The probability and predictability of dice.

[SteveZilla]

Okay, this thread is for the continuation of a quite lively discussion about dice, probability, predictability, bell curves, and partial standard effect, to name a few. The "old" thread, which was for something different, is located

here.

 

We'll be leaving that poor, derailed thread to recover and continue on it's merry way.

[Siren]

Re: The probability and predictability of dice.

 

I bought my dice from a Las Vagas Casino. They are stamped out.

I would think they are as good as you get.

[SteveZilla]

Re: The probability and predictability of dice.

 

I bought my dice from a Las Vagas Casino. They are stamped out.

I would think they are as good as you get.

 

If they are "official" dice as opposed to "slightly defective rejects" or "souvenirs", then that's very likely.

 

But what does "stamped out" mean?

[Siren]

Re: The probability and predictability of dice.

 

Yep they are offical. The stamp out means that they can no longer be used in that casino.

 

Casino dice are only used so many times, then stamped out and disguaded or sold.

[prestidigitator]

Re: The probability and predictability of dice.

 

All right. All right. Well, I'll repost (as opposed to riposte ) this, then.

 

Here's a chart of the probability distributions of {2d6-7, 4d6-14, 6d6-21, 8d6-28, 10d6-35}. Take from it what you will.

 

(Click to see larger image)

[Dust Raven]

Re: The probability and predictability of dice.

 

Fine' date=' but you're just quibbling with words. You may take me to mean "a range of absolute numbers". Is it so hard to understand the difference between an [b']absolute[/b] result/number/range and a relavite result/number/range? That's the distiction I'm trying to make.

 

If the GM says, "The bad guy hits you for 114% of the mean," you have no idea what to do with this information. What you need is for the GM to say, "The bad guy hits you for 40 STUN." Then you know what to do: subtract your appropriate defense from 40 and apply the rest to your STUN, etc. "114% of the mean" is a relative result. "40 STUN" is an absolute result. 114% or more of the mean" is a relative range. "40 or more STUN" is an absolute range.

 

The absolutes are what matter in HERO (and AFAIK, every other game that uses dice, RPG or otherwise).

 

You are right, no one is going to talk like that in a game. One you've determined the result of a random event, only the result matters. But we're not talking about final results exclusively. We're talking about the probability of getting those results, and by extension, the probability of not getting them.

 

RE: 2d6+x (where x is greater than 0) has the same probability of 2d6+0

 

I did not say these were equal. I said the probability curve is the same. I thought it was blatantly obvious the possible results on each of these examples rolls were far from equal.

[Robyn]

Re: The probability and predictability of dice.

 

RE: 2d6+x (where x is greater than 0) has the same probability of 2d6+0

 

I did not say these were equal. I said the probability curve is the same. I thought it was blatantly obvious the possible results on each of these examples rolls were far from equal.

 

2d6+0 generates a result from 2-12, spread over 36 combinations.

2d6+6 generates a result from 8-18, spread over 36 combinations.

The probability of getting any single number will vary between the two, but both have at most 36 different combinations.

 

It's a lesser problem with two dice, though, than in The Dying Earth, where a single d6 is rolled for all checks. The range of results goes even further up as more dice (than 2) are added.

[SteveZilla]

Re: The probability and predictability of dice.

 

No' date=' no, no. You've gone all sideways on me.[/quote']

 

I looked only at the data you provieded and made logical, step-by-step conclusions from them.

 

2d6 is going to give a less predictible result than 10d6 will. We know' date=' because of the rules of probabilty, normal distribution and the law of large numbers[/quote']

 

I haven't taken any classes in this, or made it a serious intellectual hobby. I don't know the "official" definition of these terms: "predictable", "rules of probabilty", "normal distribution", "the law of large numbers". And is "mean" in this context the same as "average" (i.e., the "peak" or "middle" of the bell curve)?

 

 

...that 10d6 will' date=' over time, present results much closer to the mean result than 2d6 will.[/quote']

 

The only way I can see that being the case is if the definition of "much closer" varies by (is proportional to) the number of dice used. 2D6 may vary "wildly", but "wildly" is going to be no more than +/-5 from the mean/average. But 1,000,000d6 may vary only "slightly", and still be easly more than +/- 20 away from the mean/average. (I'm pulling numbers out of the air for the 1,000,000d6 part).

 

It is, that extent, more predictible. If you change a portion of the 10d6 to a static element and roll the rest you will make it even more predictible.

 

The confusion you have is that 2d6 and 2d6+28 are 2DC and 10DC respectively (Assume an EB if you must).

 

Can we please leave game-specific designations out of this? They have no bearing on the issue. The probability curve/distribution/whatever for 3d6 or 3D6+Y will be the same in Hero System, in D&D, in GURPS, and any other system that adds the dice together (and also a fixed amount, if present) to get a result.

 

They do not have the same distribution curve or mean.

 

Can you give a definition of "distribution curve", please?

