Second, in my ACKs campaign I have a character who leads and army but I have only two companions. I decided to recruit the soldier of my army who has the highest charisma... but how can you calculate that?

In other words, how do you calculate the stats value of the character who has the highest stat of a group? What if i want to meat the most intelligent mage in a tower, or challenge the strongest warrior of an army to arm wrestling or, in this case, I want to recuit the soldier with the highest charisma.

I came up with a chart for that:

18

( 1 - (0.99)^n ) *100

17

( (0.99)^n - (0.97)^n )*100

16

( (0.97)^n - (0.95)^n )*100

15

( (0.95)^n - (0.91)^n )*100

14

( (0.91)^n - (0.84)^n )*100

Lets n be the number of people in the group, ^n means "to the Nth power"

Once you replace n in the chart, you roll a d100, the result shows the exact stat of the character with highest value.

For example, in a village with 100 inhabitants, the chart would be:

18

63

17

31

16

04

15

00

14

00

If I'm looking for the smartest person in the town and I roll a 54, that NPC has an intelligence stat of 17.

I will try to explain how I did chart and the math behind it, in the mid time, what do you think...?

So the probability of any given statblock having an 18 in a given stat rolled on 3d6 is 1/216. The probability that someone in a group of n people has that stat at 18 is 1 minus the probability that nobody has the stat at 18, which is (215/216)^n, so we get 1-(215/216)^n as the probability that someone has an 18. Doors Thief seems to have approximated the probability of an 18 at 1/100 instead of 1/216, which is kinda reasonable if you’re working with a population that selects for that stat (strength in the army, int at wizard college, …).

I’m less clear on what was done for stats lower than 18, but I think it’s “the probability that anyone has a score greater than or equal to x, minus the probability that anyone has a score greater than x”. But again, Doors Thief’s probabilities don’t quite line up with what I’d expect on 3d6.

Well, I had a bad time trying to find the "sum of three d6" distribution, so Ihad to approach it with a Normal continuous distribution with a medium value of 10.5 and standard error 2.959729.

The normal distribution tell us the probability of a result equal or lower (working with numbers ending up in 0.5; 17.5, 16.5, etc because we were trying to calculate a discrete distribution with a continuous one). Then, by raising the result to the n power, we have the possibility of having one result equal or lower in many distribution.

18 or less has a 1 of probability, exactly 18 would be 18 minus the probability of 17 or less. You get the chart by repeating the process.

A lot of the efficiency was lost because the Normal distributions uses numbers like 0.9965, I decided to use only the first two digits because it sums up the idea and keeps the chart relatively easy to use, calculate 1/216 raise to n is more efficient than 0.99, but is also more difficult and i would take more time than the appropriate. I think that the difference is not so significant, but I will upload the original chart when I find it.

I tried really hard to find a way of doing it in a discrete distribution like 1/216, but i couldn't find the cumulative distribution of the sum of three d6.

I don’t entirely understand the probability involved (my probability level is ‘I took a class on it ten years ago’), but this might help in finding the distribution of various dice.

[quote="Aryxymaraki"]
I don't entirely understand the probability involved (my probability level is 'I took a class on it ten years ago'), but this might help in finding the distribution of various dice. http://www.anydice.com Tell it "output 3d6" (without the quotes) and it will show you a variety of graphs, tables, and distributions for the sum of 3d6. (Now that I'm actually looking at the site, the 'at least' tab seems like it might be relevant here.)
[/quote]

That site is awesome... thanks!

Now that i check it, I also made a huge mistake in how to use chart, the correct one with the results in www.anydice.com would be

18

99.54^n

17

98.15^n

16

95.37^n

15

90.74^n

14

83.80^n

A character would have 18 if you rolled (99.54^n) or higher, 17 if you rolled a number between (99.54^n) and (98.15^n) and so on... You calculate the probability of rolling the number that you are looking for minus one, calculate the probability of rolling that number at most and then you calculate the n power. If you roll a higher number in a d100, you found the stat that you were looking for.

Neat! I'm always gleeful to see someone talking about distributions. Let's try making this into a slightly more usable format though. So, here's a simple chart for ingame use where you can just ask "How many dudes are hanging out?" and then know how strong the strongest dude is.

Ability Score

Minimum Population

18

300

17

100

16

50

15

15

I used a 75% as the breaking point (So, technically, one in four 300 man armies will not have an 18 strength guy) and rounded a bit to produce nice numbers, but in general, this will probably be be easier to use at the table.

Note that all the fun math aside, this hinges on the idea that every NPC rolls for stats using the same method PCs do, which I probably wouldn't do.

FWIW, a table actually entitled "How many dudes are hanging out?" seems like a game with a lot of

sleeveless shirts

motorcycles

patrick swayzes & sam elliots

baseball bats

machine guns

ninjas

dinosaurs

...and this is really going towards some sort of mashup of Dr McNinja and Axe Cop and the like?

Feels like a game that should exist. Autarch knows demographics, so all we have to do is base everything on "who is the awesomist" as opposed to "who is the richest" and all of a sudden there's a dudeconomy.

I think a reasonable question to ask is "how do the players determine which soldier or villager has the highest mental stat"?

If outsiders came to a village of 100 and asked for the most intelligent or wisest person, I imagine they would be directed toward a chief, shaman, or elder, regardless of their actual intelligence. Those people are not necessarily the smartest, and may have gained their posts from politicking or heritage. The brilliant 17-INT man that the players want may well be decorating pots in an unremarkable hut at the edge of the village, because even if he believed himself to be the smartest, merely claiming to be smarter than the elders could put his reputation and safety in real danger.

If you go to an armed group and ask for the strongest warrior, they'd probably give you the highest-level one unless you clarified that you want raw physical power and not talent or lethality. But you could probably determine the highest strength score with reasonable objectivity via a simple weight-lifting contest.

In a mage tower or academic setting, you might run into similar problems as with the village. A 24 year old arcanist (level 1) with 18 INT and few friends might not want to question the smarts of the egotistical 16 INT, level 9 wizard who runs the tower and practically owns his whole life. A more humble leader may be willing to concede that he isn't the smartest and present his favorite underling instead (again this person is not necessarily the smartest), but even that is far from a guarantee in anything resembling academia. You'd need some kind of objective measure to sort them out, and even then the lower mages may be pressured to perform worse so that they don't risk the ire of their bosses.

This suggests that most attempts to determine someone else’s stats should yield a result modified by their charisma mod, and probably some level mod (reputation?)