Probabilities#

Pounce & Pandemonium uses a pool system. We roll nd6. The basic rule for 1d6 is:

  • 1-3 is a bad outcome, probability is one half or 50%.
  • 4-5 is a partial success, probability is one third or 33.3%.
  • 6 is a full success, probability is one sixth or 16.67%.

Mathematically spoken:

$$ p_- = \frac{1}{2}; p_0 = \frac{1}{3}; p_+ = \frac{1}{6}; \qquad \mathrm{where } \, p_- + p_0 + p_+ = 1. $$

If you throw multiple dice, you can also achieve a critical success on rolling multiple sixes. If your action rating become zero or less you throw 2d6 and choose the lower one of them.

For an arbitrary number $n$ of dice the probabilities are:

$$ P_-(n) = (p_-)^n $$

$$ P_0(n) = \sum_{k=1}^n \binom{n}{k} p_0^k p_-^{n-k} $$

$$ P_+(n) = \sum_{k=1}^n \binom{n}{k} p_+^k (p_-+p_0)^{n-k} $$

The probability for a critical success is achieved from $P_+$ when starting the sum at $k = 2$. It’s a nice exercise to show that $P_-(n) + P_0(n) + P_+(n) = 1$ for all natural numbers $n \geq 1$.

For action rating $\leq 0$ we obtain the following probabilities:

$$ P_-(0) = p_-^2 + 2 p_- p_0 + 2 p_- p_+ = \frac{27}{36} $$

$$ P_0(0) = 2 p_+ p_+ + p_0^2 = \frac{8}{36} $$

$$ P_+(0) = p_+^2 = \frac{1}{36} $$

And that’s all the maths to throw into some script and produce the graph shown at the beginning of this chapter. What we essentially see: There is a comfortable range of action rating 2-4 where with around 50% probability the DM will get a chance to act.

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