Statistical probabilities involved in Luck and Active Luck?

[Calvin]

It's been several years since I last did a university-level statistics course, so I've been having a little trouble figuring out how to approach this problem, but I'm trying to work out how the Luck advantage impacts the statistical probability of a successful roll.

The way I see it, there are two related but different problem sets. The answer to which would be some sort of curve plotted on the 3-18 scale.

1. How does Active Luck, as in, luck that must be activated prior to the roll you want to use Luck on, impact the curve?

2. How does non-Active luck impact the curve? So of the three rolls you get, one has already failed by the time you decide to use luck. But there's the corollary that you aren't going to use non-active luck unless you need it.

I tried to mess around with anydice.com to find the solution, but while it can do 4d6 drop lowest, it's not really set up for this type of problem.

I don't even necessarily need an answer, though I'm not going to complain if handed one, what I'm really looking for is what principle I need to apply to find my answer.

[AlexanderHowl]

In principle, the probabilities are the same when Luck is an advantage, it is the application that varies. Luck is usually triggered in the case of a critical failure on the part of the luck user or a critical success on the part of the luck user's opponent. Active Luck is used when the user really needs to succeed or when the user really needs their opponent to fail.

The changes occur when Luck is an abilitity. Active Luck may be modified with Talent (and, since it is not an attack, Reliable), meaning that a superhuman with Probability Control 4 [20] could take an ability that was Ridiculous Luck (Active, -40%; Reliable, +10, +50; Super, -10%) [60]. Once every 15 minutes of play, they may take two extra rolls on an action and give a bonus of +14. In that case though, I would also take an ability that was Ridiculous Luck (Aspected, Combat, -20%; Defensive, -20%; Super, -10%) [30] to prevent unexpected critical hits.

[Anders]

Oh, I would not allow Reliable on Luck. Is that canon?

[Calvin]

Originally Posted by AlexanderHowl

In principle, the probabilities are the same when Luck is an advantage, it is the application that varies. Luck is usually triggered in the case of a critical failure on the part of the luck user or a critical success on the part of the luck user's opponent. Active Luck is used when the user really needs to succeed or when the user really needs their opponent to fail.

The changes occur when Luck is an abilitity. Active Luck may be modified with Talent (and, since it is not an attack, Reliable), meaning that a superhuman with Probability Control 4 [20] could take an ability that was Ridiculous Luck (Active, -40%; Reliable, +10, +50; Super, -10%) [60]. Once every 15 minutes of play, they may take two extra rolls on an action and give a bonus of +14. In that case though, I would also take an ability that was Ridiculous Luck (Aspected, Combat, -20%; Defensive, -20%; Super, -10%) [30] to prevent unexpected critical hits.

I'm not certain I understand what you mean here, particularly the first paragraph as it makes assumptions about how players use Luck. The assumptions aren't wrong, they're just not always right. I'm also not certain that Active Luck can be modified by a talent like that.

What I'm talking about is specifically a bell curve like this: https://anydice.com/program/1 but with the application of pre-roll luck (Three die sets) and post-roll luck (Two die sets, maybe some other fiddling since it's known that you're only using it against guaranteed failures.)

[thrash]

Presuming that you are rolling to beat a set target value:

1. Determine the probability of success for a single roll (e.g., from B171).
2. Subtract that value (expressed on the range 0 to 1, rather than as a percent) from 1, to get the probability of failure on any single roll.
3. Multiply the probabilities of failure for each roll together, to get the probability of failure on all rolls. (For two identical rolls, this means to square the probability of failure; for three, cube it.)
4. Subtract the result from 1 to get the combined probability that at least one of the rolls will be successful.

Does that help?

[NineDaysDead]

Originally Posted by Anders

Oh, I would not allow Reliable on Luck. Is that canon?

Yes, but...

Originally Posted by Reliable, Powers page 109

This works exactly like Talent, and can’t affect anything that Talent wouldn’t affect. It’s cumulative with Talent, where applicable.

Originally Posted by Luck, Powers page 59

Talent gives a bonus to any die roll influence by Luck – but only if that use of Luck is declared in advance.

Given that your first roll might be really good and you've just wasted your Luck and have to wait for it to recharge, this doesn't seem to unbalanced.

[NineDaysDead]

Originally Posted by AlexanderHowl

Active Luck may be modified with Talent (and, since it is not an attack, Reliable),

I don't think you need to take the active limitation to add Talent/Reliable, but you only get the benefit if you declare in advance. Active just means you have no choice about declaring in advance.

[Celjabba]

Originally Posted by thrash

Presuming that you are rolling to beat a set target value:

1. Determine the probability of success for a single roll (e.g., from B171).
2. Subtract that value (expressed on the range 0 to 1, rather than as a percent) from 1, to get the probability of failure on any single roll.
3. Multiply the probabilities of failure for each roll together, to get the probability of failure on all rolls. (For two identical rolls, this means to square the probability of failure; for three, cube it.)
4. Subtract the result from 1 to get the combined probability that at least one of the rolls will be successful.

Does that help?

And here are the probabilities, unless I made a mistake :

Target 	1 roll best of 2	best of 3
3	0,46%	0,92%	1,38%
4	1,85%	3,67%	5,45%
5	4,63%	9,04%	13,26%
6	9,26%	17,66%	25,29%
7	16,20%	29,78%	41,16%
8	25,93%	45,13%	59,36%
9	37,50%	60,94%	75,59%
10	50,00%	75,00%	87,50%
11	62,50%	85,94%	94,73%
12	74,07%	93,28%	98,26%
13	83,80%	97,37%	99,57%
14	90,74%	99,14%	99,92%
15	95,37%	99,79%	99,99%
16	98,15%	99,97%	100,00%
17	98,15%	99,97%	100,00%
18	98,15%	99,97%	100,00%

[Anthony]

As far as passive luck goes, it has the same probabilities as active luck, except that it isn't consumed if you succeed at the first roll (this is more complicated in situations where you don't know what a success is). That makes it twice as good as active luck if your modified skill is 10, and 54 times as good if your modified skill is 16.

[martinl]

Originally Posted by Calvin

2. How does non-Active luck impact the curve? So of the three rolls you get, one has already failed by the time you decide to use luck. But there's the corollary that you aren't going to use non-active luck unless you need it.

I worked this out in this thread.