Dice Spreads, Probability Curves, and Rules Writing

In advance of an upcoming article I am planning to write, where I review the Knights and Magick rules (by Arnold Hendrick, and published by Heritage back in 1980), I wanted to address a rather interesting dice mechanism that are employed in the rules.

That is the 1-15 dice roll.  This is accomplished by rolling two dice, and adding them together.  The first dice is a rather common d6, and the second dice is the typical d10, but numbered from 0-9 (the 0 face on the die is valued at zero, and not at ten).  Adding the two results together yields a number from 1 to 15.

Now, the asymmetry of the roll always bothered me.  How could you possible hope to have game feng shui when you are rolling two different dice, and adding them together?  It was preposterous!  Sort of like the damage rolls from early Runequest and Call of Cthulhu and other games, where you might have 1d8 for the weapon, plus 1d4 for your strength, and maybe +1 for a good quality blade - so your damage looked like 1d8 + 1d4 + 1.  Weird, but okay for roleplaying games.  Wargames were supposed to be above all that sort of thing.  Better.  More pure.

But here we were, with Mr. Hendrick's rules using a D6 and a D10 added together.

Let's look at the results.  To compare it, a brief examination of the (much more common) 2d6 bell curve.  Here we see a smooth progression from one result (for a Two), up to six results for a Seven.  And then back down again to one result for a Twelve.  There are a total of 36 possible dice pairings, so the frequency is a number of times out of 36.  This is pretty standard stuff, that is part of any study of probability, but also should be pretty intuitive to just about any long time gamer, and/or game designer.

Result	Frequency
2	1/36
3	2/36
4	3/36
5	4/36
6	5/36
7	6/36
8	5/36
9	4/36
10	3/36
11	2/36
12	1/36

But now lets take a look at the rather interesting D15 result.  Here we have 60 possible results, rather than 36.  There are more possible results (1-15, rather than 2-12).  Plotting the possible results we see:

Result	Frequency
1	1/60
2	2/60
3	3/60
4	4/60
5	5/60
6	6/60
7	6/60
8	6/60
9	6/60
10	6/60
11	5/60
12	4/60
13	3/60
14	2/60
15	1/60

Okay, so what do we see from this?  First, the numbers of all the middle results, from 6-10, have the same frequency - 6, our of 60 results, or a 10% chance of getting any of those numbers.

How does this behave in the game?  Well, we end up with many more middling values (half the possible results are from 6-10).  Plus, in a game that deals in + and - factors added to the dice, for a variety of different causes, this will tend to balance out and level the impact of the dice modifiers.  What I mean by that, is that in a 2d6 dice roll, a single +1 or -1 can have a very high percentage impact on the dice chance, especially if your base number is off the middle.  In this case, the middle (or stable region) is spread out, so that dice modifiers are more predictable in their impact (a minor change with a +1 or -1).  This also means that having additional plus or minus factors won't be an overwhelming impact, as it is in a 2d6 curve.

Once I pulled out the results, and looked at them, the asymmetry doesn't bother me AS MUCH, but it is still there.  I think I would prefer a 3d6 roll, or 5d4 roll, to balance out that middle and level it - but that is a different story.

Now, with the recently reviewed Hackbutt & Pike rules (written by Ben King, in the Tac-50 rules series), we see that the casualty table is driven by a dice difference.  Again, looking at the 2d6 probability table, but selecting results based on the difference between the two numbers (agreeing that doubles results have to be rerolled), we have the following:

Die Difference	Frequency
1	10/36
2	8/36
3	6/36
4	4/36
5	2/36

Of course, the total frequency of pairs with a difference only adds up to 30 out of 36, again because we tossed out the possible results where doubles are rolled on the dice (6 chances out of 36).

This is a very flat progression.  It is also interesting that on the Tac-50 tables, the least valued result is always a difference of 5, followed (in order) by 1,2,3,4.  A very interesting use of an interesting probability spread.

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One more thing - I have a plan to (very soon) write a review of the original Sword and the Flame rules.  In those rules, as in many others before and since, combat is resolved between two miniatures by rolling a dice on each side, and the high scorer wins the fight.  With no modifiers, and with Tie results not counting, there are 15/36 chances for each side to win, if using six sided dice.

• Side A wins - 15/36
• Side B wins - 15/36
• Neither Side (tie) - 6/36

But what happens when we change this by just a little bit?  Say, as in The Sword and the Flame, one side or the other gains a +1.  In this case, what if Side A gains a plus 1?  Very different results.

• Side A wins - 21/36
• Side B wins - 10/36
• Neither Side (tie) - 5/36

Looking at the results it shifts, from even chances, to a Two-One chance of winning - from just granting a +1 on the dice.

With a +2 on the dice, it is even more extreme.  In fact, looking that the following table, we see it is 5-1 in favor of the side with a +2 on the dice.

• Side A wins  - 26/36
• Side B wins - 6/36
• Neither Side (tie) - 4/36

Now, lets stretch this even further, and take a look at what happens, with no bonuses, but with one side winning Ties...  Again, lets assume that Side A has the advantage, and with no dice modifiers, will win ties.

• Side A wins - 21/36
• Side B wins - 15/36
• Neither Side (tie) - No Ties

From these results, the odds change slightly, from even chances, to 4-3 chances for the side with the advantage.  Not as extreme as a +1, but then again, conceptually, saying that one side or the other wins ties is like saying they got a +1/2.

What we see is that in the case of an opposed dice roll (found in games going back to Featherstone and Grant, of course, but still present in modern sets) getting a +1 or even the benefit of winning ties is a very large bonus.

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This idea of opposed dice rolls is common in many rulesets.  But in some others, there is the possibility of a player rolling several dice (typically more dice, for higher skilled combatants, for instance), and then selecting the highest dice, before comparing it to the opposition.  In this case, for instance, there is a much greater chance for a soldier who rolls 3d6 and selects the highest, to have a great number than his opponent, who is only rolling 2d6 to select the highest.  The component of a good number is still present, as a singleton dice can always come up as a 6, and the highest opposition roll could be any value less.  But what is the probability?

Lets construct a table, where we record the odds (and percentage chance) of getting a number, between 1 and 6, if it is the highest (or tied for the highest) out of a pool of dice.  To keep this simple, and illustrative, I am going to do it for three pools of dice - a single D6, 2d6, and 3d6.  Notice the shift in probability...

Highest Result	1d6 Prob.	2d6 Prob.	3d6 Prob.
1	1/6 (17%)	1/36 (3%)	1/216 (.5%)
2	1/6 (17%)	3/36 (8%)	7/216 (3%)
3	1/6 (17%)	5/36 (14%)	19/216 (9%)
4	1/6 (17%)	7/36 (19%)	37/216 (17%)
5	1/6 (17%)	9/36 (25%)	61/216 (28%)
6	1/6 (17%)	11/36 (31%)	91/216 (42%)

Again, there is a dramatic shift in probability, just by changing the dice rolling mechanism slightly.  By rolling 2d6, and selecting the highest number, your chance of having a 2 is halved, and your chance of having only a 1 is dropped to approximately 3% chance.

So the chances of rolling a higher number, when you can pick from a pool, are better.  That is intuitive, but looking at the table for 3d6, we see over a 40% chance that your number will be a 6.   And if you need AT LEAST a 5 (which means that the results for a 5 or a 6 will satisfy) your chance increases to 60% (42+28), and so on.

At this point, I will abandon the exercise.  It might be nice, to compare the chances of a player rolling a pool of 2d6 to beat a player rolling a pool of 3d6, but I think I would rather return to writing reviews.  And maybe lunch.