The bell-curve for Luck

[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

[Ts_]

Originally Posted by PseudoFenton

AnyDice.com

O ... M ... G! That is really sweet.

But who said it was fast? I'm still waiting for "ouput [lowest of 300d600 and 300d600]". ;)

Ts

[Peter Knutsen]

Originally Posted by Verjigorm

Could you do that with say, 4 d6, drop the lowest 2, as well as 5d6, drop lowest 2? what about roll 5 and drop the lowest three, roll 6 and drop the lowest three?

If Bruno can't, it's presumably possible to do it in the Troll dice rolling analysis programming lanuage, by computer scientist Torben Mogensen.

[Bruno]

Originally Posted by Verjigorm

Could you do that with say, 4 d6, drop the lowest 2, as well as 5d6, drop lowest 2? what about roll 5 and drop the lowest three, roll 6 and drop the lowest three?

Sure.

Here's 4d6 drop lowest 2

2:	0.08%	0.08%
3:	0.32%	0.40%
4:	1.15%	1.55%
5:	2.47%	4.02%
6:	5.05%	9.07%
7:	8.35%	17.42%
8:	13.28%	30.70%
9:	17.31%	48.01%
10:	20.08%	68.09%
11:	18.76%	86.85%
12:	13.15%	100.00%

5d6 drop lowest 2

3:	0.01%	0.01%
4:	0.07%	0.08%
5:	0.19%	0.27%
6:	0.55%	0.82%
7:	1.15%	1.98%
8:	2.19%	4.17%
9:	3.81%	7.98%
10:	6.05%	14.03%
11:	8.52%	22.54%
12:	11.35%	33.90%
13:	13.49%	47.39%
14:	14.86%	62.24%
15:	14.30%	76.54%
16:	12.05%	88.59%
17:	7.87%	96.46%
18:	3.54%	100.00%

5d6 drop lowest 3

2:	0.01%	0.01%
3:	0.07%	0.08%
4:	0.40%	0.48%
5:	1.03%	1.51%
6:	2.69%	4.20%
7:	5.19%	9.39%
8:	10.01%	19.39%
9:	15.38%	34.78%
10:	21.88%	56.65%
11:	23.69%	80.35%
12:	19.65%	100.00%

6d6 drop lowest 3

3:	0.002%	0.002%
4:	0.01%	0.01%
5:	0.05%	0.07%
6:	0.17%	0.23%
7:	0.44%	0.68%
8:	0.94%	1.62%
9:	1.95%	3.56%
10:	3.63%	7.19%
11:	5.81%	13.00%
12:	8.85%	21.85%
13:	12.08%	33.93%
14:	15.10%	49.03%
15:	16.67%	65.70%
16:	16.19%	81.89%
17:	11.89%	93.78%
18:	6.22%	100.00%

[Dragondog]

Since Bruno's numbers are worked examples, rather than the actual probabilities, they do not always match the probabilities. And since I'm doing this by hand, I'll just show an example.

4d6 drop the two lowest
2: 0.08% 0.08%
3: 0.31% 0.39%
4: 1.16% 1.55%
5: 2.47% 4.02%

Instead of
2: 0.08% 0.08%
3: 0.32% 0.40%
4: 1.15% 1.55%
5: 2.47% 4.02%

[SCAR]

Originally Posted by Bruno

That really illustrates why I do it the "lazy" way - 19 lines of code (counting blank lines for readability), then I just run the stupid thing and stop thinking about it. Five minutes later, results pop out.

Yep, the lazy way is much easier - and at 10million iterations is going to be close enough for most reasonable (rpg) scenarios. (ignoring 300d6 scenarios!)

[SCAR]

Here is a look at Success probabilities with Luck, accounting for succeeding without luck, and using Luck on a Failure.

For a given Target number, the first % column is the standard Success rate on 3d6; the second % column is the Success rate using Luck; the third % column is the % Success Rate of using Luck on an initial failure.

