
Alan, "Son of Hett"
United States The Triangle North Carolina

I conducted a search of the forums last week and finally found someone's post with a handy link to an AnyDice function for what I want. Alas #1, I did not bookmark it (and I do not have the patience to conduct that search all over again). However, alas #2, it does not matter because AnyDice times out before it ever completes the task I have in mind.
So, the basic idea is this: for any given roll of 3d6 there is a 1:216 chance of rolling an 18 (also for 3, but let us focus on one target).
What I want to know is how many times does 3d6 have to be rolled to approach a 1:1 chance of rolling 18?
More specifically, I want the formula for that calculation (because I am examining several kinds of "die" and want to test each).
Background: I do not like flat dice probabilities. I prefer triangular or bellshaped graphs. However, I recognize that players make a limited number of rolls during a campaign (that number varying wildly, of course, depending on system, length of campaign, average length of sessions, etc., etc.; nonetheless…), therefore I do not want to choose a mechanic that would make it highly unlikely or impossible for a player to roll a maximum (or minimum) during a reasonable amount of real time of play.


M. B. Downey
United States Suitland Maryland

You have a 215/216 chance of not getting an 18, and you multiply that by 215/216 for each additional roll. So just multiply that until you have a less than 0.5% chance of rolling no 18s.


M. B. Downey
United States Suitland Maryland

Quick math says 1024 times gets you at a 99.2% chance of rolling an 18, so somewhat more than that.


M. B. Downey
United States Suitland Maryland

1142 rolls gets you to 99.500498% chance of rolling at least one 18.


M. B. Downey
United States Suitland Maryland

So your formula is:
(215/216)^X=0.005


M. B. Downey
United States Suitland Maryland

downeymb wrote: So your formula is:
(215/216)^X=0.005
In an easier way to solve and for the more general case:
X = log 0.005 where the base of the log is (1  probability of rolling the number you want)


Alan, "Son of Hett"
United States The Triangle North Carolina

Oh sweet, but I could not figure out how to change the base for the log function on my calculator; so I did a search on the idea of your equation (how to solve for X if X is an exponent) and that produced
X=log(p)÷log(n) where X is the number of rolls to have a (1p%) chance of rolling the maximum roll; n is the odds of not rolling the maximum roll (on any given roll); and log is the natural log (base10)
Wow, so the 99.99% target for 3d6 is 1985 rolls; what is that, like three years of playing time? Obviously 18s get rolled, so I suppose the real answer I am looking for is somewhere between 216×N% and 1985, but I think that still makes the formula useful.
Thank you for your assistance!


William Hostman
United States Alsea OR
I've been Banished to Oregon... Gaming in Corvallis, living in Alsea... Need gamers willing to try new things...
The Splattered Imperium

Note that things get much uglier when using alternate patters of 3kept dice.
4d6k3 (pretty standard old school), you have to work out the aggregate cases... 18 is 3 sixes kept. Given dice ABCD... All 4 = 6's (1/6)^4: 1 permutation: 1/1296 3 6's and a non6 (1/6)^3 * 5/6 each: 4 permutations ABC ABD ACD BCD = 4* 5/1296 = 20/1296 So 21/1296 per roll, reducing to 7/432
We need the odds of NOT rolling it, tho.so (425/432)^x ≤ 0.05. Iterating, I get x=184 on 4d6k3
Using the 3d6k3, 1 permutation, and iterating (215/216)^x ≤ 0.05... x ≥ 646


Alan, "Son of Hett"
United States The Triangle North Carolina

aramis wrote: Note that things get much uglier when using alternate patters of 3kept dice. … Using the 3d6k3, 1 permutation, and iterating (215/216)^x ≤ 0.05... x ≥ 646 I had assumed that choosing from a pool would require fewer rolls for maximum success than the straight roll, which you have kindly shown to be true.
For the record, I am actually looking at combinations of d3s, d4s, and fate dice to figure out the range, curve, and probabilities I think will work best for what I am trying to design. I do not want the outer points of the range to be so rare as to frustrate players but not so frequent as to feel swingy.


John "Omega" Williams
United States Kentwood Michigan

JVgamer wrote: Background: I do not like flat dice probabilities. I prefer triangular or bellshaped graphs. However, I recognize that players make a limited number of rolls during a campaign (that number varying wildly, of course, depending on system, length of campaign, average length of sessions, etc., etc.; nonetheless…), therefore I do not want to choose a mechanic that would make it highly unlikely or impossible for a player to roll a maximum (or minimum) during a reasonable amount of real time of play.
Some solutions that do not require such mathematics.
A: Stat array: In AD&D onward is 15, 14, 13, 12, 10, 8. This is what 5e suggests and what you use for AL play. Or using a base 3d6 the array is something like 13, 12, 11, 10, 9, 8.
B: Point system: 5e for example allots 27 points to spend with the stats starting at 8. A 15 in a stat costs 9 points.




