Steve Jackson Games Forums Math Gurus - Help with Probability
 Register FAQ Calendar Search Today's Posts Mark Forums Read

 Notices Our primary server had issues at our hosting provider and had to be restored from backup. Posts made after 5/20/2018 11:59PM CDT (UTC -5) and before 5/22/2018 3:00pm CDT may be missing and are not expected to return. Posts made after 5/22/2018 3:00pm CDT (UTC -5) and before 5/23/2018 10:00am CDT may be missing from view until new posts are made to that thread.

 09-20-2017, 10:48 AM #1 Varyon   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 "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. __________________ GURPS Overhaul
 09-20-2017, 11:02 AM #2 PK Assistant GURPS Line Editor     Join Date: Aug 2004 Location: Dobbstown Sane Asylum Re: Math Gurus - Help with Probability Moved to RIG, as this isn't a GURPS topic. __________________ Reverend Pee Kitty of the Order Malkavian-Dobbsian (Twitter) (LJ) MyGURPS: My house rules and GURPS resources. #SJGamesLive: I answered questions about GURPS After the End and more! {Watch Video} - {Read Transcript}
 09-20-2017, 11:49 AM #3 ericthered 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); }``` 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... __________________ Worlds Beyond Earth -- my blog Last edited by ericthered; 09-20-2017 at 11:54 AM.
 09-20-2017, 12:23 PM #4 Anthony   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 __________________ My GURPS site and Blog.
09-20-2017, 12:48 PM   #5
ericthered
Hero of Democracy

Join Date: Mar 2012
Location: far from the ocean
Re: Math Gurus - Help with Probability

Quote:
 Originally Posted by Anthony 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.
__________________
Worlds Beyond Earth -- my blog

09-20-2017, 02:19 PM   #6
Anthony

Join Date: Feb 2005
Location: Berkeley, CA
Re: Math Gurus - Help with Probability

Quote:
 Originally Posted by ericthered 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.
__________________
My GURPS site and Blog.

09-20-2017, 05:01 PM   #7
Varyon

Join Date: Jun 2013
Re: Math Gurus - Help with Probability

Quote:
 Originally Posted by ericthered 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 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 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.
__________________
GURPS Overhaul

 09-20-2017, 05:31 PM #8 Anthony   Join Date: Feb 2005 Location: Berkeley, CA 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 ]" __________________ My GURPS site and Blog.
09-20-2017, 06:57 PM   #9
Varyon

Join Date: Jun 2013
Re: Math Gurus - Help with Probability

Quote:
 Originally Posted by Anthony 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!
__________________
GURPS Overhaul

09-21-2017, 09:06 AM   #10
ericthered
Hero of Democracy

Join Date: Mar 2012
Location: far from the ocean
Re: Math Gurus - Help with Probability

Quote:
 Originally Posted by Anthony 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.
__________________
Worlds Beyond Earth -- my blog

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Fnords are Off [IMG] code is Off HTML code is Off Forum Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home Illuminati Headquarters     SJ Games Discussion     Daily Illuminator     Forum Feedback and Help     MIB Demo Events Warehouse 23     Warehouse 23 General Discussion     Warehouse 23 Digital     Pyramid Munchkin     Munchkin 101     Munchkin     Munchkin Collectible Card Game     Other Munchkin Games Roleplaying     Dungeon Fantasy Roleplaying Game         DFRPG Resources     GURPS         GURPS Resources     Transhuman Space     Traveller     The Fantasy Trip     In Nomine     Roleplaying in General     Play By Post Board and Card Games     Car Wars     Ogre and G.E.V.         Ogre Video Game         Ogre Scenarios     Board and Dice Games     Card Games     Miniatures The Network     GURPS Character Assistant     GURPS Vehicle Builder     GURPS Character Builder The Gnomes of Zurich     The Industry     Conventions     Trading Post     Gamer Finder

All times are GMT -6. The time now is 08:34 PM.