Recently, it seems like I've been spending an inordinate amount of time publishing ad hoc web articles about probability in RPGs, from some basic stuff to some not-so-basic stuff. Partially based on the reaction I've got from these, I am contemplating writing a "probability for roleplayers" type book (probably a free PDF).

I'm thinking it will be about half theory/general principles, half specific examples from particular (openly licensed) games, with the examples getting progressively more complicated. I have a pretty decent list of the topics I would want to see in such a book. I also have a pretty clear idea of how I want it to look.

I'm curious about some things though, which hopefully this community can answer:

• Would you even read such a book?
• What about probability do you wish you knew more about?
• Are there particular examples of situations or mechanics in specific games that you've wondered how the probability a) works and b) can be figured out? If so, what are they?

I'd like to know how to derive from first principles the distribution of "roll 4 drop lowest" and similar distributions with comparisons built in. That would cover re-rolls as well as some parry mechanics. Congrats on the reign probs, I kept turning them into nonsense, so it's great to finally have a chance to dig out my old calcs and see where I went wrong. Every other answer I have seen was brute force.

I would certainly read such a book, yes. If it covered how to decide between aggregate and discrete events, all the better.

Slight aside, but I always liked the basic 'pyramid' chart to illustrate probalility in the Settlers of Catan rulebook. Stuff like that for different distributions would be useful and visually appealing.

I'm comfortable with basic probability. A lot of my designs use funky dice systems that are far less easily modeled.

For example, in Verge, you roll a variable pool of d6es and look for the largest grouping of like numbers. The count of that grouping is what determines your score. So if you roll 7d6 and get 1112556, then your result is 3 (three 1's). THEN, you can reroll the dice you don't keep, Yahtzee style. So you keep 111 and reroll 2556 and maybe get 1456. You move the matching 1 over so you have a result (signal) of 4 (1111) and garbage (noise) of 456.

So the variables are: number of dice rolled, number of noise rerolls. Constants are die size (d6).

Is there a formula f(dice, rerolls, result) that tells me the chance of that result occurring? For example, f(1,0,n) = 1/6, for n = {1..6}.

I use Professor Torben Mogensen's "Troll" probability modeling tool to figure this stuff out computationally. Programming Verge dice isn't trivial, but it's possible. You might consider discussing this tool in your book.

Tsanuri wrote:
Slight aside, but I always liked the basic 'pyramid' chart to illustrate probability in the Settlers of Catan rulebook. Stuff like that for different distributions would be useful and visually appealing.

Adam wrote: Is there a formula f(dice, rerolls, result) that tells me the chance of that result occurring? For example, f(1,0,n) = 1/6, for n = {1..6}.

Yeah, I've taken college classes in combinatorics, but it's been a while.  P = 1 - !P is pretty rudimentary. ;)

I understand the probabilities of exploding dice. The bit of your post that is helpful (and really, the crux of my problem) is the pointer towards an approach that doesn't involve infinite regression. More of that, please.

The catch with Verge dice is that the exploding part is rerolling the useless dice, not rerolling a useful die. The "signal" grows as you reroll and match more dice to it from the "noise" (which diminishes in dice). So if you roll 2d6, there's a 1/6 (6/36) chance of getting a match, and a 5/6 (30/36) chance of not getting a match. In the case that you do not get a match, there's a 1/6 chance of matching on a reroll and a 5/6 chance of not matching.

So you'll NOT get 2 matching dice 5/6 + (1/6*5/6) of the time. You'll always get at least 1 matching die, by default.

So for 2d6 with 1 reroll max, you have a 30/36 + 5/36 = 35/36 chance of not getting 2 matching dice, or rather a 1/36 chance of matching 2 dice.

This gets harder with two dice. How do you approach generating tables of such things? How do you go about generalizing the formula from the case-by-case walkthrough like I did above?

Wordman,
Just to respond to your initial post, I would definitely be interested in reading such a book as described, particularly if it were a free PDF.  I've actually been looking for something along the lines of "Probability and Game Design" for a while now, though I haven't found it yet. Short on time now, but I'll try to come back later and contribute further, particularly some 'prob probs' I've been curious about.

Cheers,

J

Adam wrote: How do you go about generalizing the formula from the case-by-case walkthrough like I did above?

I have a decent understanding of probability and I would enjoy reading a book that was specifically RPG related. I had thought about writing something similar but my probability tends to be more "brute force" (lots of tables) rather than formula-based and I'd love to expand my horizons.

It's amazing to me how few game designers seem to grasp the notion of how probablities work -- I spent an hour once trying to explain how rolling 1d20 was so very different than 2d10. He just never seemed to get it, yet he wanted to design a RPG with all sorts of random tables in his rules.

Yeah, I'd read it!

I've had similar discussions, trying fruitlessly to convince someone that adding +1 to a d20 was the same thing as adding +5% to a d100.

Consequently, this book may be a bit more rudimentary than some of the responders here might be after. Just a fair warning.

I don't know how feasible this is, but it would be really nice as a GM to be able to compute the percentage chance that any one PC succeeds at something assuming every PC tries it once.  This comes up a lot, especially with perception rolls in many games -- if I'm rolling one for the party, I really want a single percentage that anyone notices, not one roll for each PC.  (Some games probably give you a way to fudge this, but I know many don't.)  You can start by assuming a list of success% for the PCs and just show how to compute a single success% for the party.  But of course then the question is how to recompute it for various difficulty levels, which gets very RPG system specific.  Maybe in all practicality people would need a computer program for this.  I don't know.

You could probably make the five or six rolls faster than you could do the multiplication of probabilities.

To see if anyone succeeds, you really need to determine if everyone fails. The chance of everyone failing is the product of the chances of each of them failing.

So if you have five players, each with failure probability F1, F2, F3, F4, and F5, the chance of everyone failing is FE=F1*F2*F3*F4*F5. The chance of anyone succeeding is 1-FE.

A simple example with three players, each with an 40% chance of succeeding (60% chance of failing). The chance of all of them failing is 0.60*0.60*0.60 = 0.216%. The chance of any one of them succeeding is 1 - 0.216 = 0.784, or 78%.

I do need some help on probabilities (more working out the formulas for spread and standard deviation) for some dice rolls. I've posted the idea in a separate topic so as not to hijack this one because the explanation is a little lengthy, http://www.indie-rpgs.com/forum/index.php?topic=27533.0

OK,
  So, I am looking at otherkind and it is a mechanic were no matter what you do, you keep 4 dice. You can roll up to about 8 dice in a highly charged situation. Normally, I am pretty good about calculating averages. But I am not sure if that is the correct approach here. So, if I roll 5 dice and keep the highest 4, how does that change the probabilities of any of them being over 5? How do I calculate this (preferable in Excel)? And how do I check roll 6 and keep 4, etc.?

First, there's a book I recommend call "Statistics for the Terrified", which I believe is by John Kranzler. Most (if not all) of it is not very helpful for stats in RPG games. Nevertheless, I recommend it because I think it's one of the best stats books in terms of the writing and it's ability to educate someone about stats who is not comfortable with the subject.

I think a book like what you suggest would be interesting. I think it would be beneficial if you focused not specifically/entirely on the p-distributions of dice, but on the equations which generate these distributions. A lot of simple mechanics are pretty obvious, but there are plenty of systems out there that use exploding dice, various types of dice pools, etc. that could do with some jargon-free explanation. Whenever I tinker with a really complicated mechanic, I usually just take the lazy way out and write a little program in Python to figure things out for me, but it would be interesting to see how such things could be figured out arithmetically.