 

If you define predictibility as the likely hood to roll close to mean' date=' 2d6 will be more predictible because the possible variance is small relative to the mean.[/quote']

 

When you say "more predictable" here, I presume you mean "more predictable than a greater number of dice"?

 

The problem is that we never do your statement two. Instead of adding a number to dice' date=' we change dice to a number.[/quote']

 

I think I'm sensing a potato - potato thing here. 2d6 is two dice. 2d6+Y is two dice with Y added to any result. The bell curve for each can easily be calculated. Nobody in their right mind is going to generate a bell curve graph for 2D6+Y the hard way (rolling endlessly).

 

Think of it as a formula with a variable (that is random, but generates some calculable bell curve of probabilities) and also a fixed componenet. It seems quite similar to basic algebra to me. We all know (or can calculate) what shape and size bell curve Xd6 will produce. Adding a fixed number to that doesn't change the shape and size, just the little numbers at the bottom of the graph (using a seperate piece of paper for each graph).

[prestidigitator]

Re: The probability and predictability of dice.

 

And is "mean" in this context the same as "average" (i.e.' date=' the "peak" or "middle" of the bell curve)?[/quote']

There is no set definition for, "average." There are several things that are used for averaging, the most common one being the mean. The mean is the value for which the probability of the value falling below (or on) the mean is one half, and the probability of the the value falling above (or on) the mean is one half (or at least the two are equal for a discrete distribution). In the case of a symmetric probability density function, the mean coincides with the point of symmetry. So for a normal distribution: yes, the mean is located at the peak.

[Conduit]

Re: The probability and predictability of dice.

 

I looked only at the data you provieded and made logical, step-by-step conclusions from them.

 

 

 

I haven't taken any classes in this, or made it a serious intellectual hobby. I don't know the "official" definition of these terms: "predictable", "rules of probabilty", "normal distribution", "the law of large numbers". And is "mean" in this context the same as "average" (i.e., the "peak" or "middle" of the bell curve)?

 

Mean is one of several methods of averaging (others are median and mode, there may be more but these are the main three). All of these are often referred to as the average of a set of numbers. Mean is the method of summing all of the members of the set and dividing the sum by the number of members. So the mean of 2, 5 , 8, 10 is 6.25.

 

The law of large numbers is, in layman's terms (which is all I really know), as more possible results are added the less effect each result has on the whole. A couple of common examples are batting averages and grade point averages. In the first month of a season, a batter's average can vary wildly, going up and down 100 points or more in a single game. By the end of the season, a single game will only cause a change of less than 5 points, if that. The reason is that each result is much more important to the whole when there are few of them. If I have 3 hits in 10 at bats (a .300 batting average) and go 3 for 3 in a game my average goes to .461. If I have 90 hits in 300 at bats (again .300) and go 3 for 3 in a game my average goes to .306. So in the terms of dice rolling, each additional dice has a much smaller affect on the likely outcome than any of the previous dice rolled, so extreme results (all 6s or all 1s or even anything approaching those results) become less and less likely as the number of dice rolled becomes large.

 

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. The second thing that is true is that 68.2% of the possible results fall within one standard deviation of the mean and 95% fall within two standard devations of the mean and 99% fall within three standard diviations of the mean. It is the "Bell Curve" which we so often talk about. It is useful because it allows us to make confident predictions about the behaviour over time of the object or circumstance being measured.

 

By rules of probabilty I just mean the calculation of possible results and their chance of happening on a given roll.

 

The only way I can see that being the case is if the definition of "much closer" varies by (is proportional to) the number of dice used. 2D6 may vary "wildly", but "wildly" is going to be no more than +/-5 from the mean/average. But 1,000,000d6 may vary only "slightly", and still be easly more than +/- 20 away from the mean/average. (I'm pulling numbers out of the air for the 1,000,000d6 part).

 

Correct. 2d6 most extreme result ±5 is in absolute terms equal to the standard deviation of (if I remember correctly) 10d6. But 5 is a much more significant number to 2d6 than it is to 10d6 and completely irrelevent to 1Md6. The standard deviation gets larger as you add dice but becomes less significant to the total faster.

 

Can you give a definition of "distribution curve", please?

 

Plot on a graph the possible results of a throw of xd6 on the x-axis and the number of ways to make a result on the y-axis. The resultant curve is the distribution curve (and happens to be a normal distribution curve)

 

 

When you say "more predictable" here, I presume you mean "more predictable than a greater number of dice"?

I mis-typed, I mean to say 2d6+28

 

 

I think I'm sensing a potato - potato thing here. 2d6 is two dice. 2d6+Y is two dice with Y added to any result. The bell curve for each can easily be calculated. Nobody in their right mind is going to generate a bell curve graph for 2D6+Y the hard way (rolling endlessly).

 

Think of it as a formula with a variable (that is random, but generates some calculable bell curve of probabilities) and also a fixed componenet. It seems quite similar to basic algebra to me. We all know (or can calculate) what shape and size bell curve Xd6 will produce. Adding a fixed number to that doesn't change the shape and size, just the little numbers at the bottom of the graph (using a seperate piece of paper for each graph).

