The bell-curve for Luck

[Bruno]

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%

[Not another shrubbery]

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.

[Bruno]

Originally Posted by Not another shrubbery

Not necessarily... the chance of getting three 18s in a row is only about one in ten million.

Actually, I double checked - since I got an 18 at all in my output, it means that at least one 18 was rolled. I don't register the number in the hash table unless it comes up.

For example, I just tested the "best of 4" scenario and no 17s or 18s ever came up :)

	4x Luck	3x Luck	Regular
3:	1.8375%	1.3832%	0.5%
4:	5.3775%	4.0677%	1.389%
5:	10.0936%	7.8090%	2.740%
6:	14.9343%	12.0218%	4.616%
7:	18.5378%	15.8806%	6.945%
8:	19.2047%	18.2009%	9.736%
9:	14.7819%	16.2288%	11.564%
10:	8.9922%	11.9093%	12.473%
11:	4.2772%	7.2197%	12.509%
12:	1.5167%	3.5351%	11.649%
13:	0.3788%	1.3187%	9.736%
14:	0.0607%	0.3452%	6.920%
15:	0.0069%	0.0706%	4.626%
16:	0.0002%	0.0089%	2.795%
17:		0.0006%	1.391%
18:		0.0000%	0.458%

[Dragondog]

Getting an 18 in best of 4 happens once in about 2.18 billion rolls. And getting 17 is 15 times more likely.

Originally Posted by Bruno

Actually, I double checked - since I got an 18 at all in my output, it means that at least one 18 was rolled. I don't register the number in the hash table unless it comes up.

For example, I just tested the "best of 4" scenario and no 17s or 18s ever came up :)

	4x Luck	3x Luck	Regular
3:	1.8375%	1.3832%	0.5%
4:	5.3775%	4.0677%	1.389%
5:	10.0936%	7.8090%	2.740%
6:	14.9343%	12.0218%	4.616%
7:	18.5378%	15.8806%	6.945%
8:	19.2047%	18.2009%	9.736%
9:	14.7819%	16.2288%	11.564%
10:	8.9922%	11.9093%	12.473%
11:	4.2772%	7.2197%	12.509%
12:	1.5167%	3.5351%	11.649%
13:	0.3788%	1.3187%	9.736%
14:	0.0607%	0.3452%	6.920%
15:	0.0069%	0.0706%	4.626%
16:	0.0002%	0.0089%	2.795%
17:		0.0006%	1.391%
18:		0.0000%	0.458%

[Kuroshima]

Emily, that's not the lazy option, the lazy option is to go to anydice.com and get this:

http://anydice.com/program/c40

Oh, and it does it by probability exploration, not by doing a gazillion trials ;) Oh, and it is fast!

EDIT: Ninjaed by PseudoFenton!

EDIT2: Anydice is just so nice... you can get nice tables, bar graphs, line graphs,... check "graph" and "at most" for the curves for GURPS probabilities

[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.

[Figleaf23]

I find these results remarkable, in that the peak only shifts down to 8, and not all that decisively.

[Bruno]

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

[SCAR]

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.

[bcd]

One important subtlety about how the Luck advantage works (without modifiers) is that when you use it, you have already made a bad roll. Therefore, a more accurate probabilistic analysis would take into account that the first roll failed, and then look at the likelihood of the two next rolls having at least one success. This can get a bit complicated of course since now you can't really ignore what your effective skill was.

The probability curve given in this thread is the one you get if you choose the Active limitation on Luck, which is still a very instructive analysis. :-)