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Author Topic: Dice probabilities question and meta-question  (Read 7400 times)
Robert Bohl
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« on: August 14, 2006, 09:27:45 AM »

1) I am wondering where I can find probability distributions for various dice pools, or how I can generate them.  Most of us know the 3d6 bell curve, what I'm wondering about is what the curves for the following look like:

...
1d4, 2d6
2d6, 1d8
1d6, 2d8
...

And so on, from 1, 1, 1d2 to 1d8, 2d12.

2) Does this matter?  Am I going to get anything useful from looking at these distributions?  Should I just go with my gut and have it playtested?

(PS: On the drive home from Gen Con I wrote the complete notes/outline of a game.)
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stack0v3rflow
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« Reply #1 on: August 14, 2006, 10:18:59 AM »

I'm not sure what you are looking for, but to find the average of a die you take the number of sides, divide by two and add 0.5.  So a 1d4 will roll a 2.5 average.

The 1d4+2d6 will have a minimum of 3 (since all of the dice could roll a one), an average of 9.5 (2.5+3.5+3.5), and a maximum value of 16 (4+6+6).

I hope that helps a little.
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-Aaron
Robert Bohl
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« Reply #2 on: August 14, 2006, 10:23:15 AM »

Oh, right.  I was more thinking, like this is the 3d6 bell curve:



So what would that look like for 1d4, 2d6, and so on.  Here are all the combinations I'm thinking of, by the way:

3d1
2d1, 1d2
1d1, 2d2
3d2
2d2, 1d4
1d2, 2d4
3d4
2d4, 1d6
1d4, 2d6
3d6
2d6, 1d8
1d6, 2d8
3d8
2d8, 1d10
1d8, 2d10
3d10
2d10, 1d12
1d10, 2d12
3d12
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Robert Bohl
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« Reply #3 on: August 23, 2006, 09:56:19 AM »

I'm going to make one bump attempt on this and then I'll let it die an ignoble death if that's its fate.  Does anyone have any input on this?  Maybe there's a website that can pump out these probability diagrams?
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TonyLB
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« Reply #4 on: August 23, 2006, 10:26:05 AM »

I sort of figured that someone would get you an interactive page for generating histograms ... but surprisingly, even when I went searching, I couldn't find one.  Is there seriously no tool out there that will map out the outcomes of 1d4+2d12?  That seems ... odd.

Anyway ... if all you're doing is adding independent variables then you're going to get shapes that look like the gaussian bell curve you're familiar with.  Generally speaking:

  • Mixing small dice with large dice, the results will not be very steeply slanted away from the edges.  For 1d4+1d20, any value from 5 to 21 is equally likely (I think) and the only places where the probabilities will be different are from 2-4 and 22-24.
  • Mixing same-size dice will tend to fiddle with the distribution across the whole range.  For 2d6, 7 is more likely than 6 or 8, and so on all the way out to 2 and 12.
  • More dice will create a larger field of possibilities (100d6 can go from 100 to 600) so the actual variation away from the centerline will be huge:  100d6 is not that much less likely to generate 345 than 350 ... that's five points away ... the difference in odds between 2d6 generating 7 and generating 2 (also five points away) is immense.  But, proportionate to the scale that you're operating, the probabilities will be much more tightly clustered.  The difference in odds between 100d6 generating 300 vs. 350 is a much bigger difference than the difference in odds between 2d6 generating 6 vs. 7.  When you scale down to the same overall size, more dice will result in a much tighter, sharper peak at their most probable point.

I hope that helps some.  There's a whole branch of mathematics, y'know, and it's hard to know what pieces you need.
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Valamir
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« Reply #5 on: August 23, 2006, 11:27:55 AM »

You know for the small number of dice you're looking at, it wouldn't be too hard to just do it the hard way and bang out an excel spreadsheet that lists all possible combinations...count each combination up and graph the result.
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Robert Bohl
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« Reply #6 on: August 23, 2006, 02:15:46 PM »

Tony,

Thanks for your attempts.  Actually what I'm searching for is the single thing you didn't mention, which is keeping the dice pool size the same, and the dice only ever varying by one step.

Valamir,

I might try that.  I started and then gave up because I wasn't sure what it would look like.  Also because if you look at the list of dice pool possibilities, that seemed like a lot of tables to write up.

