I'm working on a fantasy game to replace the often frustrating experience that is D&D (in any edition). My "core mechanic" functions like so:

Characters roll 2d if untrained or  3d if trained, in addition, they may gain bonus dice for special abilities or skills, but they only keep and total 2d if untrained and 3d if trained. The character cannot roll dice than twice the base (so 4d if untrained, 6d if trained). These are then compared to a DC... so far, so good. It gets a little more complicated when you add in Edge. A character's total dice pool size is increased by his Edge and in addition he may keep and add any dice in his pool that come up below his Edge without having them count against his total dice kept.

Can anyone out there help me determine the probabilities of various results as the dice pool sizes and Edges increase... or  perhaps they could tell me how to figure these things out myself.

What is the die size?

Good question. Six sided.

Some simple rules for probability.. obvious, but having them spelled out may be instructive.

Notation:
        P(A) = probability of event A
        P(B|A) = probability of event B given that event A has already happened

Statistical Laws:

        P(A) = number of outcomes where A happens / number of possible outcomes
                For example, P(drawing an ace from a full 52 card deck) =  4 / 52,  from having four aces in fifty-two cards

        P(B|A) = P(B) * P(A) if and only if the events are independent.
                For example, P(rolling another 6 on a d6 | rolled a 6 on your first roll) = 1/6 * 1/6
                This is not true for a deck of cards, because the possible outcomes of drawing a 2nd card depend on what
                you drew for the first. If you drew an Ace, your chances of drawing another ace are only 3/51.

        n! = factorial, which is mathspeak for n*(n-1)*(n-2)*..*1 .. for example 3! = 3*2*1 = 6. It is taken by convention
                that 0! = 1, though 0*anything is meaningless. Factorials are useful in dealing with probabilities in a fixed-
                sized set, like playing cards.

        (n choose x) another math term. It means n! divided by the product x! * (n-x)!  What it MEANS means, is when
                you have a certain number n of items, like tokens, how many ways can you choose an x size subset of
                those items.
                For example, from a handsize of five Magic Cards, how many ways can I choose three of those cards?
                    (5 choose 3) = 5! / (3! * 2!) = 120 / (6 * 2) = 120/12 = 10

Your game:
        In the case where you're rolling some and dropping the lowest dies to keep only a certain max number of dice, the probabilities are no longer independent so you can break the problem down to easier-to-calculate probabilities. You'd have to do it in one go.
EG For an Untrained character (ie 2d), the total number of possible results is 11 (2..12). If the PC has four dice to roll but drops the two lowest, you'd have to figure out the number of ways each total from 2..12 can be rolled with four drop two dice.

Calculating for 12 is easiest: given that 6 is the best you can roll, if you roll two sixes, the other two dice are irrelevent.
Therefore, the first question is how many ways can you roll at LEAST two sixes? Answer: (4 choose 2) = 4! / (2!*2!) = 24/4 = 6
Next: how many ways are there to roll the remaining two dice, given that their results don't matter? 6*6 = 36

Therefore, the probability of rolling a 12 in a situation where you roll four d6's and drop the two lowest is:
        P(A) = (6 * 36) / (36*36*36*36)
                = 6 / (36*36*36)
                = 1 /  (6*36*36)

Calculating the other results just gets trickier from there. Good luck! :-D

P.S. if the "Edge" dice are simply added on top of the original rolls, their probabilities don't really matter because the Edge dice are independent. By the statistical law for independent probabilities I mentioned above, the probabilities can be broken down into simpler terms, ie you calculate the probabilities for the Edge dice separately and just multiply. You'll get a table, one axis is the original set of results (eg 2..12 in my example), the other axis is Edge dice result.

Whoops, n! and (n choose x) shoulda been under Notation

This system is very difficult to calculate probabilities.

If you want player calculation of odds to be part of the game, you are going to have to think of another way.

It's probably not that Masquerade wants players to calculate probabilities; more likely he'll want to do it himself during design.

This is a very complex dicing set-up, however, and I don't have the time myself to calculate this right now. I suggest using the Monte Carlo method: write a computer program to roll the dice for you a couple thousand times and see what sort of results come up on average.

