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Hey all,

I'm writing a ruleset based around D6s, and I'm running into a couple of elementary maths problems. If anyone can point out where I'm making the mistake, I'd appreciate it!

The rules are based around the humble D6, with 1 being good and 6 being bad. When making a test, you try to equal or beat your stat (also from 1 to 6) – so a model with resilience of 4 would need to roll a 4, 5 or 6 to pass a resilience test. Simple enough, right?

However, there are a number of modifiers:
*, which indicates you may re-roll the die for the test if you fail. If either the first or the second roll are successful, you pass.

+, which indicates you roll an additional die for the test. If either or both are successful, you pass. If both are failures, you fail.

!, which indicates you always test against the basic stat. This doesn't make any difference to my question, but I'll keep it in for completeness in case I need some help with modifiers etc.

So, what are the formulae required to work out the survival rates of the various permutations?

The basic stats are easy:
6 = 1/6 chance of passing
5 = 1/3
4 = 1/2
3 = 2/3
2 = 5/6
1 = 6/6 (i.e. automatic)

The reroll(*) stats, I think you add the chance of success to the chance of failure multiplied by the chance of success: a + (b x a), where b is chance of failure (since you need to fail to attempt the second test).
6* = 1/6 + (5/6 x 1/6) = 11/36
5* = 1/3 + (2/3 x 1/3) = 5/9
4* = 1/2 + (1/2 x 1/2) = 3/4
3* = 2/3 + (1/3 x 2/3) = 8/9
2* = 5/6 + (1/6 x 5/6) = 35/36
1* = 6/6 + (0 x 6/6) = 6/6

For the extra die (+)  tests, do you just add the two chances together?
6+ = 1/6 + 1/6 = 1/3
5+ = 1/3 + 1/3 = 2/3
4+ = 1/2 + 1/2 = 1
3+ = 2/3 + 2/3 = 1 1/3
2+ = 5/6 + 5/6 = 1 2/3
1+ = 6/6 + 6/6 = 2
This doesn't look right at all – surely there should be a chance of failure! What am I missing?

As far as I can tell, * and + are identical. In both cases you roll 2d6. At one time you roll them simultaneously, at the other time one after the other which should not impinge on the percentages.

Ah... that might be it. Hmm, might need a rethink:

The mistake I made was the equivalent of comparing a combi-bolter to a stormbolter (at 24" but of course a stormbolter has the chance to hit twice.

Any statisticians that can confirm that?

...And any suggestions for a similar modifier?

+++Edit+++

Please note that the mechanic has to revolve around a single die – to assist with speed of gameplay: using two dice means that you have to identify pairs within a batch of rolling – too difficult!

Of course, what options are there for rolling dice?

Roll a die

Re-roll a die

Roll more than one die... hmm

Ah Statistics... *rolls my eyes*

To answer your question (if I am understanding your question correctly), your * and + are both trying to obtain the same result. You want 1 success, and you get two chances. The only thing you are changing is when / how you roll the second chance.

In your * example, you wait to roll a failure before trying to re-roll a success.

In your + example, you roll both of your chances at the same time, but you just need 1 sucess.

Anyway, CPL Saint, instead of a d6, you might want to try a d20. You get a bit better of a bell curve, and you can actually have tables and abilities that matter.

Also, take a look at the d20 systems that WOTC came out with. In short, all your rolls you want to be HIGH numbers. Your bonuses add to your d20 roll. You never have to remember if you want a high roll or a low one, as you always want a high roll.

In my opinion, one of Warhammer's biggest problems is it is all d6 based. But, half the time you want numbers above something, the other half the time you want numbers below.

If you want to stay with the d6 idea, take a look at the GRUPS stats tables that Steve Jackson games has. 3d6 still has problems (as 10 - 11 tends to be your average, but you have values from 3 - 18), but you do get a bit wider of a bell curve than d6 or 2d6 gives you. And, as an added bonus, Steve Jackson Games has already done all the math.

Or take a look at the L5R rules, which are 2d10. Again, AEG already calculated the various chances of something occuring.

Overall, if you just want to keep it to being one roll, I'd suggest something like this.

To Hit formula = 10 - bonus + penalties

For example, lets say your trooper has "accurate" (which gives him +1 hit). Your target is behind hard cover (which gives a -2 to hit), and has BIG ARMA (-2 to hit).

So, using the formula you get

To hit = 10 - 1(+1 accurate) +1( -2 hard cover - 2 BIG ARMA)

To hit = 10 -1 +2 +2 = 13.

Now, you need a 13 or better to hit.

Another big advantage of using a d20 is that you can combine a lot of rolls into one roll. For example, to actually remove an enemy model you need some 3 or 4 rolls (to hit, to wound, armor save, and then possibly a ward save). If you combine all those into a single roll, you do speed up the game.

Anyway, just some quick thoughts.

+++EDIT+++

Well, phoey, looks like someone already informed you of the "re-roll" thingy at least.

Thanks for the input. I'm sticking with d6s for a number of reasons:

1) Availability and familiarity.
2) I prefer the 'jumps' to a smooth bell curve – I think it adds to the 'game' feel, rather than a 'simulation'.

Weirdly, I'm thinking that the reroll and two dice approach, while giving identical results, will still 'feel' different enough in play.

Ded 'ard orkses get two dice with their shootas, while tricksy eldar can reroll to hit.

Identical chances, different 'feel'.

Well, this applies to all dice – combining D20 rolls just gives you a more gradated approach than D6s. In practise, when I've played D10 or D20-based games, bigger dice slow things down for the simple reason they're harder to read!

Thoughts?

Posted By Cpl_Saint on 04/19/2007 6:55 AM

In practise, when I've played D10 or D20-based games, bigger dice slow things down for the simple reason they're harder to read!

Are you serious?  The only way this could reasonably be a factor is if everyone in your group wears coke-bottle glasses. 

CPL Saint,

Something else to think about. If you want to keep the d6 system, you might also want to take a look at Star Fleet Battles. That system has been around for like... 30 years, and has a lot of time to test out what works and what doesn't. It is usually considered a solid, and very fair game. It uses single d6s a lot, coupled with tables. Anyway, it's another d6 system that might give you some ideas on game balance and results and whatnot.

My maths is a bit rusty but when caclulating 2 die results:

Calculating 6+
Make a matrix 6 x 6.
Numer rows and columns 1-6
Write the sum of row + column in each cell.


1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12


Count the numbers above 6. = 21

Divide with total 36 which gives 21/36

Or... 7/12?
So...?

I'm not mathmagician but rolling two dice vs. re-rolling for success will give you the same chance unless the re-roll mechanic isn't automatic and is one of those things where you have to decide whether or not to use it before you try and if you elect not to use it then you're stuck with the result. In that case it's not as powerful as roling two dice. The only way that mechanic works is to have a finite limit on re-rolls other wise you'd always do it.