Making graphs of die roll probabilities

[Frozen_Feet]

So, I'm thinking of resuming work on an RPG system of my own. It will be a fairly generic fantasy game, with inspiration taken mostly from Basic D&D, Praedor, CODA and couple of CRPG roguelikes.

One thing I'd like to get right from the start would be solid system math. And for that, I need a bit help with counting probabilities.

To give a specific example, I'd like to have a probability distribution and graph for 2d12 roll. Like this.

Ideally, I'd like a crash course on how to do the maths so I can make these via Excell or by hand.

[jaybird]

IIRC the best normal distribution approximation of the d20 system is generated by 4d6-3 or 3d8-2

Beyond that, what sort of die sizes/pools are you looking at?

[Frozen_Feet]

But I'm not approximating d20 system, I'm making another one. I though of using 2d10 first, since that's easy enough to count in my head, but decided I like the slightly wider range of 2d12 more.

A curve created by throwing 3 or more dice is a bit too "high" for my tastes to use as a core mechanic, ie. middle numbers are too likely and numbers at the ends too rare.

[Reaper_Monkey]

A curve created by throwing 3 or more dice is a bit too "high" for my tastes to use as a core mechanic, ie. middle numbers are too likely and numbers at the ends too rare.

Unless you're playing GURPS, in which case that rarity difference between middle and edge numbers is the basis for the entire system.

[LCP]

Someone has already coded it for you:

http://www.anydice.com/

[Frozen_Feet]

@LCP: Hah! Thanks! That's exactly what I need.

@Reaper_Monkey: Wouldn't know about that, never played GURPS and it's not on my list of inspirations.

[dobu]

my favorite approach for getting sime kind of curve into the d20 system was using 3d20 take middle. if you roll 6, 14, 18 you'd take 14.
the curve is some kind of paraboloid.

the only stuff you would have to adjust would be the crit ranges for weapons...

[TuggyNE]

Someone has already coded it for you:

http://www.anydice.com/

Awesome. I have yoinked this link for future use. Thank you, kind sir or madam! (Also, thanks Frozen_Feet for asking! )

[Skyrunner]

*bookmarks*

My theory seems to be right! :D

Max damage from a Disintegrate is almost impossible.

[Siosilvar]

*bookmarks*

My theory seems to be right! :D

Max damage from a Disintegrate is almost impossible.

(1/6)^X, where X is the number of dice you roll, is fairly trivial. Multiply by 100 to get a percentage.

That's .0000000000000000760% chance to roll max damage on 22d6, by the way. Add another 14 zeroes in front to approximate 40d6.

So not only is it "almost impossible", it's "this is never going to happen in your lifetime". One in a hundred quadrillion for the first, one in 10 nonillion for the second (if my math is right).

[Skyrunner]

Whoops, wrong theory.

I meant 'randint(20,400)' and sum(1->40)(rad(1,6)) were different ^^^;

[Siosilvar]

Whoops, wrong theory.

I meant 'randint(20,400)' and sum(1->20)(randint(1,20)) were different ^^^;

Fixed that for you. And yeah, that's basically what happens when you do a bunch of things - even if they're evenly distributed, together they tend to cluster towards the middle. Stats 101 (okay, the class was actually 1410, but nevertheless).

[warty goblin]

Fixed that for you. And yeah, that's basically what happens when you do a bunch of things - even if they're evenly distributed, together they tend to cluster towards the middle. Stats 101 (okay, the class was actually 1410, but nevertheless).

The central limit theorem is rather inexorable that way - the sum of anything that's remotely well behaved goes to a normal distribution at infinity. If you really want things not to cluster, you're going to need to start with a skewed distribution. Adding up too many of them will become mostly normal again, but if you stick to a few, it'll be non-normal.

Without crunching the numbers, I'd imagine you could get some fun results with, say a d6 labeled 1, 1, 2, 3, 4, 5.

[Mystify]

The central limit theorem is the key to dice based systems. Even when you are rolling a single die, like with an attack roll, that single die roll is not the determining factor in everything. You have many attack rolls, many damage rolls, and so while the moment to moment fluctuations are unpredictable and provide interest to the game, the overall trends follow probabilistic norms.
Which is why things start to break down when you start determining things based on a single roll, for instance a save or die effect. The probabilities cannot average out, and instead you have a random fluctuation kill your BBEG, things end up unsatisfying. Or a player rolls poorly, and is killed, more because of the fall of the die than his own decisions and tactical choices. For a similar reason, a character dealing enough damage to kill something with one blow is not a desirable mechanic, unless that one thing is one of many minions, so the progress in whittling down their numbers is its own amortization.


What, I don't spend hours contemplating game theory, what you are talking about?

[TheOOB]

What, I don't spend hours contemplating game theory, what you are talking about?

Game theory is a very important field of study for anyone who wants to make good RPG's(or games in general), and I've seen quite a few games do good or bad based solely on their dice mechanics.

D&D shows it's wargaming roots strongly with it's dice system. It's all about statistics and bonuses, there are very few ways to actually influence the random part of the equation, and their is no curve to the probability at all, the game system relying on the sheer number of times you'll be rolling that d20 to average things out.

Shadowrun, on the other hand, goes in the other direction. You only roll d6's, and a 5 or a 6 on a die is a hit. As your character becomes more skilled you roll more and more dice. This makes the numbers fairly interesting. While anyone who's taken middle school math can tell you you'll average .33 successes per die, the curve the numbers make is very interesting, as you add more dice not only do you increase your chances at getting more successes, but you also dramatically reduce the likelihood you'll get a bad roll. It creates a bigger disparity between those who are skilled and those who are not.

[Mystify]

Game theory is a very important field of study for anyone who wants to make good RPG's(or games in general), and I've seen quite a few games do good or bad based solely on their dice mechanics.

D&D shows it's wargaming roots strongly with it's dice system. It's all about statistics and bonuses, there are very few ways to actually influence the random part of the equation, and their is no curve to the probability at all, the game system relying on the sheer number of times you'll be rolling that d20 to average things out.

Shadowrun, on the other hand, goes in the other direction. You only roll d6's, and a 5 or a 6 on a die is a hit. As your character becomes more skilled you roll more and more dice. This makes the numbers fairly interesting. While anyone who's taken middle school math can tell you you'll average .33 successes per die, the curve the numbers make is very interesting, as you add more dice not only do you increase your chances at getting more successes, but you also dramatically reduce the likelihood you'll get a bad roll. It creates a bigger disparity between those who are skilled and those who are not.

Indeed, the spread of 20 in your capability is always odd. If you take 10 above your skill as your point of competence(aka what you get when you take 10), and you have 1 skill point increase per level(there are other factors, but they are sporadic), then you range from failing to do something someone ten levels below you can do competently do pulling off something somebody 10 levels above you should be doing. In essence, you have a 20 level spread in how well you will do. In a game with 20 levels. This is simply an absurd range. It kinda works when you have fixed DC to make, but when you have a sliding scale(such as with jump checks), all of that variance comes into play. And jump checks really illustrate the absurdity of that spread; a bad roll, and you can fail to jump over a small 5ft gap, or roll well and you can go sailing 20 ft. Imagine watching jumping along. They hop forward a few feat, then are launched a couple dozen feat, then 7 feat, and just randomly fluctuate. All skills work like that, its just normally less graphic and hidden behind a single pass/fail DC. There is not even a normalizing aspect from multiple dice, which wouldn't necessarily cut down on the range, would at least make it more consistent.