The probability and predictability of dice.

[Anatosuchus]

Re: The probability and predictability of dice.

 

If someone games for 72 years' date=' for 23 hours each day, and rolls once every second, they will just make it. Jolt Cola, anyone? [/quote']

Not necessarily. It could just as easily be their first roll of the 2 billion as their last.

[Guyon]

Re: The probability and predictability of dice.

 

Not necessarily. It could just as easily be their first roll of the 2 billion as their last.

 

But probably won't.

[prestidigitator]

Re: The probability and predictability of dice.

 

Normal distribution is a situation where several things are true. The first thing that is true is that there are as many possible results greater than the mean as less than the mean.

Which, for a dice roll, means that the bell curve is symmetrical about the mean, right?

Not quite. The very definition of the mean is that there is an equal probability of rolling above or below it. That a distribution has a mean does not imply it is symmetrical about any point. The reverse happens to be true though: if a distribution is symmetrical about a point, then that point has to be the mean. And it just so happens that both normal distributions and all the die-related distributions we have in Hero have a point of symmetry.

[Silbeg]

Re: The probability and predictability of dice.

 

?

 

Not nearly enough samples? Go ahead and work out a couple more examples for yourself then. I guarantee you that they all fit materially within a normal distrbution. The reason that I did 2d6 as the example is that it is the furthest away from a zero-tolerance normal distribution.

 

By the way the odds of rolling 18 four times in a row on 3d6 are 1 in 2,176,782,336. Despite somewhat robust HERO play I don't think I have come anywhere near 2 billion rolls.

 

I have seen an 18, 3, 18 sequence, which has the same probability. It was truly bizarre... and was a campaign changing event.

[Conduit]

Re: The probability and predictability of dice.

 

I have seen an 18' date=' 3, 18 sequence, which has the same probability. It was truly bizarre... and was a campaign changing event. [/quote']

 

Just a quibble but that is far more likely at only 1 in 10,077,696 You would have to added another 3 or 18 for the win.

[Silbeg]

Re: The probability and predictability of dice.

 

Just a quibble but that is far more likely at only 1 in 10' date='077,696 You would have to added another 3 or 18 for the win.[/quote']

 

Well, the final roll (for 2d6 damage) was 12, so the odds decrease to 1 in 362,797,056 - still I grok your drift.

[Cancer]

Re: The probability and predictability of dice.

 

Not quite. The very definition of the mean is that there is an equal probability of rolling above or below it. That a distribution has a mean does not imply it is symmetrical about any point. The reverse happens to be true though: if a distribution is symmetrical about a point' date=' then that point has to be the mean. And it just so [i']happens[/i] that both normal distributions and all the die-related distributions we have in Hero have a point of symmetry.

Minor quibble: that first sentence is the definition of median, not mean. For a symmetric distribution, the mean and median are equal, and what you say there is correct.

[Dust Raven]

Re: The probability and predictability of dice.

 

Have you seen the semi-transparent 6-sided ones' date=' with a smaller die inside of a hollow larger one?[/quote']

 

I have a d6 that was picked up in Vegas by a family member on vacation that is made of clear hollow plastic, and inside are two red opaque d6. That's 3d6 and exactly what I need to make any Skill or Attack roll! It's awsome!

[prestidigitator]

Re: The probability and predictability of dice.

 

Minor quibble: that first sentence is the definition of median' date=' not mean. For a symmetric distribution, the mean and median are equal, and what you say there is correct.

Ah! Quite right. Sorry about that.

 

In fact, for a discrete probability distribution it can quite often be the case that the median is not a well-defined value. For the distribution of a perfect d6, we can take the median to be anywhere in the range of values (3, 4), whereas the mean (which is actually defined as the expectation value or first moment of the probability density function) is exactly 7/2. The mean is well-defined for a generally wider set of distributions (there are a few cases--e.g. the Cauchy distribution--in which the mean is not technically, "defined," though there is IMO still usually an obvious and practical choice for it).

 

Again for a symmetic distribution they are basically the same (ignoring some weird distributions like Cauchy), except that for a discrete distribution like we have with dice there might be a range of possible values for the theoretical median, but a well-defined value for the theoretical mean.

[prestidigitator]

Re: The probability and predictability of dice.

 

I have a d6 that was picked up in Vegas by a family member on vacation that is made of clear hollow plastic' date=' and inside are two red opaque d6. That's 3d6 and exactly what I need to make any Skill or Attack roll! It's awsome![/quote']

Wow. I'd imagine the outer one is not extremely easy to read, but still: cool!

[PhilFleischmann]

Re: The probability and predictability of dice.

 

I don't recall ever seeing the word "Average" used to mean "Median" or "Mode", at least not correctly. When it's used to refer to a specific mathematical statistic, it always refers to the Arithmetic Mean, as far as I know. There is also the Geometric Mean, which no one has yet mentioned: the Nth root of the product of the N samples. The Geometric Mean has no use in HERO that I can think of.

 

And BTW, IMO, the original thread was never derailed - it just got over-heated, but it was always on topic. The topic was "What can one do if one wants to not roll so many dice?" Several different solutions were given. All of those solutions came with various "side-effects". The discussion of those side effects is very much relevent to the topic.

