Dice math – What’s the difference?

Content warning: Sloppy statistics and a chance of horrible mistakes.

One interesting question regarding various mechanics is ‘How long does it take to figure out who is better?’ Characters in the world don’t observe each other’s bonuses but rather the results of the checks and checks have a random component in them. So it is very much possible that a more skilled character loses multiple times to a less skilled one. Luckily, we can use (or abuse) a few methods from statistics to take a stab at this problem.

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Dice Math – Mathematics of PURGE.bat

I recently found an in-development cyberpunk RPG called PURGE.bat. If you want to check it yourselves, you can find it here. The game itself is currently quite rough (completely understandable), but we can still explore the skill mechanic a little. The combat in the game uses a simple linear dice roll which is very easy to grasp, so I’ll be focusing on the skill system.

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Dice Math III – Summing dice

It’s been a while since the last time (part I and part II). Besides dice pools and linear RNGs, a common way to get a random number is to roll a few dice and sum their results. This results in a curved distribution where some results are more probable than others. For example, rolling 3d6, you can only get result 3 by getting three ones but there are multiple ways of adding to e.g. 7 using numbers between 1 and 6.  Continue reading “Dice Math III – Summing dice”

Dice pools – Addendum

In the last post, I mentioned a few examples of common modifications to the “pure” dice pool mechanic. I’ll cover a couple of them here.

Exploding dice

The fist case is the so called exploding dice, i.e. if a die comes up with the (usually) maximum possible result, then another die is rolled and added to the dice pool. Eg. a player rolls 5 d6s, and gets 1,1,3,6,6. The player then rolls two more d6s, say 2,6. The second six again adds another die which gives 2. Given the target number of 5, the final result is then 3 hits. So how does this affect the probability distribution? Well, I’m glad that you asked! First consider the expected number of hits for each die. With standard dicepool, the probability of getting a hit is p. For the exploding dice, we need to account for the possibility of getting another roll. Let’s call p_{max} the probability of exploding. So first there is a chance of getting a hit. Then there is a  p_{max} chance of getting another roll that gives a hit with probability of p and again there is a probability p_{max} of the additional die exploding and giving another roll with probability p of getting a hit and so on. This can be expressed as p + p_{max}\left(p + p_{max}(p + p_{max}(\ldots))\right) This can be expanded asp +p \cdot p_{max} + p \cdot p_{max}^{2} + p \cdot p_{max}^{3} + \ldots Continue reading “Dice pools – Addendum”

Dice math – Part II: Dice pools

In part 1, I covered some properties of a linear dice mechanic using d20 system as an example. Another very common mechanic is the dice pool. The basic idea is that the player rolls multiple similar dice and counts the number of dice that have result equal to or greater than a target number. The number of ‘hits’ is then the result of the roll. Games that use it are numerous: Shadowrun, World of Darkness, Burning wheel, Warhammer (the miniature game!). For example, in Shadowrun, the dice are d6s and the target number is five. So rolling 7 dice and getting numbers 2,3,3,5,6,1,4 would result in two hits. Sometimes there are additional complications, like exploding dice, re-rolls or botches. I’ll cover these on a separate post. The next part is going to be a little math heavy.

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Dice math, part 1 of N

Most RPGs feature at least some mechanics with uncertainty and from both the designer and player perspective, it is important to know what you get out of them. Often their properties are intuitively clear, but I thought it’d be fun to go through them in more detail than usually happens in a gaming session. Even though pieces like this might be available somewhere, I wanted to go through the math and simulations myself. This is part 1 of N of series where I’ll go through the properties of different mechanics. I’ll start with a staple: A linear random number generator (RNG, usually a die / dice). The most popular example of this is probably D&D and it’s spawn (eg pathfinder).

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