Dice pool probabilities

Used under license from Wikimedia Commons

Various tabletop RPG systems use dice pools to resolve conflicts. Particularly, for some the mechanic is

1. Roll X dice
2. Count the number of successes (where a "success" is a roll over some number, generally fixed for the game system)
3. You win if the number of successes is greater than some target number

Now, that may or may not be a good system. Either way, it's not a system that I could look at and intuitively grasp my chance of success. For example, in Lady Blackbird, you roll d6s and a success is on a 4, 5, or 6 (so a one in two chance). So if I need 3 successes, how many dice should I roll? Not at all obvious.

For the mathematically inclined, it is, of course, a binomial distribution with p = 0.5. To help me understand it, I wrote a little javascript application. It will also hopefully be helpful if you need to look something up, either mid-design, or mid-play.

Here it is:

This is showing you success probabilities for Lady Blackbird (d6, success on a 4, 5, or 6). Down the left-hand side is how many dice you are rolling, and across the top is the target number of successes. The probabilities are of getting at least that many.

I'll carry on to talk about things I've understood by using the tool, but first, give it a play yourself. The cog button on the top-right will give you a set of options. You can change everything I've mentioned so far and explore other dice setups - or if you are mid-play, get it to show you the probabilities for the situation you are in.

So what things can we see? Firstly, for a given target number N, you can get a 0.5 probability of success by using 2N - 1 dice. So for N = 3, you'll need 5 dice just to get an even chance.

Now take a gander at this graph, which shows you how the probability improves as you add more dice for targets of 2, 3, 4, and 5:

It's clear there's a diminishing return on adding more dice. I haven't spotted any more precise mathematical relationships, but for these low target numbers, 2N + 1 gives you a rougly 3 in 4 chance of success.

The code for the dice pool application is licensed under this MIT license. Feel free to use it on your own site under those terms (although it'll always be here if you need it).

In other news: Lady Blackbird looks great, I can't wait to give it a try.