When doing a mechanic for a game I want to know the odds of the outcomes that result in success, failure, etc.

Single die, linear probability rolls are easy, even with conditionals, as are multi-die add them all together rolls.   But if people want could start adding in some of the math here.  

However, how do you do dice pool mechanics for large number of dice?  If there is a brute force program on line that can brute force generate the odds that would be interesting.

I've taken to using a combination of probability and generating functions.  Generating functions are basically combinatorial math.

Here is a simple example say I have a d6 dice pool system and get a success on a 6, but have a +1 I can add to any one die.   I get to roll n d6 dice.  A generating function for this situation is:

(4x+y+z)^n


Where x represents the numerals 1-4, y the number 5 and z the number 6.   The coefficient in front of the term in the polynomial x^i * y^j * z^k, is the number of i dice that read 1-4, j dice that read 5, and k dice that read 6.

An example, roll 2D6

(4x+y+z)^2 = 16x^2 + 8xy + 8xz + y^2 + 2yz +z^2

Which means there are 16 rolls with no successes, 9 with one success if you use your +1, 8 with one natural success, 2 with two success if you use your +1, and 1 with two natural successes.

  How do you calculate the odds for your game mechanics?

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Quote from: Xanther;1014770How do you calculate the odds for your game mechanics?

For straight odds for a single roll, I use an online dice simulator like http://anydice.com.

For odds for lots of rolls, like an entire combat, being a software developer, I write something in C++ or Java to do similar simulations of different situations, to see how long combat takes and how often each side wins. I'll run "lots" (tens to hundreds of thousands of permutations) to get mean, median, and standard deviation.

The first is useful for single roll resolutions and how often you hit in combat. The second is good for "how long will this take?" My player's like killing monsters, but we don't want to spend most of the session in a single combat, so I looked at how many rolls (turns) it took to resolve combat given differing numbers of combatants and differing levels and hit points.