Dice Math: Rolling Doubles on Arbitrary Dice

As I was finalizing the mechanics on Runecycle I needed a semi-frequent and easy to read way to trigger an event: a way to regain resources by playing. I tried a couple different ways until I settled on my current approach: when you roll doubles. In the game you always roll two dice of varying sizes, similar to how the Year Zero Engine step-dice variant works. A lot can be said about the math in this system but that is a story for another day. Here I want to talk about rolling doubles, or in other words rolling the same number on both dice. You might notice this is how crits work in Daggerheart too. Topical, huh.

This isn’t something new, this probability has been known for a loooong time. But I wager many don’t know about it especially considering how little the mechanic is used in ttrpgs.

The probability of rolling the same number on two arbitrarily sized dice is

graph LR 
    A["$$\frac{x}{all}$$"]

where x is the number of desired outcomes (dice with same value) and all is every possible outcome.

We know that the number of all outcomes is die size 1 x die size 2. Every combination of both dice. For desired outcomes if both dice are the same size this is easy as we can just take each pair. For 2d6 that would be (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6). Total of 6 desired outcomes.

But if the dice are arbitrarily sized? That’s surprisingly easy as well. Since the larger of the two dice can never make a pair with numbers that are higher than the possible value of the smaller die, the total possible amount of desired outcomes is still the size of the smaller die, same as with a pair of same sized dice, right? So that leaves us with the following formula:

graph LR 
    A["$$\frac{smaller die}{smaller die * larger die} = \frac{1}{larger die}$$"]

In other words the probability of rolling doubles on two arbitrary dice is 1 divided by the size of the larger die. Crazy and convinient. It’s the same as rolling the smaller die to determine which number needs to be rolled and then rolling the bigger die. The probability of rolling any one number on the bigger die is of course 1 / (die size).

Now comes the fun part. List all of the probabilities based on the larger die for d4 to d20:

As you can see, whatever pair of dice you roll, you know the probability of rolling doubles. And assuming you are using a pair between d6 and d12, you know the probability is approximately 10-15%. If larger die is better a player might work it out that your chance of rolling doubles is twice as low for a pair that has a d12 than it is for 2d6. But if succeeding with 2d6 is way less likely than say d12+d6 then no-one is willing to take that bet for a 16% chance to roll doubles, when the alternative is a high probability to succeed and 8% chance to roll doubles. Especially in my game where rolling doubles does not make you succeed at the roll like Daggerheart does.

That’s it for today! Go make cool stuff now.