth about a +3 on average, it's just higher in the middle of the range. It counts most when you need it, while still allowing for unlikely results, which I suspect is why they adopted it so much in 5e.

It's actually advantage on the tens only, then. The tens die is 0-9. Which means it's modeled pretty well by "output [highest 1 of 2d10]-1" on anydice, which winds up giving you mean of 6.15 and SD of 2.35. Given the mean is usually 4.5, the advantage is about +1.65; given that it's the tens die, it would be about a +16 on average.

But how about in that middle of the range where it's most effective? Well, as per anydice you have a 84% chance of rolling at least a 4, versus 60% for a straight roll, whereas for a 5 it's 75%, versus 50% for a straight roll, and for a 4 it's 64%, versus 40% for a straight roll. So it winds up being about a +25% bonus in the middle of the range.

#### overgeeked

##### B/X Known World
Advantage on a d20 is worth about a +3 on average, it's just higher in the middle of the range. It counts most when you need it, while still allowing for unlikely results, which I suspect is why they adopted it so much in 5e.

It's actually advantage on the tens only, then. The tens die is 0-9. Which means it's modeled pretty well by "output [highest 1 of 2d10]-1" on anydice, which winds up giving you mean of 6.15 and SD of 2.35. Given the mean is usually 4.5, the advantage is about +1.65; given that it's the tens die, it would be about a +16 on average.

But how about in that middle of the range where it's most effective? Well, as per anydice you have a 84% chance of rolling at least a 4, versus 60% for a straight roll, whereas for a 5 it's 75%, versus 50% for a straight roll, and for a 4 it's 64%, versus 40% for a straight roll. So it winds up being about a +25% bonus in the middle of the range.
Awesome, thanks.

It's interesting that it's about the same benefit. Advantage on a d20 is a +3 to a +5, or about +15-25%. Advantage on the tens digit of a d100 roll is about +15-25%.

It's interesting that it's about the same benefit. Advantage on a d20 is a +3 to a +5, or about +15-25%. Advantage on the tens digit of a d100 roll is about +15-25%.
A d20 is just a d100/5.

#### Blue Orange

##### Hero
That is a good point, and I wondered about that.

If you roll a d4 with advantage (1e DM being generous with wizard HP?), you have a 100% chance of rolling at least a 1 (no change), a 94% (rounded) chance of rolling at least a 2 (versus 75), a 75% chance of rolling at least a 3 (versus 50), and a 44% chance of rolling at least a 4 (versus 25). Here it ranges from 20-25%. So it looks like advantage is somewhat stronger for smaller dice, but the effect is small.

A d20 is just a d100/5.
The probability distribution's slightly different though--the d100 is more granular. Also the mean of a d20 is 10.5, the mean of a d100 is 10.1=50.5/5.

Last edited:

#### overgeeked

##### B/X Known World
A d20 is just a d100/5.
Yes, I'm aware. That wasn't very helpful, NPC Thom.
That is a good point, and I wondered about that.

If you roll a d4 with advantage (1e DM being generous with wizard HP?), you have a 100% chance of rolling at least a 1 (no change), a 94% (rounded) chance of rolling at least a 2 (versus 75), a 75% chance of rolling at least a 3 (versus 50), and a 44% chance of rolling at least a 4 (versus 25). Here it ranges from 20-25%. So it looks like advantage is somewhat stronger for smaller dice, but the effect is small.

The probability distribution's slightly different though--the d100 is more granular.
Right. Which is why I assumed advantage would be different, mathematically, on the d10 vs the d20. I wonder if that range of benefit holds for the rest of the dice. At a guess I'd say yes it's probably in that same range as it only moves to +15-25% when you get to a d20. Weird.

#### Blue Orange

##### Hero
This actually has a statistical concept (it's an order statistic), and someone has gone after this before me:

P(rolling at least n)= 1-((n-1)/20)^2 = 1-(1/400)(n^2-2n+1) = 399/400+(n/200)-(n^2/400).
Compare that with a regular d20 roll, where
P(rolling at least n)= 1-(n-1)/20 = 1-(1/20)(n-1) = 19/20-n/20.

Theoretically we should be able to sub in x for 20, giving us
P(rolling at least n on dx with advantage) = 1-((n-1)/x)^2 = 1-1/x^2 + (2n/x^2)-(n^2/x^2)
P(rolling at least n on dx) = (1-1/x)-n/x.

You can simplify a little to (1/x-1/x^2) + (2n/x^2)-n/x -(n^2/x^2) = ((x-1)/x^2) + ((2n-nx)/x^2) -(n^2/x^2) =
((x-1 + 2x-nx-n^2)/(x^2)) = ((3-n)x-1-n^2)/(x^2).

Where you go with that is your own best guess.

#### aramis erak

##### Legend
The math on a given number N exact on advantage is (2N-1)/(sides²)

So, for example: 20...
N = 20, A(N)=(40-1)=39/400, and D(N) = (2(20+1-20)-1)/(20²) = 2(1)-1/400 = 1/400
For a given roll of X+, just sum the values for X to sides.

(the visual proof is on an episode of mathologer for this week, tho' it's part of a proof in a different algorithm.
For the d6, the advantage results look like this:
6 B(N)=1/6, A(N)= 11/36, D(N) = 1/36
5 B(N)=1/6, A(N)= 9/36, D(N) = 3/36
4 B(N)=1/6, A(N)= 7/36, D(N) = 5/36
3 B(N)=1/6, A(N)= 5/36, D(N) = 7/36
2 B(N)=1/6, A(N)= 3/36, D(N) = 9/36
1 B(N)=1/6, A(N)= 1/36, D(N) = 11/36
And the visual proof
 Advantage 6 5 4 3 2 1 6 6 6 6 6 6 6 5 6 5 5 5 5 5 4 6 5 4 4 4 4 3 6 5 4 3 3 3 2 6 5 4 3 2 2 1 6 5 4 3 2 1

For the odds of, say, 3+ on advantaged d6 (5+7+9+11)/36 or 32/36, or 8/9; the 2- is (3+1)/36. or 1/9.

The same methodology works for any arbitrary 2 dice.
The 9 is 17 & 3
the 8 is 15 and 5
the 7 is 13 and 7
and so on.

Replies
4
Views
4K
Replies
35
Views
2K
Replies
9
Views
846
Replies
9
Views
836
Replies
54
Views
2K