More on Dice Games, Probability and Distributions

In a previous post I've analysed probability distributions from rolling two dice: looking at the sum of two dice (a well-known distribution), the difference; the largest of two dice and the smallest.  This article comes about following some home-schooling I've been doing with my children, looking at distributions of results for throwing one or two dice - this was an extension activity we looked at (and I'm examining it in more depth here).

In this article I'm going to examine a different distribution: rolling three dice and summing the two largest dice. This is clearly going to be skewed towards larger totals than the normal two-dice distribution- but by how much, why, and what does this distribution look like?

For example, there is only one way to get a sum of 2, which is to roll 1 1 1. However there are many ways to get 12, including 6 6 3, 6 6 2, and 6 6 1.

To abbreviate the analysis, I'd like to show and use the following results:

There is only one way to roll 1 1 1, 2 2 2 or 3 3 3.  This is self-evident but worth mentioning.

When one of the dice has a different value to the other two, there are three equivalent combinations: 4 4 5 is equivalent to 4 5 4 and 5 4 4. For any a a b, there are three equivalent combinations.

When the dice have three different values,  there are six equivalent combinations: a b c, a c b, b a c, b c a, c b a and c a b.

Onto the actual distribution, then:

Score	n	List
2	1	111	1
3	3	211	3
4	7	222	1	221	3	311	3
5	12	321	6	322	3	411	3
6	19	333	1	332	3	331	3	422	3	421	6	511	3
7	27	433	3	432	6	431	6	522	3	521	6	611	3
8	34	444	1	443	3	442	3	441	3	533	3	532	6	531	6	622	3	621	6
9	36	544	3	543	6	542	6	541	6	633	3	632	6	631	6
10	34	555	1	554	3	553	3	552	3	551	3	644	3	643	6	642	6	641	6
11	27	655	3	654	6	653	6	652	6	651	6
12	16	666	1	665	3	664	3	663	3	662	3	661	3

Total:  216   = 6*6*6 

The distribution is clearly skewed to the higher values, compared to the two dice distribution.  This makes sense, we're selecting the two largest values from three dice. Interestingly, it's not even symmetrical either side of the mode of 9 - the values for 8 and 10 are the same, as are 7 and 11, but there are fewer ways of getting 12 compared to 6.

Here's the breakdown, which compares to "seven for everything" for the normal two-dice distribution.

Mode = 9
Median = 9
Mean 8.458

Comments:

In game design, if you want a character, figure or unit to move generally faster or perform better than a typical two dice distribution, then taking the two highest values from three dice will certainly achieve this, while still retaining a range of 2-12 spaces or points.  Your unit will hit harder, but can there's still a possibility that it will be outperformed by a two-dice unit.

If you're interested in probabilities, I have a few articles you may be interested in:  the probability of getting a set of toys from blind bags; which expanded over four articles; and Isaac Newton's Random Walk.