The bell-curve for Luck

[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. :-)

[Grouchy Chris]

Originally Posted by 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".

That's very simple. The probability of rolling at or below a target number T is the same as 1 minus the probability of rolling above T. So the probability of getting at least one roll at T or below is 1 minus the odds of exceeding T on all rolls.

So, if p is the probability of rolling T or less, the probability of getting T or less at least once in 3 rolls is 1-(1-p)^3.

(As usual when I'm working with probabilities, I checked this with my wife, who teaches college statistics.)

Using that formula, your probability for at least one success, expressed at a percentage, are:

	Luck	Regular
3:	1.38%	0.5%
4:	5.45%	1.8%
5:	13.26%	4.6%
6:	25.29%	9.2%
7:	41.16%	16.1%
8:	59.36%	25.9%
9:	75.59%	37.4%
10:	87.50%	49.9%
11:	94.73%	62.4%
12:	98.26%	74.1%
13:	99.57%	83.8%
14:	99.92%	90.7%
15:	99.99%	95.4%
16:	99.9994%	98.2%

So your method agrees with my calculations to three decimal places.

However, you usually use luck when you've already failed once (edit: bcd is a ninja), so it's usually more useful to ask what the odds of succeeding with the two rolls you have left are. The formula in this case is 1-(1-p)^2.

	Luck	Regular
3:	0.92%	0.5%
4:	3.67%	1.8%
5:	9.04%	4.6%
6:	17.66%	9.2%
7:	29.78%	16.1%
8:	45.13%	25.9%
9:	60.94%	37.4%
10:	75.00%	49.9%
11:	85.94%	62.4%
12:	93.28%	74.1%
13:	97.37%	83.8%
14:	99.14%	90.7%
15:	99.79%	95.4%
16:	99.97%	98.2%

[Ts_]

I guess I could check the book myself, but doesn't Luck give you 4 rolls in total? 1 failed original roll and 3 rerolls?

So, I'd rather look at the curve for 4 rolls, exaggerating it even further. This is your chance when you're willing to use Luck. (How often will you then actually use the advantage? Easy: Failure rate at effective skill.)

Ts

[Bruno]

Originally Posted by 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 confess my approach comes from a combination of

• ignorance (I know how to roll dice, record the result, and get some interesting answers out of it, theoreticals of dice make my brain hurt)
• stubborn pride (I rewrote a dice-parsing module for my in-progress DF8 table roller, I'm damn well going to find uses for it)
• familiarity (I've worked on that DF8 roller enough that this is a very familiar approach).

Fortunately my computer at home and at work are good enough that I can do a substantial statistical sampling conveniently. If I were at home, I'd have based this on my multithreaded mass-dice-roller that I have there, and then since I'd be using all six cores, I probably would have kicked it up to 1k million rolls since it would still run pretty quickly. It's easy to just slap a few zeroes onto the end of my stop condition. It's hard to figure out probability.

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.

Cool.

[bcd]

It seems a bit unfair to give probabilities for Luck on the premise "the first roll has already failed", side by side with probabilities for normal use without that premise. Properly therefore, the "Regular" column should be all zeros to be comparable. :P

[Bruno]

Originally Posted by Ts_

I guess I could check the book myself, but doesn't Luck give you 4 rolls in total? 1 failed original roll and 3 rerolls?

Nope.

Originally Posted by Basic Set

Luck: Once per hour of play, you
may reroll a single bad die roll twice
and take the best of the three rolls!

Originally Posted by Ts_

So, I'd rather look at the curve for 4 rolls, exaggerating it even further. This is your chance when you're willing to use Luck. (How often will you then actually use the advantage? Easy: Failure rate at effective skill.)

I'm running the 10 million for a 4 roll scenario now. I'll get back to you in 5-10 minutes.

[Bruno]

Originally Posted by bcd

It seems a bit unfair to give probabilities for Luck on the premise "the first roll has already failed", side by side with probabilities for normal use without that premise.

Good job I didn't do that then. :)

What I did, was show you what "best of three rolls" looks like. The first roll could be the best. These are not skill checks, these are the curve for "best of three rolls".

The actual luck usage case is, as you've noted, different. I think it might actually skew things further towards the end.

[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%

[Ts_]

Originally Posted by Bruno

Nope.

Oh. Again, I'm completely amazed at what is actually written in those books. Carry on with your 3 rolls in total then! No need for the 4th.

[Verjigorm]

How do you people do this stuff?!