09202017, 10:48 AM  #1 
Join Date: Jun 2013

Math Gurus  Help with Probability
So, we've got a lot of people on these boards who are pretty good with more complex probability problems (or, failing that, know how to make a quick automated system to work it out for them). So, I have an exploding d6 scheme that I like more than the typical. Rather than rolling a 6 meaning you roll again and add that to 6, I have it as rolling a 6 translates into "Roll 1d+3, minimum 6," where this roll can similarly explode (becoming "1d+9, minimum 12," and so forth). This also works on the other side, where a roll of 1 becomes "1d3, maximum 1."
The probability for 1d is easy. 25 are just the normal 1/6, 1 and 6 are 1/12, 40 and 711 are 1/36 each, and so forth. I have no idea how to work out the probabilities for higher numbers of dice  particularly, 3d6, so I can determine if I want to replace the standard success roll with this exploding variant, and where the criticals should be. I actually worked this out before, going IIRC out to only 1 "explosion" (so using dice from 5 to 9, for 3d results from 15 to 27), but did so by semimanually working out the probabilities of each result and looking at the trend. I probably made a mistake somewhere in there, and also I've lost said spreadsheet, so I can't determine how well the rules I came up with* would actually work. Thus, I'm here asking for assistance. *For those curious, you always need at least MoS 5 for a Critical Success, and must roll a 1 or lower for skill below 10. Higher skills have a higher threshold  2 or lower for skill 10, 3 or lower for skill 13, 4 or lower for skill 16, and 5 or lower for skill 19  this trend doesn't continue (5 or lower is the best you can get to). Critical Failure always occurs with MoS 10, a roll of 20 or higher is a Critical Failure with MoS 5 or worse, and a roll of 24 or higher is a Critical Failure with MoF 1 or worse. Any roll of 20 is also always at least a Failure, if not a Critical Failure. There is no "always succeeds" threshold.
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09202017, 11:02 AM  #2 
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Join Date: Aug 2004
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Re: Math Gurus  Help with Probability
<MOD> Moved to RIG, as this isn't a GURPS topic. </MOD>
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09202017, 11:49 AM  #3 
Hero of Democracy
Join Date: Mar 2012
Location: far from the ocean

Re: Math Gurus  Help with Probability
Ok, so I have a function that will give probability for any single die roll:
Code:
function probability(a){ var b = Math.abs(a 3.5)+.5; return Math.pow(6,Math.ceil(b/3))/ (b%3==0?2:1); } Now we can deal with that infinite series you lined up for us to calculate...
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09202017, 12:23 PM  #4 
Join Date: Feb 2005
Location: Berkeley, CA

Re: Math Gurus  Help with Probability
I'm confused by the exploding mechanism; I would expect a second explosion to be 1d+6 (minimum 9).
For the simple exploding die with a minimum value of 3, the average value A is equal to (3 + 3 + 3 + 4 + 5 + (A+3))/6, or 3.5 + A/6, or 5/6A = 3.5, or A = 4.2. As the first die does not have a minimum of 1, its actual average is (1 + 2 + 3 + 4 + 5 + (A+3))/6, or 3 + A/6, or 3.7. If exploding dice go up by 6 instead of 3, A becomes 4.8 and the final average is 3.8 
09202017, 12:48 PM  #5  
Hero of Democracy
Join Date: Mar 2012
Location: far from the ocean

Re: Math Gurus  Help with Probability
Quote:
He's exploding in both directions, so the average is as for normal dice.
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09202017, 02:19 PM  #6 
Join Date: Feb 2005
Location: Berkeley, CA

Re: Math Gurus  Help with Probability
Ah right, missed that. Yes, assuming they both work the same way the average will be 3.5 and the explosion details don't matter.

09202017, 05:01 PM  #7  
Join Date: Jun 2013

Re: Math Gurus  Help with Probability
Quote:
Code:
Result Probability 5 1/216 4 1/216 3 1/216 2 1/72 1 1/36 0 1/36 1 1/12 2 1/6 3 1/6 4 1/6 5 1/6 6 1/12 7 1/36 8 1/36 9 1/72 10 1/216 11 1/216 12 1/216 Quote:
Indeed. I'm trying to get a handle on exactly what the spread looks like (I know the curve gets flattened out a bit) to decide if this is something I'd like to actually use in a game or not.
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09202017, 06:57 PM  #9  
Join Date: Jun 2013

Re: Math Gurus  Help with Probability
Quote:
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09212017, 09:06 AM  #10  
Hero of Democracy
Join Date: Mar 2012
Location: far from the ocean

Re: Math Gurus  Help with Probability
Quote:
For the record, the infinite series is actually very managable: in every case it boils down to SUM(6^ni) where n is the number of dice and i goes from 1 to infinity. And that's the best known infinite series there is. And then you have another number in front I didn't figure out the pattern for, and you possibly have to cut out a few middle terms. So yeah, use any dice.
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