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Old 09-20-2017, 11:48 AM   #1
Varyon
 
Join Date: Jun 2013
Default 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 "1d-3, maximum 1."

The probability for 1d is easy. 2-5 are just the normal 1/6, 1 and 6 are 1/12, -4-0 and 7-11 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 semi-manually 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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Old 09-20-2017, 12:02 PM   #2
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Default Re: Math Gurus - Help with Probability

<MOD> Moved to RIG, as this isn't a GURPS topic. </MOD>
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Old 09-20-2017, 12:49 PM   #3
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Default 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);
}
Yes, its steps up in series of three, going outwards from the 3-4 center, with the third element being half the value of the other two, and each set of three decreasing in likelihood by 6.

Now we can deal with that infinite series you lined up for us to calculate...
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Old 09-20-2017, 01:23 PM   #4
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Default 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
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Old 09-20-2017, 01:48 PM   #5
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Default Re: Math Gurus - Help with Probability

Quote:
Originally Posted by Anthony View Post
I'm confused by the exploding mechanism; I would expect a second explosion to be 1d+6 (minimum 9).

If exploding dice go up by 6 instead of 3, A becomes 4.8 and the final average is 3.8
Arg! you're right, I calculated for the second explosion to be 1d+6 (minimum 9), and going up and down by 3's. There go those numbers...

He's exploding in both directions, so the average is as for normal dice.
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Old 09-20-2017, 03:19 PM   #6
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Default Re: Math Gurus - Help with Probability

Quote:
Originally Posted by ericthered View Post
Arg! you're right, I calculated for the second explosion to be 1d+6 (minimum 9), and going up and down by 3's. There go those numbers...

He's exploding in both directions, so the average is as for normal dice.
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.
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Old 09-20-2017, 06:01 PM   #7
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Default Re: Math Gurus - Help with Probability

Quote:
Originally Posted by ericthered View Post
Now we can deal with that infinite series you lined up for us to calculate...
If a stopping point makes things easier, assume each die can only explode twice. Anything more than that I consider to be sufficiently unlikely to occur and thus ignorable. If it helps, here's the probabilities of each individual die roll.
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
With infinite explosions, -5 and 12 would actually be 1/432 each, of course.

Quote:
Originally Posted by Anthony View Post
I'm confused by the exploding mechanism; I would expect a second explosion to be 1d+6 (minimum 9).
Err... right. Sorry, bit of a brainfart when I was writing that. You are correct - the pattern is +3, +6, +9, and minimum 6, 9, 12, for 1, 2, and 3 explosions, respectively.

Quote:
Originally Posted by ericthered View Post
He's exploding in both directions, so the average is as for normal dice.
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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Old 09-20-2017, 06:31 PM   #8
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Default Re: Math Gurus - Help with Probability

If you want to play around, here's an anydice. If you want to try adding dice up, change the output statement to something like "output 2d[ dexplode d6 ]"
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Old 09-20-2017, 07:57 PM   #9
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Default Re: Math Gurus - Help with Probability

Quote:
Originally Posted by Anthony View Post
If you want to play around, here's an anydice. If you want to try adding dice up, change the output statement to something like "output 2d[ dexplode d6 ]"
This? This is exactly the sort of thing I was looking for. Thanks Anthony, you're awesome!
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Old 09-21-2017, 10:06 AM   #10
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Default Re: Math Gurus - Help with Probability

Quote:
Originally Posted by Anthony View Post
If you want to play around, here's an anydice. If you want to try adding dice up, change the output statement to something like "output 2d[ dexplode d6 ]"
Yep, that's the way to do it.

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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