"Buckets of dice - NOT!" Topic

So in this thread I didn't explain myself well:

TMP link

Here's what i want to do. Let's say you are rolling 12 dice and need a 3+. The odds of one success are x%, the odds of two successes are y%…the odds of 12 successes are z%

If I convert that so that my 2D6 gives me the same spread of results, I get the same game with less dice.

So let me ask this: can someone give me the formula that:

For a given number of dice (nD6) the chance of 0 successes, 1 success, 2 successes, 3 successes….n successes?

catlikecoding.com/anydice

They make tools for this sort of thing. But the short answer is, the probability curves are going to be totally different, and you won't be able to cleanly fit the results into 2d6 or even 1d100.

link Results for your dice mechanic above.

Bucket of chicken YES!

Why are you rolling 12 dice? That is a mechnic that might require the 12 dice or the needed result % wise may seem to require the use of fewer dice.

Is it the number of firers? Then 12 dice need to be rolled since it is a chance for each firer to hit. Is it the artillery and you have a low troop ratio which means a lot of castings are attacking you? Then you still to roll 12 dice as that indicates the number of possible casualties with 3+ indicating the chance that a hit is acheived.

If your troop scale is a larger (1:30 or higher) then different mechanics can be used to lessen the number of dice needed to be rolled.

You explained it just fine the first time.

In essence if you roll two dice looking for a sum of 2-12, or even a 11-66 distribution, you are dealing with a single result.

If on the other hand you have one player rolling 10 dice that hit on a roll of 4+ you would, on the average get 5 hits. If there is an armor save roll of 5+ on the average 1/3 or in this case 1 2/3 (lets make it 2 to keep the math easy) of the time, the hit is saved by the armor. So . . . a kill is achieved 3 of 10 times on the average.

Oh boy, you could just roll a 10 sider or better yet, not roll at all. Just collect your three kills. No need to waste time rolling dice.

The problem arises the next time you shoot. Now with only 7 models needing a 4+ you get 3.5 hits and the save is 1/3, so 3.5/3 hits--or just to make the math easy a single hit??

That's right -- 10 models get you three hits and 7 models get you 1 hit? What's with that? I could make up more silly examples all night.

That is why your 'conversion' doesn't work with a system like this. Each model has a single roll and it's own unique chance to 'hit'. There are a lot of times you roll different colored dice in FOW because you want to single out that LMG or what ever because the odds are a bit different either in terms of 'to hit' or perhaps 'to save'.

I mean just to take this to ridiculous extremes, why not reduce all the rolls needed in a Napoleonic game to three; the opening battle roll, the decisive moment of the battle roll and the pursuit roll. There are board games that do just this kind of thing. With a miniatures game this is somehow less than satisfying.

mjc

You explained it fine and Bill's table gives you all the data you need.

But 2D6 might not be enough since that gives you ~3% increments. If you look at Bill's 12D6 3+ example, the prob of getting 0-4 hits is 1.88%, so are all covered by the 2D6 roll of 2.

Rolling 3D6 gets you ~0.5% increments but even then, the prob of getting 0-3 hits is 0.39% so triple 1s covers all of them.

Essentially using a table and rolling fewer dice will mean you can't replicated the chances of rolling extreme results. No more rolling 5 sixes on 10 dice kinda thing.

Never tell me the odds….Han Solo

This is a binomial distribution. If you have excel or Open Office Calc you can use the BINOMDIST function to get the probabilty (0 to 1) of x successes from n trials, where p is the probabilty of success for an individual trial.

BINOMDIST(x, n, p, cumulative)

For "to hit" value h (from 1+ to 7+), p = (7 – h)/6.

Example: 4+ to hit is p = (7 – 4)/6 = 0.5

Yeah, I decided I'd try 2D6 but read them like percentiles.

Thanks bill that was exactly the data I needed!

Again, the idea is to recreate the possible results and distribution but just do so with a lot fewer dice. Less visual impact and speedier play. No more counting dice, finding a place to roll, checking results, etc. It's always 2D6…

Of course, this is only because I like FoW and hate buckets o'dice.

I like Buckets O'Dice, because It is a faint simulation of each individual man firing or fighting. There is an "average" number of hits, but there can also be a higher or lower number.
I dislike games where after you tote up the factors, you "kill 4 and have a 38% chance of killing a 5th." That seems leaden and sterile. I like the approach where every man can miss, or every man hit, and with no certainties.