 

I want to achieve X.

To do so, I will use method Y.

This method has the additional effect of Z.

Do I consider Z a good thing or a bad thing?

If Z is a bad thing, is it worse than not having X? Can I live with Z as long as I achieve X? If not, is there some other method, besides Y, that will also achieve X?

 

The discussion of Z is important to the topic of X. Not a derailment at all.

 

X = not having to roll so many dice

Y = use partial standard effect (other methods were mentioned as well)

Z = results of rolls will be more predictable.

 

The argument was over the nature of Z. Some were saying that Z was that the results would be less predictable. There seemed to be much confusion over what "predictable" means, in general, and specifically in regard to the HERO System.

[Dust Raven]

Re: The probability and predictability of dice.

 

Wow. I'd imagine the outer one is not extremely easy to read' date=' but still: cool![/quote']

 

It's a big clear die with big red dots, with too little red dice with white dots. Not that hard to read, though I might have to tilt my head if part of one of the inner dice fell behind a dot.

[Dust Raven]

Re: The probability and predictability of dice.

 

And BTW, IMO, the original thread was never derailed - it just got over-heated, but it was always on topic. The topic was "What can one do if one wants to not roll so many dice?" Several different solutions were given. All of those solutions came with various "side-effects". The discussion of those side effects is very much relevent to the topic.

 

I want to achieve X.

To do so, I will use method Y.

This method has the additional effect of Z.

Do I consider Z a good thing or a bad thing?

If Z is a bad thing, is it worse than not having X? Can I live with Z as long as I achieve X? If not, is there some other method, besides Y, that will also achieve X?

 

The discussion of Z is important to the topic of X. Not a derailment at all.

 

X = not having to roll so many dice

Y = use partial standard effect (other methods were mentioned as well)

Z = results of rolls will be more predictable.

 

The argument was over the nature of Z. Some were saying that Z was that the results would be less predictable. There seemed to be much confusion over what "predictable" means, in general, and specifically in regard to the HERO System.

 

Well, then it got pushed past Z into Z also means N, and whether Z did or did not mean N had absolutely no relevance to the topic.

 

N = a roll of fewer absolute dice result in more preditable results.

[PhilFleischmann]

Re: The probability and predictability of dice.

 

Well, then it got pushed past Z into Z also means N, and whether Z did or did not mean N had absolutely no relevance to the topic.

 

N = a roll of fewer absolute dice result in more preditable results.

Huh? What is the difference between Z and N as you see it? They look the same to me.

 

Oh. By "absolute dice", do you mean dice without adding the standard effect value for the dice not rolled? I guess that must be what you mean.

 

Yes, there was also some discussion of whether or not 2d6 was equally predictable as 2d6+28. To me it seemed obvious beyond the need of discussion that they were, but others were using different definitions of "predictable".

 

And so the discussion turned to the meaning of "predictable" as it pertains to the HERO System. This seems to me to be nothing more than a clarification of the meaning of Z, and therefore still relevent to the topic of the thread.

[Dust Raven]

Re: The probability and predictability of dice.

 

Huh? What is the difference between Z and N as you see it? They look the same to me.

 

Oh. By "absolute dice", do you mean dice without adding the standard effect value for the dice not rolled? I guess that must be what you mean.

Sure, I guess, whatever.

 

Yes, there was also some discussion of whether or not 2d6 was equally predictable as 2d6+28. To me it seemed obvious beyond the need of discussion that they were, but others were using different definitions of "predictable".

Equally predictable, but not of equal value.

 

And so the discussion turned to the meaning of "predictable" as it pertains to the HERO System. This seems to me to be nothing more than a clarification of the meaning of Z, and therefore still relevent to the topic of the thread.

Not really.

[prestidigitator]

Re: The probability and predictability of dice.

 

Z? N? I'm totally lost.

[Dust Raven]

Re: The probability and predictability of dice.

 

Z? N? I'm totally lost.

 

My work here is done.

[SteveZilla]

Re: The probability and predictability of dice.

 

Not necessarily. It could just as easily be their first roll of the 2 billion as their last.

 

I meant to say (I thought I implied) that I was referring to just the number of rolls. I know that it's theoretically possible for an astronomical roll to occur early on. You beginner luck!

[SteveZilla]

Re: The probability and predictability of dice.

 

I'm still trying to digest what everybody has said, and I've been doing some coding to try to help the process.

 

I came up with a web-based program (in PHP) that will show the distribution of probabilities (the chance of each result possible) for any group from dice from 1d6 to 30d6. One of the neat things is that I didn't have to run though every combination with nested for loops. That way would have taken forever for 30 nested loops!

 

I did have to write my own subroutines for addition, subtraction, and division of integers. Some as large as 24 digits! Not only was that beyond the scope of an integer in PHP (32 bits), but also beyond the scope of a LARGEINT in MySQL (64 bits)! To store such a large number in binary form, I'd need a 78 bit unsigned number.

 

Here's the link.

 

The scale used for the graph remains constant regardless of how many dice are chosen (max on the scale corresponds to a 16.6667% chance). That way, it's possible to make direct visual comparisons.