You could learn and use TROLL - a scripting language explicitly designed for simulating the intricate dice-rolling procedures of RPGs and calculating probabilities for the chosen method.
I don't have a bookmark handy on the PC I'm writing from right now, but I do remember there was a thread on story-games.com no more than 2 months ago.

Give me clarification on what an Edge does:

An Edge gives you extra dice (which can blow past your cap of twice the number of dice you keep) equal to its value. In addition, you keep all dice equal or lower than the value of the edge. Is this correct?

What is the range of Edge values, practically? 1-3?

How likely is someone to be "at cap" before applying an Edge?

yrs--
--Ben

I also recommend Torben Morgensen's dice probability calculator Troll.

http://www.diku.dk/~torbenm/Troll/

There is a simple Windows XP set up package, as for Moscow ML, as well as for Troll.

It can't do dice where you have mixed numbers of sides, but it can do most everythign else having to do with dice.

chance.thirteen wrote:
I also recommend Torben Morgensen's dice probability calculator Troll.

http://www.diku.dk/~torbenm/Troll/

There is a simple Windows XP set up package, as for Moscow ML, as well as for Troll.

It can't do dice where you have mixed numbers of sides, but it can do most everythign else having to do with dice.

By a rather amazing coincidence, I have a PDF out (for about two months now) which also uses a "roll many / keep one" system for DnD, mine uses 20 sided dice though, could you show some stats on that?

I'm particularly interested in whether a one die at a time strategy would work better or worse than a multi die (roll many / keep one) strategy since our system allows both.

G.

The chance of rolling a target number x on a d20 with n rolling attempts is

1 - ((x-1)/20) [sup]n[/sup]

I have a long standing respect for Troll. Sadly, I can get it up and running on a Windows machine, but I am a Mac user, and have had no luck getting it to work there. I end up using in on other peoples computers.

Eero wrote:
It's probably not that Masquerade wants players to calculate probabilities; more likely he'll want to do it himself during design.

This is a very complex dicing set-up, however, and I don't have the time myself to calculate this right now. I suggest using the Monte Carlo method: write a computer program to roll the dice for you a couple thousand times and see what sort of results come up on average.

Ben: No, Edge does two things... one it increases your maximum potential die pool by its value, not your actual die pool and it allows you to keep (add the value of) more dice than you would normally be allowed (which is fixed at 2 untrained, 3 trained) as long as the value showing on those dice is under your Edge. So, if you have an Edge of 1 and you roll 7 dice (your maximum pool size at Edge one if trained in the roll your making) the best possible result would be 6,6,6,1, 1, 1,1 for a total of 22.

Using a similar to D&D level progression (3e's 1-20), you gain +1 Edge on all rolls at every level that's divisible by three (3, 6, 9, 12 etc...). You also get Edge bonuses for very specific specialization (like say, a fighter's Weapon Specialization) and for superhuman abilities, but you can't more than double your Level-based Edge by this method.

Sense I haven't play tested, not sure how much time is spent at various Edges, but the development system should have the characters spending more time at low Edges as lower levels are easier to achieve than higher ones.

Also, the "at cap" before Edge part should be very common... a characters who's focused in an area should frequently being rolling the maximum amount of dice available at his Edge.

Try out http://hamete.org/dserver/dice its a free dice server that allows you to roll any combonation of dice (some of them rather unusual) up to 50 times at a go.  It also gives you cumulative scores, averages, PBEM certification and more. 

I ran 2D6 rolled 50 times; 20 times then averaged the averages (6.661), and whan I use my QDDAF* I get 7, which is close enough for government work.  3D6 only will result in an average of roughly 10.5, 4 = 14, 5 = 17.5 and 6D6, roughly 21. 

I find that unless you're planning on breaking the bank at Monte Carlo, save yourself some skull sweat and brain pain, use the QDDAF.

*Quick and Dirty Dice Averaging Formula (A+Z)/2 = avg
Where A is the lowest possible result of a roll, and Z is the highest possible result.