Steve Jackson Games Forums The bell-curve for Luck
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 11-24-2011, 01:01 PM #1 Bruno     Join Date: Sep 2004 Location: Canada The bell-curve for Luck I don't have a clue how to mathematically figure out what the bell-curve looks like for a Luck'd reroll - in this case looking at "roll three times, take the lowest". So I wrote a little perl script and had it roll 10 million times and crunch the results while I was working on other, less interesting XML things. I hope someone else finds this interesting, or possibly even useful. And hey, five minutes later, here are the results (expressed as decimals rather than %ages - 1 = 100%, 0.5 = 50%, etc): 3: 0.013832 4: 0.040677 5: 0.078090 6: 0.120218 7: 0.158806 8: 0.182009 9: 0.162288 10: 0.119093 11: 0.072197 12: 0.035351 13: 0.013187 14: 0.003452 15: 0.000706 16: 0.000089 17: 0.000006 18: 0.000000 Note: I'm fairly sure there were 18s in there, they're just falling below the precision in my output. EDIT: Here it is again expressed as %ages for folks that prefer that: 3: 1.3832% 4: 4.0677% 5: 7.8090% 6: 12.0218% 7: 15.8806% 8: 18.2009% 9: 16.2288% 10: 11.9093% 11: 7.2197% 12: 3.5351% 13: 1.3187% 14: 0.3452% 15: 0.0706% 16: 0.0089% 17: 0.0006% 18: 0.0000% __________________ All about Size Modifier; Unified Hit Location Table A Wiki for my F2F Group A neglected GURPS blog
11-24-2011, 01:09 PM   #2
Not another shrubbery

Join Date: Aug 2004
Re: The bell-curve for Luck

Quote:
 Originally Posted by Bruno 18: 0.000000 Note: I'm fairly sure there were 18s in there, they're just falling below the precision in my output.
Not necessarily... the chance of getting three 18s in a row is only about one in ten million.

 11-24-2011, 01:10 PM #3 Icelander     Join Date: Mar 2006 Location: Iceland* Re: The bell-curve for Luck You want to tell us what this means in terms of rolling this result or lower on a 3d, i.e. whether or not the character succeeds? Adding up all the lower results ought to do it and it would be a much handier table to reference if one didn't have to mentally do that math every time. __________________ Za uspiekh nashevo beznadiozhnovo diela!
 11-24-2011, 01:17 PM #4 Bruno     Join Date: Sep 2004 Location: Canada Re: The bell-curve for Luck Some further commentary: On average using Luck on a skill check improves your absolute odds of success by +15% - but because of the bellcurve the exact improvement strongly depends on what number you're checking against. Similarly, on average using Luck on a skill check is roughly equivalent to applying a +3 to your effective skill... but again that bellcurve means the actual "bonus" equates to everything from +1 to +5 depending on what effective skill you're checking against. Code: ``` Absolute Skill "Bonus" improvement 3 +1 +0.9% 4 +2 +3.6% 5 +2 +8.7% 6 +2 +16.1% 7 +3 +25.0% 8 +3 +33.5% 9 +4 +38.1% 10 +4 +37.6% 11 +4 +32.3% 12 +5 +24.2% 13 +5 +15.8% 14 +4 +9.2% 15 +3 +4.6% 16 +2 +1.8%``` __________________ All about Size Modifier; Unified Hit Location Table A Wiki for my F2F Group A neglected GURPS blog
 11-24-2011, 01:17 PM #5 Figleaf23 Banned     Join Date: Apr 2008 Re: The bell-curve for Luck I find these results remarkable, in that the peak only shifts down to 8, and not all that decisively.
11-24-2011, 01:19 PM   #6
Bruno

Join Date: Sep 2004
Re: The bell-curve for Luck

Quote:
 Originally Posted by Icelander You want to tell us what this means in terms of rolling this result or lower on a 3d, i.e. whether or not the character succeeds? Adding up all the lower results ought to do it and it would be a much handier table to reference if one didn't have to mentally do that math every time.
How's this?

Code:
```	Luck	Regular
3:	1.4%	0.5%
4:	5.5%	1.8%
5:	13.3%	4.6%
6:	25.3%	9.2%
7:	41.2%	16.1%
8:	59.4%	25.9%
9:	75.6%	37.4%
10:	87.5%	49.9%
11:	94.7%	62.4%
12:	98.3%	74.1%
13:	99.6%	83.8%
14:	99.9%	90.7%
15:	99.99%	95.4%
16:	99.9995%	98.2%```
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All about Size Modifier; Unified Hit Location Table
A Wiki for my F2F Group
A neglected GURPS blog

11-24-2011, 01:20 PM   #7
Icelander

Join Date: Mar 2006
Location: Iceland*
Re: The bell-curve for Luck

Quote:
 Originally Posted by Bruno How's this?
Wunderbar. Thanks a bunch.
__________________

11-24-2011, 01:24 PM   #8
Bruno

Join Date: Sep 2004
Re: The bell-curve for Luck

Quote:
 Originally Posted by Figleaf23 I find these results remarkable, in that the peak only shifts down to 8, and not all that decisively.
What it does, along with shifting the peak, is exaggerate the strength of the bellcurve around the peak. The clustering around that point is much more pronounced.

Or to put it another way, on a standard 3d6 roll, 9-11 come up ~36% of the time, but on a 3d6-luck roll, 7-9 come up ~50% of the time.

I'll make a chart, it's kind of cool in a dorky sort of way.

Done:

https://secure.flickr.com/photos/jerril/6396036681
__________________
All about Size Modifier; Unified Hit Location Table
A Wiki for my F2F Group
A neglected GURPS blog

Last edited by Bruno; 11-24-2011 at 01:45 PM.

 11-24-2011, 01:37 PM #9 SCAR   Join Date: Oct 2004 Location: Yorkshire, UK Re: The bell-curve for Luck Rather than random sampling, you could have run all possible combinations of rolls. 3d6 have 216 possible combinations, which with 3 rolls is 10,077,696 total outcomes, just a little more than your 10 million random samples. Or you could have just done the 16x16x16 actual result combinations with the probabilities of each outcome. If I was near a computer I'd run it, but to be honest, your random sample is large enough to be close enough. I have a feeling I did crunch some Luck variants a year or so ago, I'll maybe look that up tomorrow.
11-24-2011, 01:55 PM   #10
Figleaf23
Banned

Join Date: Apr 2008
Re: The bell-curve for Luck

Quote:
 Originally Posted by Bruno What it does, along with shifting the peak, is exaggerate the strength of the bellcurve around the peak. The clustering around that point is much more pronounced. Or to put it another way, on a standard 3d6 roll, 9-11 come up ~36% of the time, but on a 3d6-luck roll, 7-9 come up ~50% of the time. I'll make a chart, it's kind of cool in a dorky sort of way. Done: https://secure.flickr.com/photos/jerril/6396036681
Interesting. Seeing it graphed really adds to an intuitive appreciation for how the effect works out.

It seems like 6 is an interesting break point too. 6 and below have their frequency approximately tripled.

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