11-24-2011, 12:01 PM | #1 |
Join Date: Sep 2004
Location: Canada
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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%
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11-24-2011, 12:09 PM | #2 | |
Join Date: Aug 2004
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Re: The bell-curve for Luck
Quote:
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11-24-2011, 12:10 PM | #3 |
Join Date: Mar 2006
Location: Iceland*
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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.
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11-24-2011, 12:17 PM | #4 |
Join Date: Sep 2004
Location: Canada
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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%
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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, 12:17 PM | #5 |
Banned
Join Date: Apr 2008
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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.
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11-24-2011, 12:19 PM | #6 | |
Join Date: Sep 2004
Location: Canada
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Re: The bell-curve for Luck
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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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11-24-2011, 12:20 PM | #7 |
Join Date: Mar 2006
Location: Iceland*
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Re: The bell-curve for Luck
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Za uspiekh nashevo beznadiozhnovo diela! |
11-24-2011, 12:24 PM | #8 | |
Join Date: Sep 2004
Location: Canada
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Re: The bell-curve for Luck
Quote:
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
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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 12:45 PM. |
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11-24-2011, 12:37 PM | #9 |
Join Date: Oct 2004
Location: Yorkshire, UK
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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, 12:55 PM | #10 | |
Banned
Join Date: Apr 2008
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Re: The bell-curve for Luck
Quote:
It seems like 6 is an interesting break point too. 6 and below have their frequency approximately tripled. |
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