3d6 opposed roll probability

[Rethalgamon]

Hey guys. I've been looking at making the switch in my homebrew from 1d20 to 3d6, per the section in Unearthed Arcana. I'm building an excel spreadsheet for "playtesting", and I was wondering if the formula I've got now, which calculates opposed d20 rolls, would work for 3d6 rolls, if I just changed the numbers around; i.e., is the math the same save for the numbers?

The formula I'm using now found in a wonderful PDF about opposed rolls specifically, which I can no longer locate . It's basically:
(20-R)*(20-R-1)/(20^2)
where R is the difference of the first roller's roll modifier and the second roller's roll modifier.

My hangup is that "1"; I don't know a lot about statistics/probability, so I'm worried that my formula will come out wrong/be incorrect if I leave it as a 1 and not as a 3. I'm assuming that the "20" and the "1" in the formula represent the maximum and minimum range for the die rolls; the range with 3d6 is, of course, 3-18, and not 1-20, so I'm thinking changing the formula to:
(18-R)*(18-R-3)/(18^2)
would work, but I just need someone with a little more experience with probability to verify/correct me. I appreciate the help, guys!

 

[MerricB]

It won't work. There are very big differences in the maths. A d20 gives a linear result, but 3d6 is a bell-curve, and the intersections become wacky.

I'll see what I can dig up.

 

[Warbringer]

So, that formula doesn't look correct: try 1-(((20-R)*(20-R+1))/(2*20*20)), where R= a-b and a>b

This only works for the for 1dY (eg 1d20) as the probability steps between each result are equal, eg its linear (as MerricB pointed out). Probabilities of the form XdY (eg 3d6) are mutinominal distributions. However, good news, easy to program a simulator in excel. The below if for 2000 simulations.

Note how at lower values of R the difference has more impact with 3d6 vs d20, peaking at +7 then decaying as R approaches 20. Basically, small bonuses are more important on 3d6.

R 3d6 d20
1 55% 54%
2 64% 56%
3 73% 62%
4 79% 66%
5 86% 70%
6 89% 74%
7 94% 78%
8 96% 81%
9 98% 82%
10 99% 86%
12 100% 91%
14 100% 94%
16 100% 97%
18 100% 99%
20 100% 100%

 

[billd91]

Hey guys. I've been looking at making the switch in my homebrew from 1d20 to 3d6, per the section in Unearthed Arcana. I'm building an excel spreadsheet for "playtesting", and I was wondering if the formula I've got now, which calculates opposed d20 rolls, would work for 3d6 rolls, if I just changed the numbers around; i.e., is the math the same save for the numbers?

The formula I'm using now found in a wonderful PDF about opposed rolls specifically, which I can no longer locate . It's basically:
(20-R)*(20-R-1)/(20^2)
where R is the difference of the first roller's roll modifier and the second roller's roll modifier.

My hangup is that "1"; I don't know a lot about statistics/probability, so I'm worried that my formula will come out wrong/be incorrect if I leave it as a 1 and not as a 3. I'm assuming that the "20" and the "1" in the formula represent the maximum and minimum range for the die rolls; the range with 3d6 is, of course, 3-18, and not 1-20, so I'm thinking changing the formula to:
(18-R)*(18-R-3)/(18^2)
would work, but I just need someone with a little more experience with probability to verify/correct me. I appreciate the help, guys!

I'll reiterate that the formula won't work. The denominator (20^2) works for d20 because there are 400 outcomes of rolling two d20s - 20 for the first x 20 for the second = 400. For 3d6 rolls, there are 216 outcomes for just one set alone. When comparing two sets of 3d6 rolls, there are 216^2 (or 6^6) or 46,656 distinct outcomes for those dice. Clearly 18^2 won't cut it. That would work if you were rolling a pair of 1d18 dice that had a uniform probability of every result coming up.

To illustrate a bit more, if you were to list all of the outcomes of a 3d6 roll, you could roll 1,1,1 for a value of 3; 6,6,6 for a value of 18; 1,2,1 or 1,1,2, or 2,1,1 for a value of 4, and so on. If you were to actually figure out the numbers, you'd get a table something like this:



Value__#_____probability of rolling that value
3______1_____0.00462963
4______3_____0.013888889
5______6_____0.027777778
6_____10_____0.046296296
7_____15_____0.069444444
8_____21_____0.097222222
9_____25_____0.115740741
10____27_____0.125
11____27_____0.125
12____25_____0.115740741
13____21_____0.097222222
14____15_____0.069444444
15____10_____0.046296296
16_____6_____0.027777778
17_____3_____0.013888889
18_____1_____0.00462963

Accounting for all possible outcomes of rolling 3d6, there is exactly one way of getting a 3. There are three ways to get a 4, six ways to roll a 5, ten ways to roll a 6, and so on. At the highest point on the bell curve, you've got 27 ways to come up with a 10 or an 11. So you can see how rolling multiple dice to generate a number really changes the distribution compared to rolling a d20 in which there is exactly one way to come up with any outcome on that die and they all have the same probability of 1/20 = .05.

 

[Tuft]

There is a very nice dice probability calculator at http://anydice.com

Just put this as two separate lines into the big input field:
output 3d6-3d6
output d20-d20
and press first the "Graph" and then the "Calculate" button.

This will give you a nice comparison graph of the probabilities of these two opposed rolls.


Edit: this is a direct link to the above calculation:
http://anydice.com/program/2806/graph

 

[N'raac]

3d6 is the Hero System model, so you may find some d20 vs 3d6 discussions on those boards. There were some in the past, but they had a site problem some months back, so they may not all be there any more.

The bell curve trends to the middle much more than d20, as others have noted. Adding in the opposed roll aspect will deviate from the Hero structure as well, but the "3d6 vs d20" discussions may still provide some useful info.

 

[Starfox]

If it is a bell curve you want, using opposed d20 rolls should give you a nice pyramidal probability distribution. The more dice you add, the more bell-curved the probability distribution is, with results very strongly centered in the middle. I find that 2d6 works very well for me, and by a similar token I think 2d20 (opposed if you like, it really changes nothing) would give you a pretty good curve, centered but not too centered.

 

[Cadence]

There is a very nice dice probability calculator at http://anydice.com

Great link, thanks! I think you just saved me custom coding some stuff up for a friend using a VtM-like rolling system.