White Wolf's game mechanics typically look as though they were created by coked-up baboons. I experienced a lot of frustration creating the probability tables for their 2004 version of the World of Darkness because of the mechanic of re-rolling 10s. The whole game is like rolling damage for the arquebus in old D&D. Technically, it is possible to roll forever, so I set some limits. I stop calculating when the probability of an outcome drops below 0.001, or after the third roll. It was relatively easy to find the probabilities of up to three successes with one die, then much more complicated for two dice, then after some struggles I noticed a pattern in how many separate equations must be solved and combined for each possible outcome (that meets my criteria) for a given number of dice:

So, instead of trying to solve over two thousand equations or trying to figure out how to write a computer program that would do it for me, I brute forced the problem. I used Excel to whip up 10,000 sets of 3 rolls, then mixed them up twenty different ways, then averaged the occurrences of each number of successes. The values in the following tables should be correct to about +/- 0.002. (Click on the pictures to see bigger versions. I'm annoyed with blogspot's layouts.)

And, of course, by adding from the right we can find the probabilities of rolling at least any specific number of successes. Remember that these probabilities include up to two re-rolls of 10s.

I hope that these tables will be useful to game masters (storytellers) in determining difficulty levels of challenges for characters in their campaigns. They may also be helpful to players in deciding how much to increase a skill or attribute.

I really suck at stats, but I'm pretty sure you should be able to handle the re-rolling more correctly by just using a limit as number of re-rolls approaches infinity.

ReplyDeleteI came to some similar estimated conclusions about the rolling structure myself, but I always thought it was intentional. The possibility of rolling forever means that there is always a desperate chance of success on your part and always a chance for things to spiral out of control against you. It's a horror setting, so that seems appropriate.

ReplyDeleteHave you looked at their LARP system? Essentially you take the same stats, but you add your pool to a drawn card with a value of 1-10. Tens draw again and add the total. If the total is over 10 you've got a success, with additional successes for each increment of 3 past 10. As you'll see right away, it completely changes the probability, and that's been cause for a lot of argument amongst gaming groups. I keep trying to come up with a card-based variant that will lead to some sort of closer probability. I just don't have the math background for it.