Dicing with Probability

How many different ways can you make 7 from two dice rolled at random? The answer, obviously is six different ways. 1 plus 6, 2 plus 5, 3 plus 4, 4 plus 3, 5 plus 2 and 6 plus 1. I know some people disagree over counting the combinations in both orders, but the bottom line is that out of 36 possible sums of numbers from two d6 dice, 7 is the most common outcome, with one sixth of throws giving this value.

In fact, scores between 5 and 9 account for two thirds of all two dice rolls.

Assuming you’re only adding them together.

If you give the player some leeway, you open a whole new range of possibilities.

For a recent piece of work I set out to map the possible outcomes for rolling two dice. the biggest single value I could generate was 46,656 which is 6 to the sixth power. The smallest value (assuming you always subtract the smaller value from the larger) was zero.

If you use addition, subtraction, multiplication, division, and elevating to powers (squared, cubed etc) there are 180 different outcomes, of which 125 are integers in the range 1-12. Of that range (there’s no way to get 13 in these five methods) the most likely is 1. 23 different ways to arrive at 1 as a result from rolling two dice.

I’ll admit right now, 1 times 1 is the same as 1 divided by 1, and 1 to the power of 1. There are fewer than 23 unique combinations of numbers rolled, but 23 out of 125 possible results is a good average. By the time you get to 11, there are only two ways to make 11 from two dice. It’s a prime number, so multiplication won’t work. It’s too big a result to apply subtraction or division. Only adding 5 to 6 or 6 to 5 will do the job. 2/125 possible results.

This has had a profound impact on the way I’ve seen a recent game design challenge. I want highly probable outcomes for the first few target numbers (1,2,3 etc) with decreasing likelihood as they get larger so it becomes much more about luck than good judgement. For school children, levelling the field between students is a good thing and that’s the target audience of the game.

Of course, being mathematically minded I’ve elevated the original game concept to something a bit more devious. You can play from 1 to 12 for the basic game with two dice, or from 1 to 24 with three dice. It’ll take a bit of play testing to iron out kinks and unintended outcomes but it should be ready for market in a few weeks.

2 thoughts on “Dicing with Probability

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  1. Andrew – you mention you need low numbers to be highly probable, and higher ones unlikely. Have you considered using non-standard d6’s in which you’ve changed which numbers can appear to do this? I’ve had great results with averaged dice, which contain 2,3,3,4,4,5 on their faces.

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    1. So called “average” dice are a great way to skew outcomes, but the numbers I ran used standard d6 because they’re easy to get hold of in most settings. I have literally hundreds of them around the house.

      I’m in favour of using standard components wherever I can, because it’s easier to maintain or replace bits to keep a game in play. In this case a game which could be reproduced across a classroom need to be simple to set up and pack down, and use basic parts that can be easily sorted out and/or reused elsewhere.

      I’m interested in a d6 vs d10 mechanic for the mini RPG I’m working on. It’s more specialised than just d6 but the percentages for success are beautiful when worked out because of the maths involved in rolling a ten sided die.

      In the case of 60, a game I Kickstarted last year, the game comes with a bag of dice: d4, 4 d6, d8, d10, d12. Within the world of tabletop gaming, they’re fairly common but it’s still nice to see people getting used to them for the first time.


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