Condensing Multiple Rolls with a Chart? [Math Help Needed]

[GreatWyrmGold]

I decided I'd like to hack together a quick mass combat system, one for fights between groups too small and individuals too varied for the default mass combat system to work well (e.g. supervillain gangs). I figured that a good first step would be to have a way to simulate multiple rolls with one.
For instance, if someone made 10 attacks with skill 12 in a given timeframe, they would hit (on average) with seven or eight of them, but would hit with all 10 roughly 5% of the time. Since there's a roughly 5% chance of rolling 3-5 on 3d6, you could make it so rolling 3-5 means they hit with all ten...and, conversely, that 16-18 means they miss with all ten. And so on for every number between 1 and 9 as well.

But, of course, I want a way to easily calculate that sort of table for any combination of skill level and number of rolls. Ideally, I'd like a big chart, with 3-18 on both edges, with the cells containing the percentage of rolls which succeed for a given die roll and skill level.
It seems like the kind of thing there should be a simple formula I can plug into each square to figure out; stick in the probabilities for rolling each number on the die, multiply or exponent them in the right way, and fill the table. But I haven't been able to figure out what.
Or if that's not an option, I guess other options for not-quite-mass-combat rules could be helpful. But this is the kind of problem that keeps bugging certain types of people until they have a solution, and I am one of those types...even though probability isn't really my forte. I hope I didn't nerd-snipe anyone.

[jeff_wilson]

Originally Posted by GreatWyrmGold

But, of course, I want a way to easily calculate that sort of table for any combination of skill level and number of rolls. Ideally, I'd like a big chart, with 3-18 on both edges, with the cells containing the percentage of rolls which succeed for a given die roll and skill level.

3D6	1 roll	2 rolls	3	4	5	10	15	20	3D6
3	0.005	0.000	0.000	0.000	0.000	0.000	0.000	0.000	3
4	0.019	0.000	0.000	0.000	0.000	0.000	0.000	0.000	4
5	0.046	0.002	0.000	0.000	0.000	0.000	0.000	0.000	5
6	0.093	0.009	0.001	0.000	0.000	0.000	0.000	0.000	6
7	0.162	0.026	0.004	0.001	0.000	0.000	0.000	0.000	7
8	0.259	0.067	0.017	0.005	0.001	0.000	0.000	0.000	8
9	0.375	0.141	0.053	0.020	0.007	0.000	0.000	0.000	9
10	0.500	0.250	0.125	0.063	0.031	0.001	0.000	0.000	10
11	0.625	0.391	0.244	0.153	0.095	0.009	0.001	0.000	11
12	0.741	0.549	0.406	0.301	0.223	0.050	0.011	0.002	12
13	0.838	0.702	0.588	0.493	0.413	0.171	0.071	0.029	13
14	0.907	0.823	0.747	0.678	0.615	0.378	0.233	0.143	14
15	0.954	0.910	0.867	0.827	0.789	0.622	0.491	0.387	15
16	0.981	0.963	0.945	0.928	0.911	0.830	0.755	0.688	16
17	0.995	0.991	0.986	0.982	0.977	0.955	0.933	0.911	17
18	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	18

How does this look? If you need a probability figure for a number of rolls not given, like 17 rolls of 13 or less, use the figures given for the same number of rolls added together only multiply the probability figures given. 10 rolls of 13 or less has a probability of 0.171, 5 rolls of 13- has a probability 0.413, and 2 rolls of 13- has probability 0.702. 0.171 * 0.413 * 0.702 = 0.496 That's ~0.500, so you could roll 10- once to see if they were succesfull.

[GreatWyrmGold]

Okay, I am bad at explaining my ideas. So let's try taking a simpler example and seeing if I can explain it better with that.

Let's say that instead of GURPS, we have a game where you roll d4's for success. (Like GURPS, the lower the better.) And we're also rolling a d4 for the simplified roll.
If you need to roll a 1 for success, you would expect to succeed about 25% of the time. However, if 100 people like that rolled for success, you wouldn't get 25 people rolling successes at the same time; you would get on average 25% of them rolling successes, but each individual lot would vary.
Let's say that there was a 3/4 chance of getting at least 15 successes out of a hundred , a 2/4 chance of getting at least 25 successes, and a 1/4 chance of getting at least 35 successes. Then, when representing this chance of success as a d4, might be something like:
1—5%
2—15%
3—25%
4—35%
When rolling a d4 and getting a 2, 15% of those hundred rolls (ie, 15, because we picked nice round numbers) would succeed, and 85 would fail. And if the probabilities are right, on average, about 25% of the rolls would be successes.

