Exploding-Diepool

Little scripts related to exploding die pools, also known as open-ended dice pools.

Say we role n identical, fair dice, each with d sides (every side comes up with the same probability 1/d). On each die, the sides are numbered from 1 to d with no repeating number, as you would expect from an ordinary d sided die.

Every die in the outcome that shows a number equal or higher than the threshold number t is said to show a hit (also known as success). Every die that shows the maximum result of d is rolled again, which we call "exploding". We say the roll is "open-ended". If the re-rolled dice show hits, the number of hits is added to the hit count. Dice that show the maximum after re-rolling are rolled again and their hits counted until none show a maximum result. Given the values of

• d ... Number of sides on each die d > 0
• n ... Number of dies rolled n ≥ 0
• h ... Number of hits, we want the probability for
• t ... Threshold value for a die to roll a hit 0 < t ≤ d

what is the probability to get exactly exactly h hits?

Example:

We roll 7 six-sided dice and count those as hits that show a 5 or a 6. In this example, d = 6, n = 7, t = 5. The outcome of such a roll may be 6,5,1,2,3,6,1. That's three hits so far, but we have to roll the two sixes again (they explode). This time it's 6, 2. One more hit, and one more die to roll. We are at four hits at this point. The last die to be re-rolled shows 6 again, we re-roll it yet another time. On the last re-roll it shows a 4 - no more hits. That gives five hits in total and the roll is complete. So, for this roll h = 5.

The theory says:

Here is a (buggy) html page that might be useful to some P&P games. For an example of a python plot see the image above.

Acknowledgements

Many thanks to Brian Tung and the other folks at Mathematics Stackexchange that were kind enough to ponder (and answer) my question.

Savage Worlds

Here is the die roll situation for Savage Worlds, where you take the higher result of an exploding d6 ("Wild Die") and that of one of a d4, d6, d8, d10 or d12 that also explodes. The steps in mean reslut from one die to the next are: 0.5, 0.7, 0.7, 0.9.

Also note the gaps at common products in the graph below.