A little help on probability curves please...

Maybe.

We did a run of information on curves from mismatched dice in the Multiverser appendix on dice curves. Not sure why we did it, as they're seldom used, but we wanted to try to cover all the bases.

With two matched dice, the combinations mirror each other; that is, if you can roll 3, 4, you can also roll 4, 3. The total number of possible permutations for each number rises by one for each number to the peak, then declines. That is, for 2d4, you've got something like this:

[*]2=1+1, 1/16[*]3=1+2, 2+1, 2/16[*]4=1+3, 2+2, 3+1, 3/16[*]5=1+4, 2+3, 3+2, 4+1, 4/16[*]6=2+4, 3+3, 4+2, 3/16[*]7=3+4, 4+3, 2/16[*]8=4+4, 4/16[/list:u]
The total number of permutations is the product of the sides of the dice, that is, for 2d4, 4x4=16 total permutations. The most common return will be total sides of one die plus one, and its frequency of occurence will be equal to total sides of one die over total number of permutations.

Now, when you make the dice uneven, you get that flattened pyramid outcome. Let's suppose that we still want to generate numbers from two to eight (as we did with 2d4) but we use d3+d5 instead.

The first thing that happens is obvious. We have (3x5=)15 possible permutations rather than (4x4=)16. Because for any combination of a+b=x where x is fixed, a*b will be greatest when a=b, uneven dice always create a smaller pool of permutations. That means the ends are going to be more heavily weighted up front. the odds of rolling 1, 1 are now 1/15 rather than 1/16, a slight increase in probability even though there's still only one chance to roll it. Next, note that in the pattern laid out above, each decrease on one die was met by an increase on the other. Thus with this new die combination, we'd find that 5+1=6, 4+2=6, 3+3=6--but now we can't go further on the d3. Because the d3 cannot roll 4, there is no way to roll 6 if the d5 comes up 1 or 2. That gives you the flattened top.

Adding another die is certainly complicated. Even with all the same number of sides, the curve loses linear form and becomes a bell. It's the result of the overlap of two curves. Any time you would have rolled 5 on 2d4, you now roll a number between six and nine--equal chances of each. A roll of 4 would give you equal chances of a number between 5 and 8, but the 4 was less likely than the 5 initially, so it boosts the odds on those numbers differently.

Overlaying the flattened pyramid curves is yet more complicated. You still have the total permutations as the product of the sides of the individual dice, and the highest possible outcome equal to the sum of the sides of the dice, with the centerpoint being the most common return; but you've got flattenings in the progress where the dice break. If you're using d3+d4+d5 instead of 3d4, you lose something whenever the d3 would have to exceed 3 for balance, and something again when the d4 would have to exceed 4. That gives you the odd shape.

My uncle could tell you how to calculate that in calculus; in cases like this, I usually just set up a program with nested loops and have it run all the permutations into a matrix and spit out the relationships. There are some examples in the aforementioned appendix, and a simple Basic program that does the job if you've got Basic on your system (MicroSoft included it in most version of DOS, but has left it out of the newer Windows stuff). I could probably run something for you, as I've got the program on my computer, if you need it.

--M. J. Young