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Exploding Dice Mathematics - a challenge for Math-hammerers!  [RSS] Share on facebook Share on Twitter Submit to Reddit
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Made in gb
Longtime Dakkanaut






Hello everyone!

I'm trying to work out the maths behind exploding dice, and it's doing my nut in.

The basic premise I'm looking at is:

Attacker rolls XD6, needing Y+ and exploding on 6's. count up the number of dice which rolled a Y+.

Defender then rolls AD6, needing B+, and exploding on 6's. count the number of dice which rolled a B+.

Exploding dice count as one success and you get one additional dice, which can also explode.

if the defender rolls as many or more B+'s as the attacker rolled Y+'s, then they are saved.

EG:
Attacker rolls 3 dice, needing 4+. Defender rolls 2 dice, needing 3+.

attacker rolls 3, 4, 6, then another 6, then a 5. result = 4.
defender rolls 1, 6, then another 6, then a 4. result = 3, so the attack goes through.

I've made an excel macro which runs 100 simulations on each attack & defence combo, but it's not the most efficient way of working it out!

Can anyone break this down into a formula so I can plug the X, Y, A & B values into it and get the probability of the attack succeeding?

I know this will probably be simultaneous equations, but I've always sucked at probability.

Can anyone work this out?

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Made in fr
Longtime Dakkanaut




The number of successes for the attacker (or the defender) without explosions follows a simple binomial distribution.
A good approximation of the final result (if the attacker had more success than the defender) could then be obtained with the Hoeffding's inequality.

Now the problem is with the explosions.
The number of successes for the attacker (or the defender) generated by the first explosions also follows a simple binomial distributions (with a probability that is just the success rate times 1/6). For the second explosion it's also a binomial, but with the probability multiplied by 1/36, and so on.
The total number of successes is therefore the sum of these binomial distributions (in theory an infinite number of them, but you could easily truncate quite early), but the variables aren't independent, so summing them isn't that easy.
There might be a way to still work it out, but I don't know how.
Alternatively, you could use normal approximations for the distributions, which would make things much easier. But I have no clue how accurate that would be, since I guess you're interested in the results when throwing just a few dice.
   
Made in us
Decrepit Dakkanaut






SoCal, USA!

Quite frankly, proper "Exploding" dice (where you keep adding the results, so 6+6+3 = 15) is one of the worst mechanics available. It's slow and non-intuitive to most players, and the numerics are not easily managed by most designers.

A far better mechanic is Critical 6s, where a bonus effect is triggered. This keeps the math simple for the player, and the rules clear to the designer. In your example, 6s should simply generate an extra success. If the target is a 4, and you roll 1-3-6, you have 2 successes; if the target were 3, you'd have 3 successes. Simple and fast!

That said, if your heart is set on Exploding 6s, go for it. Just be sure that you are using the mechanic for the correct reason, not because it's a novelty.

This message was edited 1 time. Last update was at 2018/10/01 19:30:18


   
Made in us
Dakka Veteran




Seattle, WA USA

Exploding dice are a little weird mathematically, and some folks really like the mechanic and some really hate it.

Anyhow, if you haven't seen anydice.com, it's a great site for various dice probability math. In fact, just looking this morning, they have an article about exploding dice that might be of help: https://anydice.com/articles/exploding-dice/
   
 
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