Probability Curves - Calling All Mathematicians!

[RyvenCedrylle]

I'm working on some probability models for a homebrew system. Basic roll a handful of dice and count successes as any individual roll over a target value. Example: 3d6 vs 4 rolling 1, 4, 5 is two successes. I've been able to work out equations so far to calculate the odds of rolling X number of successes on Y dice with Z target value, varying number of dice, size of dice, number of successes and target value.

I'm now trying to determine how static bonuses affect the results; i.e. do you get more successes by adding +1 to any die you choose vs. rolling a bigger dice vs. rolling more dice? I've done a couple such bonus 'calculations' the hard way by writing out all possible die results and counting successes by hand with the bonus included and know that it flattens the curve out and seriously ups your odds of getting ANY successes, but I'd like to find a function for this rather than writing out dice combos. Does anyone have experience doing this that could point me in the right direction? I made it through Calc III in college, so I'm not afraid of the heavy stuff but it has admittedly been about 6 years, so I'm a little rusty. Thanks in advance.

 

[ggroy]

AnyDice Calculator

 

[ExploderWizard]

DPC


I am lazy and this has been my best friend for my own project.

 

[RyvenCedrylle]

Those are cool, but they're adding the rolls together GURPS-style. I'm looking for something that would be for say, Shadowrun or World of Darkness where each die counts individually, not added together for a final result. Thanks for the start, though wizard and ggroy

 

[ggroy]

Look up the binomial distribution and the multinomial distribution in a probability textbook. More generally, it is a huge exercise in combinatorics (ie. permutations and combinations).

 

[kitsune9]

Wow, GG and Exploder. Those are really cool. Now if I ever want to create my own 24 hour rpg, I know where to go.

 

[RyvenCedrylle]

Well that's closer.

Multinomial distribution - Wikipedia, the free encyclopedia

Still not quite there though. The binomial (or Bernoulli) distribution covers multiple independent trials in a single set with identical probabilities of a binary outcome. The multinomial is used for a single set of trials with more than 2 outcomes but identical probabilities. My 'trials' are dependent on one another though - the first success has a probability of x, but once that occurs, subsequent trials have success probabilities of y where x>y; maybe even a third probability z where y>z.

The answer might still be in there. Thanks for the pointer, ggroy.

 

[Stormonu]

Is there anything similar for drawing cards? I've been designing a game that uses something like a poker hand and trying to figure out the odds, but trying to calculate it has been difficult (mainly because you only get 4 uses of any one number over the entire deck).

 

[ggroy]

If you want to find the number of ways to roll a particular number with any combination of dice of different denominations, there is a systematic way of getting the number of combinations and probabilities.

As a hypothetical example: 3d4 + d6 + 2.

The possible sums produced by the dice and static modifiers sum of 3d4 + d6 + 2 is represented by the polynomial:

(x^2)*( x+x^2+x^3+x^4+x^5+x^6)*(x+x^2+x^3+x^4)^3

where the number of dice combinations giving a particular sum S, is the coefficient of x^S.


Go to

Wolfram|Alpha

and enter in the string

Expand[(x^2)*( x+x^2+x^3+x^4+x^5+x^6)*(x+x^2+x^3+x^4)^3]

which will expand out the polynomial.

If you do 3d4 + d6 + 2 at

AnyDice Calculator

the number of dice combinations of sum S, will be the same as the coefficients of x^S in the above polynomial expanded out.

To get the actual probabilities from the polynomial, just divide the coefficients by (4^3)*6.


One can figure out the polynomial from the above example, for any combination sum of dice and static modifiers.

A die dN with values 1, 2, 3, ... N is represented by a polynomial: x + x^2 + x^3 + ... + x^N.

A static modifier k added to the dice roll is represented by a polynomial: x^k.

Multiply together the polynomials for the dice and static modifier, and expand.

 

[pawsplay]

Essentially, you are recreating coin flips, except the 50/50 odds are shifted one way or another.