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.

[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.

[Bruno]

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.

	 	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%

[Figleaf23]

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

[Bruno]

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?

	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%

[Icelander]

Originally Posted by Bruno

How's this?

Wunderbar. Thanks a bunch.

[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.

[Figleaf23]

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.