 

However, for large numbers of dice the "legs" have so small a chance that they all resolve to a 1 pixel long bar. This is unavoidable without making the scale so large that the graph would be (IMO) useless. This is why I also included the number data.

[prestidigitator]

Re: The probability and predictability of dice.

 

I'm still trying to digest what everybody has said, and I've been doing some coding to try to help the process.

 

I came up with a web-based program (in PHP) that will show the distribution of probabilities (the chance of each result possible) for any group from dice from 1d6 to 30d6....

Hmm. I believe there may be a defect of some kind. If I choose 12d6, a couple of the bins on the outskirts appear larger than those closer to the mean. Only visually, though; the numbers don't reflect it.

[SteveZilla]

Re: The probability and predictability of dice.

 

Hmm. I believe there may be a defect of some kind. If I choose 12d6' date=' a couple of the bins on the outskirts appear larger than those closer to the mean. Only visually, though; the numbers don't reflect it.[/quote']

 

I'm not seeing anything out of the ordinary when I pull it up. The script defines the width of the bar directly from the % chance value. If the # is correct, so should the length of the bar. Can you send me a screenshot of what you're seeing?

 

 

...

 

 

Also, anyone want to guess how I generated the curves for all 30 groups in under a minute?

[prestidigitator]

Re: The probability and predictability of dice.

 

I'm not seeing anything out of the ordinary when I pull it up. The script defines the width of the bar directly from the % chance value. If the # is correct' date=' so should the length of the bar. Can you send me a screenshot of what you're seeing?[/quote']

Perhaps it is another accuracy ieeue.

 

(Click for larger image; twice for full size, depending on your screen resolution.)

 

Also, anyone want to guess how I generated the curves for all 30 groups in under a minute?

Sure. You calculated each curve based upon the one before it.

C(D, N) = sum C(D-1, N-i) for i=1..6

C(1, N) = 1 for N=1..6

C(1, N) = 0 for 16

 

C(D, N) - The number of ways of rolling N on D six-sided dice.

[SteveZilla]

Re: The probability and predictability of dice.

 

Perhaps it is another accuracy issue.

 

(Click for larger image; twice for full size, depending on your screen resolution.)

 

The width for the first two results on 12d6 is expressed as a scientific notation number (2.8333860343576E-006, and 3.4000632412291E-005, respectively), while the others are just small decimals (0.00022100411067989, for ex.). Perhaps your browser translates exponenets and small decimals differently than I.E.6?

 

I could easly include a comparison so that if the width <0, width=1. That would likely fix it.

 

 

A few questions:

 

Are you only seeing this on the 12d6 graph?

 

Are you seeing an identical patter on the opposite end of the graph?

 

What browser are you using to access the page (and version, please)?

 

Did you try refreshing the page?

 

Sure. You calculated each curve based upon the one before it.

C(D, N) = sum C(D-1, N-i) for i=1..6

C(1, N) = 1 for N=1..6

C(1, N) = 0 for 16

 

C(D, N) - The number of ways of rolling N on D six-sided dice.

 

That's exactly what I did. I came to that realization on my own -- no help from a person or book. I find it interesting that the two different methods can take so vastly different times to compute.

[prestidigitator]

Re: The probability and predictability of dice.

 

Are you only seeing this on the 12d6 graph?

I seem to be seeing it on all graphs from 11d6 up, though I haven't checked them all. Here's what I see at the start for 18d6. Almost looks random. I wonder if least significant digits are being used or something.

 

(Click for larger size; again for full size, depending on screen resolution)

 

Are you seeing an identical patter on the opposite end of the graph?

Yes. The ends are symmetric as far as I can tell.

 

What browser are you using to access the page (and version, please)?

Mozilla/5.0 (X11; U; Linux i686 (x86_64); en-US; rv:1.8.0.1)

Gecko/20060313 Fedora/1.5.0.1-9 Firefox/1.5.0.1 pango-text

 

Did you try refreshing the page?

After a warning about POSTDATA, it appears the same after a refresh, or if I submit a second time.

 

That's exactly what I did. I came to that realization on my own -- no help from a person or book. I find it interesting that the two different methods can take so vastly different times to compute.

I love counting and algorithms.

[SteveZilla]

Re: The probability and predictability of dice.

 

I seem to be seeing it on all graphs from 11d6 up' date=' though I haven't checked them all. Here's what I see at the start for 18d6. Almost looks random. I wonder if least significant digits are being used or something.[/quote']

Yes. The ends are symmetric as far as I can tell.

 

That makes sense. The bigger the die pool, the lower those numbers will be.

 

Mozilla/5.0 (X11; U; Linux i686 (x86_64); en-US; rv:1.8.0.1)

Gecko/20060313 Fedora/1.5.0.1-9 Firefox/1.5.0.1 pango-text

 

That's significatly different than the systems I used to write the code. Both were WinXP with I.E.6.

 

I've modified the code to limit the low end to a width of one, and to also round the widths to the nearest integer. That should clear up any weirdness.

 

I love counting and algorithms.

 

I'm fond of algorithms, and especially so of coding. I prefer to let the computer do my counting for me, however.