Random Elements that Aren't

Statisticians tell us that adding multiple random generators isn't necessarily a good idea, as it tends not to provide additional randomness, but rather have the contrary effect. (I learned that in an encryption lecture. Go figure.)

Anyway, it seems clear that if you roll 1d6, there are 6 results, each equally likely. If you roll 2d6, there are 11 results, but a result of 7 is 6 times as likely as a result of 2. It is fairly easy to grasp the notion that adding lots of dice gives nice, average results with little deviation.

What isn't quite so obvious is that dice pools and sequences of rolls suffer from the same effect.

Let's consider Storyteller yet again, and assume a roll of difficulty 6, with a 1 cancelling a success on another die. When rolling 1 die, it's obvious that you have a 10% chance of botching, 40% of failing, and 50% of success. The percentages for 2 dice are somewhere below, in my analysis of armour. (Note that I ignore the reroll-10's rule from Storyteller, since it totally screws up the statistics.)

What isn't quite so clear is that each die has a net value of 0.4 (-1*0.1 + 0*0.4 + 1*0.5). When rolling lots of dice, you can get a rough estimate of the number of successes by mutliplying the number of dice by 0.4. Hence, rolling 40 dice will give 16 successes with reasonable consistency (40*0.4 = 16). Naturally, this only works with lots of dice, and the more dice you have, the better it works.

The Exalted system of causing 1 automatic damage for every 3 dice reflects this. (Note that Exalted, with a fixed difficulty of 7, gives each die a value of 0.3.)

Now, let's consider a character with a to-hit pool of 40, using a weapon with a base damage of 25. This is, of course, an absurd example, but I'm using appropriately large numbers to give reasonably good statistics. As we've seen above, 40 dice at 0.4 weighting gives 16 successes pretty consistently. Since excess successes add to the damage pool we roll 25+15 dice for damage. So, we roll another 40 dice, to eventually get an end result of 16 damage.

It's dangerous to try to scale down this method for small dice pools. With only a few dice being rolled, there's still a fairly wide deviation. And, if the opponent has some sort of automatic damage reduction, that deviation may be enough to change the environment from one where you never cause damage to one where you occasionally cause damage. Still, over the course of a campaign, you can expect to see characters with a given dice pool operating at a certain level of effectiveness.

To get to a point, in Storyteller combat we typically have four rolls to make: to-hit, dodge, damage and soak. Each roll is likely to involve several dice. Each takes time, and the end result is quite consistent - small amounts of damage will be taken by each side until eventually one side falls over from attrition, or someone lands a really good blow. The same result could be achieved with a single roll, with appropriate modifiers.