Target   Before Luck     After Luck   (% Successful Luck Usage)
  3       0.4630%         1.3825%       (  0.9238%)
  4       1.8519%         5.4533%       (  3.6694%)
  5       4.6296%        13.2558%       (  9.0449%)
  6       9.2593%        25.2851%       ( 17.6612%)
  7      16.2037%        41.1598%       ( 29.7818%)
  8      25.9259%        59.3558%       ( 45.1303%)
  9      37.5000%        75.5859%       ( 60.9375%)
 10      50.0000%        87.5000%       ( 75.0000%)
 11      62.5000%        94.7266%       ( 85.9375%)
 12      74.0741%        98.2574%       ( 93.2785%)
 13      83.7963%        99.5746%       ( 97.3744%)
 14      90.7407%        99.9206%       ( 99.1427%)
 15      95.3704%        99.9901%       ( 99.7857%)
 16      98.1481%        99.9994%       ( 99.9657%)

Note: These are fully calculate probabilities, not Monte Carlo'ed (large random sample!)

[SCAR]

Originally Posted by SCAR

I have a feeling I did crunch some Luck variants a year or so ago, I'll maybe look that up tomorrow.

It seems I was looking at alternate Luck mechanisms, can't remember exactly why:

I looked at rolling an extra 'Luck' d6 with the main 3d6; and Luck could be used to swap the Luck die for one of the 3d6; it's not as good as standard Luck (although pretty close for a Target number of 3 to 6), but you know before you use Luck if it's going to work for you!

Charted as per standard Luck above:

Target   Before Luck     After Luck   (% Successful Luck Usage)
  3       0.4630%         1.6204%       (  1.1628%)
  4       1.8519%         5.7870%       (  4.0094%)
  5       4.6296%        13.0401%       (  8.8188%)
  6       9.2593%        23.1481%       ( 15.3061%)
  7      16.2037%        35.4938%       ( 23.0203%)
  8      25.9259%        48.7654%       ( 30.8333%)
  9      37.5000%        61.6512%       ( 38.6420%)
 10      50.0000%        73.0710%       ( 46.1420%)
 11      62.5000%        82.4846%       ( 53.2922%)
 12      74.0741%        89.5062%       ( 59.5238%)
 13      83.7963%        94.2901%       ( 64.7619%)
 14      90.7407%        97.2222%       ( 70.0000%)
 15      95.3704%        98.8426%       ( 75.0000%)
 16      98.1481%        99.6142%       ( 79.1667%)

I also tried 2 Luck die, which is better than standard Luck at lower target numbers (3-8), comparable around 10, and a little worse above 10 but you're into 90+% success with luck anyway!)

Target   Before Luck     After Luck   (% Successful Luck Usage)
  3       0.4630%         3.5494%       (  3.1008%)
  4       1.8519%        11.3940%       (  9.7222%)
  5       4.6296%        23.4182%       ( 19.7006%)
  6       9.2593%        37.7058%       ( 31.3492%)
  7      16.2037%        52.5592%       ( 43.3855%)
  8      25.9259%        66.1265%       ( 54.2708%)
  9      37.5000%        77.4563%       ( 63.9300%)
 10      50.0000%        86.0082%       ( 72.0165%)
 11      62.5000%        92.0525%       ( 78.8066%)
 12      74.0741%        95.8591%       ( 84.0278%)
 13      83.7963%        98.0453%       ( 87.9365%)
 14      90.7407%        99.2027%       ( 91.3889%)
 15      95.3704%        99.7299%       ( 94.1667%)
 16      98.1481%        99.9228%       ( 95.8333%)

I don't remember actually using these alternates though!

[PseudoFenton]

Originally Posted by SCAR

I don't remember actually using these alternates though!

Heh, they seem interesting though. I quite like doing the maths behind various exploding and imploding die methods to see how they'd alter the outcome of probabilities - although I can't say I've done this for Luck yet. It'd be fun to draw up a few alternative "Luck" advantages that had different statistical outcomes and mechanics and base the CP cost around the probability distribution accordingly. Fun in an utterly geeky sort of way, of cause.

[SCAR]

Originally Posted by PseudoFenton

Heh, they seem interesting though. I quite like doing the maths behind various exploding and imploding die methods to see how they'd alter the outcome of probabilities - although I can't say I've done this for Luck yet. It'd be fun to draw up a few alternative "Luck" advantages that had different statistical outcomes and mechanics and base the CP cost around the probability distribution accordingly. Fun in an utterly geeky sort of way, of cause.

I think the 3d6+1d6 variant is fairly balanced with standard luck - it's not quite as good, but you know before you use it if it's going to work, so you only use it when it's going to make the difference.

I think I was trying to avoid additional rolling, roll once (3d6+1d6) instead of: roll once then if using luck rolling twice more.