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RPG» Forums » General Discussion » RPG Design

Subject: Dice Probability: no. of rolls to 'guarantee' maximum rss

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Alan, "Son of Hett"
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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 bell-shaped 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.
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M. B. Downey
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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.
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M. B. Downey
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Quick math says 1024 times gets you at a 99.2% chance of rolling an 18, so somewhat more than that.
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M. B. Downey
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1142 rolls gets you to 99.500498% chance of rolling at least one 18.
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M. B. Downey
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So your formula is:

(215/216)^X=0.005
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M. B. Downey
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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)
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Alan, "Son of Hett"
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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 (1-p%) 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 (base-10)

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!
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William Hostman
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Note that things get much uglier when using alternate patters of 3-kept 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 non-6 (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

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Alan, "Son of Hett"
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aramis wrote:
Note that things get much uglier when using alternate patters of 3-kept 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.
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John "Omega" Williams
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JVgamer wrote:
Background: I do not like flat dice probabilities. I prefer triangular or bell-shaped 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.
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