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.  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.