Math question about bonus/penalty dice

[overgeeked]

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]

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]

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%.

 

[Helpful NPC Thom]

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]

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.

 

[overgeeked]

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]

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

How does rolling two d20 and taking the higher affect the average outcome?

D&D 5E has an "advantage" concept where instead of rolling 1d20, you roll 2d20 and take the higher. Likewise, disadvantage means rolling 2d20 and taking the lower. How does this affec...

rpg.stackexchange.com

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.

So the advantage from advantage is [1-1/x^2 + (2n/x^2)-(n^2/x^2)]-[(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]

The math on a given number N exact on advantage is (2N-1)/(sides²)
N on disadvantage is (2(sides+1-N)-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.
So, the 10 on d10 is 19/100 Advantage, and 1/100 disad
The 9 is 17 & 3
the 8 is 15 and 5
the 7 is 13 and 7
and so on.