Thanks for the attempts, guys.  Basically what I know about these is:

1) As the dice used go up, the central node of the bell curve increases, and vice versa.
2) Bigger dice make higher curves, smaller dice make flatter curves.
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jerry
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« Reply #7 on: August 23, 2006, 04:05:11 PM »

Tony: I think the problem is that most of us who program, end up doing this sort of thing on the fly and never wrap it up into a nicer piece of software.

Rob: You might look at statistical software. 'R' is free:

http://www.r-project.org/

and can generate histograms, though it's easier to use to generate random tests than full charts (it most likely can do the latter, since it does have a repeat function).

I wrote about it for dice checking:

http://www.hoboes.com/html/RPG/Gods/?ART=88

If you look towards the bottom, you'll see how to generate the standard 3d6 bell curve. If you wanted to instead plot 2d8+d4, for example, you could use:

Code:
samplesize = 500000
roll1 = sample(8, samplesize, replace=TRUE)
roll2 = sample(8, samplesize, replace=TRUE)
roll3 = sample(4, samplesize, replace=TRUE)
rolls = roll1+roll2+roll3
frequency = table(rolls)
plot(frequency, type="l", xlab="2d8+d4 rolls")

That'll generate 50 random rolls of 2d8+d4. Given that it's statistic software, I'm sure it can generate standard deviations and such, if that's what you mean by probability distributions.

Jerry
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Jerry
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jerry
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« Reply #8 on: August 23, 2006, 04:06:20 PM »

That would actually be "500,000 random rolls", not 50.
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Jerry
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Robert Bohl
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« Reply #9 on: August 23, 2006, 04:33:13 PM »

Thanks, Jerry.  I'll give that a try.  I also have access to SPSS and SAS through my job, so maybe I can work out something that way too if R doesn't work for me.  But it looks like it will.

Perhaps a broader question, what about original question 2?  Do people stress out over the statistical spread their randomizers present?
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Hans
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« Reply #10 on: August 24, 2006, 05:23:26 AM »

Rob:

Did a brute force excel spreadsheet that should meet your requirements.  See:

http://files.meetup.com/180731/ThreeDiceRoller.xls

Let me know if the link is bad or there are other problems.  Put the die types in the yellow boxes, and read the histogram and probabilities.

Hans
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Hans
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« Reply #11 on: August 24, 2006, 05:24:44 AM »

Did a brute force excel spreadsheet that should meet your requirements.  See:

Ah, one other thing, you will need to trust me enough to let the Visual Basic in the spreadsheet run...that's where the brute force is applied. :)
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Robert Bohl
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« Reply #12 on: August 24, 2006, 06:21:34 AM »

This is awesome, Hans.  Thanks.
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Lxndr
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« Reply #13 on: August 24, 2006, 06:30:02 AM »

On question #2:

A lot of people go with their gut, and are well-served by it.  Any really obvious statistical anomalies generally show up in playtesting. 

I'm a math-numbers-guy who's currently taking a course in probability, and I have fun figuring out things like that, and sometimes, if you have really complex things, it's good to know which selections are more common (like, rolling 4 dice, what will be more common: two pair, or a triple?).  But in the end, no matter what the numbers tell you, even I say 'go with your gut.'   Sometimes, the dice (or whatever you use) will take you places you weren't expecting.
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David Artman
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« Reply #14 on: August 24, 2006, 08:04:17 AM »

I'm not sure what you are looking for, but to find the average of a die you take the number of sides, divide by two and add 0.5.  So a 1d4 will roll a 2.5 average.
That gets you the median (mean?) value, not the average. For the average, I think one sums the lowest value and highest value, then divides by two.

Suppose a die rolls from 0 to 9? Your method delivers an average of 5.5, when it's actually 4.5.

Or am I totally wrong? I am a bit sucky at math (so I stopped after second year Calculus, changed majors from Computer Science to English and Philosophy, and never looked back!).

By the way, if you want your head to explode, read this PDF article:
Roll: A Language for Specifying Die-Rolls.
I don't know if it will tell you the stuff you want to know--my ears were smoking by about page three--but if you understand the whole PDF, then you should be able to own all of us WRT die mechanics and probability. ;)

David
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