Dice Probabilities

When you are desigining games, you are almost certainly going to be using a lot of dice. Not always, and sometimes you won’t use any, but when you are, you need to have a basic understanding of the probabilities of certain results occuring.

You’ll never be able to replace playtesting to understand how well the dice you’ve chosen will work, but knowing the probabilities will help you find a sensible starting point.

The standard dice you have available are four, six, eight, ten, twelve and twenty sided. They are commonly referred to as d4, d6 and so on. When you need to roll a number of similar dice you will see a number preceeding the dice symbol, and if you need to add or subtract a value it is included on the end. For example, 3d6 + 2 means roll three six sided dice, total the score and then then add two.

Now we need to understand the probabilities, and the two most useful things to consider are your expected value, or the mean, and the variance (which is a measure of how the values you get spread around the mean).

To calculate the expected value of a dice roll you sum all of the available values, and divide them by the number of sides. For the standard dice above, this makes the mean equal to the mid-point of the possible scores (so for a d6 the expected value is 3.5). This is often written as E(d6) = 3.5.

There are a few simple rules to deal with expected values, the expected value of a constant is the value of the constant, and the expected value of the sum of two dice is the sum of the expected values of the individual dice. So, for our earlier example, E(3d6 + 2) = E(3d6) + E(2) = 3E(d6) + 2 = 3 * 3.5 + 2 = 12.5. This should allow you to get a fair idea of what sort of value you expect people to roll.

Variance is less easy to calculate, one way is to take each possible value, subtract the mean, square it, sum the squares and divide by the number of values. A fuller explanation is given at the Mathworld site, but for our purposes it’s enough to know that adding a constant does not alter the variance. Similarly, the sum of two dice produces a sum of a variance (as the rolls are unrelated), so, given the variance of a d6 is 2.92, the variance of our example 3d6 + 2, V(3d6 + 2) = 3V(d6) = 3 * 2.92, which is 8.76.

The main thing we learn here is the more dice you roll, the more spread out your results can be, so bear this in mind when requiring lots of dice rolls.

There is once last thing to remember, that is not strictly linked to probability, which is the range. Range is measure of what values could actually be rolled, with a d6 it’s simply 1-6, but with our favourite example R(3d6 + 2) is 5-20.

If we consider the Expected value, the Variance and the Range when deciding which dice to use, then we can start with a sensible base, and reduce the amount of fiddling around with dice during playtesting, leading to quicker development and better games in a shorter period of time.

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