The quick averagers in the audience will note that this does not actually average out to a 25% rate of success. This is because the probabilities I cited are a load of hooey. I'm trying to figure out what the right probabilities to put in the actual list of probabilities, for both skill 1 and skills 2-4. Only with GURPS instead of the example system.
Does that help people understand the question I'm trying to ask?

[smurf]

A quick fix is to use the ROF rules. First work out the average to hit roll and then add a bonus to x attacks (similar to the ROF rules) and for every one point under that to hit roll is another hit.

[Anthony]

Originally Posted by GreatWyrmGold

Does that help people understand the question I'm trying to ask?

Assuming you have n independent tests at probability p, the average result will be np, and the standard deviation will be sqrt( n * p * (1-p)). It doesn't take a very large value of n before the curve is visually very close to a standard bell curve.

However, I don't recommend doing this, because, while GURPS mostly treats the tests as independent, they aren't.

[Anaraxes]

Originally Posted by GreatWyrmGold

Let's say that i... you roll d4's for success. If you need to roll a 1 for success, you would expect to succeed about 25% of the time. However, if 100 people like that rolled for success, you wouldn't get 25 people rolling successes at the same time; you would get on average 25% of them rolling successes, but each individual lot would vary.

The math jargon for the thing you describe is a "Bernoulli trial" (also "binomial"). There are lots of calculators online you could use to build a table to get k successes (or at least k, or less than k) in N trials.

https://www.wolframalpha.com/example...noulli-trials/
https://stattrek.com/online-calculator/binomial.aspx

I'd probably structure the table as a cumulative one, so you could roll percentiles once and then read up to the largest value your rolls beats, and take that as the number of successes in that batch of rolls.

[GreatWyrmGold]

Originally Posted by smurf

A quick fix is to use the ROF rules. First work out the average to hit roll and then add a bonus to x attacks (similar to the ROF rules) and for every one point under that to hit roll is another hit.

I suppose that's a possibility, but the number of successes would drop off drastically with more rolls. If we take the "Shots" column as "Number of people shooting at once" (remember, this is intended for a mid-mass combat system), then the difference between 1-5 people in a group shooting and 9-12 (a difference of roughly 2x to 10x) would be the same as between a group of 9-12 and 17-24 (roughly 1.5x to 2.5x). That really doesn't feel right. Not to mention that the number of average/expected hits doesn't even come close to scaling linearly with the number of people shooting, which makes a lot more sense for one guy firing X bullets than for X guys firing one bullet (each).

Originally Posted by Anthony

Assuming you have n independent tests at probability p, the average result will be np, and the standard deviation will be sqrt( n * p * (1-p)). It doesn't take a very large value of n before the curve is visually very close to a standard bell curve.

That sounds like "No, it still sounds like you just want to learn the probability of rolling any given number, which is provided in the GURPS rulebook among other places so I don't know why you'd need that".
Hm.

However, I don't recommend doing this, because, while GURPS mostly treats the tests as independent, they aren't.

...Huh? Which tests?

Originally Posted by Anaraxes

The math jargon for the thing you describe is a "Bernoulli trial" (also "binomial"). There are lots of calculators online you could use to build a table to get k successes (or at least k, or less than k) in N trials.

https://www.wolframalpha.com/example...noulli-trials/
https://stattrek.com/online-calculator/binomial.aspx

I'd probably structure the table as a cumulative one, so you could roll percentiles once and then read up to the largest value your rolls beats, and take that as the number of successes in that batch of rolls.

That sounds like a good idea! I'll try it out on my previous, simplified example. I'll fiddle around with the second calculator and find success numbers for trials that get the percentages I'm looking for, which should average out right.
Table of raw numbers, table of percentages. A column from the latter (minus the row for number of trials) is closest to what I'd want to put in column 1 of my hypothetical chart. However...which column? The percentage for rolling a 1 doesn't change much from 10 trials to 1,000 (20%->not quite 24%), but the percentage of successes you'd get for rolling a 4 drops from 40% to barely over 26%.
...Hm. Starting to see some problems with my methodology.