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All right stop, collaborate and listen!

The topic of Dice probabilities seems to come up a lot in this forum, so let's get a nice thread out on the subject. Here is a graph to get us started about rolling 2d6....



You can see a nice shaped Bell Curve, and chances are if you are rolling and combination of dice you will see the same bell curve develop over the number of options you can generate. Multiple dice tend to aggregate towards an average result.

For a single dice, you see the following results.



You can see the difference. Discuss.

2dX isn't even close to a bell curve - it's triangular, but doesn't flatten at the peak, nor taper at the ends.

You need 3dX, preferably 4dX before it starts looking like a bell. The key property of a bell is that the shape holds constant as you process it against itself:
- 2dX v 2dX does not look like a 2dX curve; however,
- 4dX v 4dX still looks pretty much like a 4dX curve.

Also, procedurally, 2+dX needs to process model-by-model. You lose the ability to process models simultaneously, unless you are tweaking the curve or target.

Yeah, 2d6 gives a more normalized distribution curve with outliers to the ends, but doesn't reach bell curve. If you want to see what adding more dice does to probabilities, here you go: https://wizardofodds.com/gambling/dice/2/

Now let's get to the nitty gritty- do you want a more normalized distribution? It depends. With a linear distribution, for example, a +/- is always a set percentage. So with 1 6 sided dice, a modifier of +1 is always +16.67%. With 2 dice, it not so!

 jmurph wrote:
Now let's get to the nitty gritty- do you want a more normalized distribution? It depends. With a linear distribution, for example, a +/- is always a set percentage. So with 1 6 sided dice, a modifier of +1 is always +16.67%. With 2 dice, it not so!

If the core mechanic is QdX, the correct modifier isn't +/- N, but rather to add N dice, removing highest/lowest. This preserves the curve.

OTOH, if you're still looking at success as +/- 1 StDev, then you might as well be rolling 1d6 against 3+ to 5+ targets, as the variance is a single d20 pip. Faster and simpler and cheaper and easier.

It also depends on how you set your definer of success. In craps, you will if you roll a 7 (a 1 in 6 chance on 2d6). But if you won on a 7+ your odds would be quite different. See the basic dice mechanic in Traveller - rolling 8+ on 2d6.

For varying odds, you can set a success requirement of, say, 2, 2 or 12; 2, 3 or 12; 2, 3, 11 or 12 etc on 2d6.

Precinct, you used d12, but also a cancelling mechanism. Care to talk through your decision making with that?

I did a *lot* of math-y scribbling before I went firm, but the mechanic originally came to me out of pure intuition.

I was already a fan of d12s. They're wonderfully versatile, whilst still being really legible. After d6s, they're really the most easily legible of all the polyhedral dice, and base 12 is a lovely way to do maths. Working out probabilities is much easier when you can instantly do one-half, one-third, one-quarter, two-thirds, three-quarters... all in your head.

The idea of exactly matching arose from some thought experimenting I was doing with the preliminary mechanics. I'd already built Armour into the rules as a range modifier, but it's a trope of miniatures game design that it's good to give the opponent something to do while you're shooting at him! This gave me the idea for a Defence stat, separate from Armour and some sort of opposed roll.

I was already wedded to a "your stat equals the number of dice you roll" mechanic from Firepower and thought about something similar. But a straight-up "beat your opponent's roll" swung the odds too far in the defender's favour whilst also shutting the door on cancelling natural 12s. Natural 12s provide critical hits and I wanted there to be a sense of drama around those moments when your opponent cancels a crit with the Defence roll.

Because of the grouping mechanic in shooting (you group dice together to create clusters of dice, the combined value of which beats the target number), I realized that cancelling just a single dice (even a paltry 1) could have a significant impact on the number of successful hits.

After much scribbling and analysis, I also worked out that the value of a Defence dice is geometric as you add to it. D1 is pretty rubbish, but D2 is more than twice as good and D3 is really quite good indeed. When you get up to D4, your odds of cancelling at least one of your opponent's dice are really good. This meant that investing stat points into Defence was a legitimate decision for commanders. If you're deciding whether to put one or two, it's not a big decision. But if you want to put 3 or even 4 points into D that makes a big impact of the element's survivability (D4 Light Infantry are harder to kill that you expect, despite their vulnerability to critical hits!).

So as well as fitting with the statistical distribution of hits and misses, it also meant forcing commanders into difficult decisions in the design process, which was very much what I wanted.

Overall, although there are a few things about Horizon Wars with which I'm not 100% happy, the mechanics for shooting are probably its strongest legacy, mechanically.