But, that is not the question you asked, is it?

If I convert that so that my 2D6 gives me the same spread of results, I get the same game with less dice.

Only approximately, not exactly the same.

@The OFM: If you don't like my question of course you;re always free to answer one you like better…

@Jim:

Yeah, but I'm okay with it.

After working it out I don't like all the charts I'd end up with either.

So it's buckets o'dice for now…

See if you can approximate 1 chart by shifting another chart in one or both axis. Can prob do it all in excel.

Example
Try 4+ as your basic chart with 2-12 on 1 axis and 3-30 on the other.
Take the 5+ chart and shift it 1 (or 2 etc) column left or 1 etc rows up.
Calculate the error (sum of mod of differences might work)

If the error is reasonable you can use a column or row shift on the 4+ table to approximate the 3+, 5+ and 6+ tables:-)

If the error is not reasonable you've wasted a bunch of time:-(

So why use D6 anyway, especially 2D6 (which gives you something of a binomial curve)? And why use 2D6 reading each die for 11-66 (which means is you roll 16 and add+2 the final result 22 which is not intuitive)? Why not use percentile dice for a true 1-100? I could never figure out why people went away from % dice. Or if you feel that having a +1% or +2% isn't enough of an impact to the odds, then use 1D20 which gives you nice 5% increments.

The problem with d20 is that they never stop rolling, usually off the table where the cat pursues them under the piano.

I don't care for D20 as they are hard to read and often end up "cocked" on undulating terrain. I could use two D10 I suppose. But I like using D6 as they are so easy to find.

Using 2D6 as digit dice does not end up in a curve – it is a straight line. Basically you use them exactly as percentile dice. So red 6 and white 2 = 62 (not 8). Red 3 and white five =35, and so on.

Essentially using a table and rolling fewer dice will mean you can't replicated the chances of rolling extreme results.

True, but some might think it's worth giving up the possibility of highly improbable flukes, to gain something else (like fewer dice on the table). But then others might not--there's no "right" answer.

Have you convinced your opponents to do this?

If you want to experiment with something really different with charts, you might be able to work something out using Nomography. See:

projectrho.com/nomogram

Somebody here on TMP -- I'd give attribution if the search function worked well -- mentioned it as a solution for "precalculating" complex equations for lookup on paper.

There seem to be Nomographs for binomial distribution, but I don't have the math skills to tell you how to apply them. This site:

link
(note: SSL certificate used, browser may fuss)

Seems to be touching on a subject you might be interested in.

That'll keep you busy for a bit!

- Bob

Sorry EC looks like I posted my thoughts on the wrong thread. It does seem out of whack.

As far as the opponents go…I only play with my gaming group and since I bring troops for both sides, my house my rules so to speak!

If I played a "pick up" game or with someone I didn't know I'd just bring my regular 6500 dice…

"Normal range" has specific meaning in probability. For most tests the majority fall into a peak in the middle. If you plot the result on the X-axis and quantity of each on the Y-axis, you get a bell-shaped curve. When outlying results start being significant, they can be broken into bands of less and less likely probability, referred to as standard deviations from the normal range.

The advantage of 2D6, the reason game designers like it, is its results table looks like the bell curve described above. This is a very effective design tool to abstract any result that is expected to follow a normal range.

So … assuming you expect your test to fall into a normal range, what you want to do is easy. Just pick how many standard deviations you want to model and their proportion to the expected average result. Apply that to your 2D6 results table and you are done.

So suppose you want to model up to 2 standard deviations. The 2D6 table looks like this:

2D6:
2-3 -2 SD
4-5 -1 SD
6-8 Normal
9-10 +1 SD
11-12 +2 SD

For normal apply the statistically average result. For Standard Deviations, add or subtract the proportion you want to model.

So for your example of 30 dice hitting on 3+, normal is 20 hits. Plus or minus an SD of 15% changes that by an increment of 3. Roll a 2, result is 20 – 2x3 = 14 hits. Roll a 9, result is 20 + 1x3 = 23.

For more chunky results use larger percentages for the SD. For more granular use smaller percentages and break the chart out more to give more SDs. If every possible result is its own standard deviation band, you get 7 is average, and 2 or 12 is minus or plus 6 standard deviations.

Now you also mention using 11-66. That is just a flat distribution. To get a normal range you then have to model it in your results chart, instead of just letting the dice do it for you. It's more work in other words.