User:Aprogressivist/Nwoddice

An investigation into the standard dice-rolling probabilities involved in the nWOD.

Description
In the nWOD game, multiple ten-sided dice are rolled in dicepools to determine the success of actions; each dice coming up 1 to 7 counts as a failure, while each dice coming up 8 to 10 counts as a success. Moreover, each 10 rolled allows an extra dice to be rolled, which can, in turn, cause yet another dice to be rolled if it comes up 10; this is referred to as "exploding 10s".

The number of successes is then tallied up, and has an effect upon the course of the game.

If a dicepool is zero or negative, a chance dice is rolled instead, where only 10 counts as a success, and a 1 counts as a dramatic failure. The consequences of dramatic failures will be ignored for now.

Definition:

$$\ P(x,n)$$ describes the probability of rolling x successes on an initial dicepool of n dice.

Note, therefore, that n does not increase as a result of exploding 10s.

Observations

 * Since 10s explode, there is theoretically no upper limit to x; even a chance dice can roll a million successes. Practically, we are only really interested in the first order of magnitude of x. Indeed, in the vast majority of cases it is somewhat absurd, in the context of the game, to deal with more than the second order of magnitude of x.
 * P(0,0) = 0.9, P(x,0) = 0.1x. Similarly, P(0,1) = 0.7, P(x,1) = (0.1x-1)0.27. This is by simple observation; however, these values may become useful as variables or constants for recursive algorithms.

Definition:

The following constants are used for simplicity:


 * f = 0.7, probability of rolling a failure on one dice.
 * s = 0.27, probability of rolling one success on one dice.
 * t = 0.1, probability of rolling an exploding 10 on one dice.
 * g = 1 - f - t = 0.2, probability of rolling an 8 or 9 on one dice.

Hence: P(0,1) = f and P(x,1) = tx-1s.

Expansion of P(x,n)
Calculating the probability of achieving x successes on n dice is complicated by the fact that exploding 10s include a recursive probability of generating more successes beyond the fact that a 10 is one success already.