 

Right, it is a different curve because it is shifted to the right. It doesn't vary in any other way.

[Cancer]

Re: The probability and predictability of dice.

 

Y'know, you guys, there are scads of statistics textbooks and courses that address these questions.

[Robyn]

Re: The probability and predictability of dice.

 

Could people please stop deleting their posts after making them? It's confusing to read through and see replies to posts that are missing by the time I get there.

[Rapier]

Re: The probability and predictability of dice.

 

I believe these are all fair definitions:

 

An average is when you add up all the numbers and then divide the sum by the number of members. If there are 6 numbers you add them all up and divide by 6.

 

A mean is the statistical center. If you have 5 numbers the mean is the 3rd (the one in the middle). If you have an off number of members it's real easy, just pick the middle one. If you have an even number of members, the mean is halfway between the middle two (eg if you have 6 numbers, add 3 and 4 and divide by the number 2). You could say if you have an even number of values average the two middles for the mean.

 

The Law of Large Numbers, and this gets kind of non-laymen, is that if you have a whole series of values and randomly pick out a group of of them. The average for those numbers you picked out will be very close to the mean of the entire list (including the ones you removed). IIRC (and it's been a little while) it's the Law of Large Numbers that is the basis of Probability Theory. If you roll 1000d6 (odd number of items so its easy to find the median) and write all the numbers down in order of value (all the 1s go first). You will find the median (average of 500 and 501) should be about 3.5 (sound like a familiar value?). Now, if you pull out any truly random sampling and average them the result should be near 3.5.

 

Predictable is just what you think it means. Predictable means that at some point you can guess (eg predict) what is going to happen with a certain degree of certainty (no wild guesses).

 

There is a 1 in 6 chances of rolling any value on a d6. Normal distribution is a term to say that if you roll 6000d6, you would have about 1000 1s, 1000 2s, 1000 3s etc. That would be a normal distribution. If one of your dice was off and tended to have odd results and you rolled 737 1s, 645 2s, 997 3s, 756 4s, 2859 5s...that would NOT be a normal result.

[Conduit]

Re: The probability and predictability of dice.

 

I believe these are all fair definitions:

 

An average is when you add up all the numbers and then divide the sum by the number of members. If there are 6 numbers you add them all up and divide by 6.

 

A mean is the statistical center. If you have 5 numbers the mean is the 3rd (the one in the middle). If you have an off number of members it's real easy, just pick the middle one. If you have an even number of members, the mean is halfway between the middle two (eg if you have 6 numbers, add 3 and 4 and divide by the number 2). You could say if you have an even number of values average the two middles for the mean.

 

 

I have to disagree. Average is a shorthand term or mean, median or mode.

 

Mean is the sum of a set divided by the number of members in the set.

 

Median is that number which when the members of a set have been arranged in numerical order has the same number of members less than and more than that number. For a set with an odd number of members it will actually be one of the numbers (the 3rd member of a set consisting of 5 members). For a set with an even number of members it will be the mean of the two members (The mean of the 3rd and 4th member in a 6 member set). You mistakenly refer to this as the mean.

 

Mode is that number which occurs most frequently in a set.

 

Given the set [3,5,11,11,18] we have the following result:

Mean: 9.6

Median: 11

Mode: 11

[Conduit]

Re: The probability and predictability of dice.

 

Could people please stop deleting their posts after making them? It's confusing to read through and see replies to posts that are missing by the time I get there.

 

Some of the quotes on the first page where taken from the orginating thread.

[prestidigitator]

Re: The probability and predictability of dice.

 

Plot on a graph the possible results of a throw of xd6 on the x-axis and the number of ways to make a result on the y-axis. The resultant curve is the distribution curve (and happens to be a normal distribution curve).

See post #5.

 

Approaches a normal distribution as you add more and more dice.

[Guyon]

Re: The probability and predictability of dice.

 

I also have a set of casino dice, although I bought mine on ebay. They have great sharp corners, and love the weight, size, and feel. While I have not done a study on there curve. They have pulled me out of hopless situations and left me flat when I needed them most. What more could you ask? LOL

[Conduit]

Re: The probability and predictability of dice.

 

See post #5.

 

Approaches a normal distribution as you add more and more dice.

 

I ran the number on 2d6.

 

Total Results:36

Mean: 7

Standard Deviation: 2.45

68.2% of Total Results: 24.6 (25)

Mean - 1 StD: 4.55 (5)

Mean + 1 StD: 9.45 (9)

Number of Results between 5 and 9: 24

Percentage of results within 1 StD: 66.7%

 

I think this is close enough to call a normal distribution at this level of granularity.

[Siren]

Re: The probability and predictability of dice.

 

@Conduit That is not nearly enough samples to make an experimental determination.

Though in my Discreet math class in collage they had said the more samples you take the greater the odd behaviors and straight runs you will have ex: With 3d6 getting the sum 18, 4 times in a row.

[prestidigitator]

Re: The probability and predictability of dice.

 

Err...I think he was, "counting," himself, not performing an experiment. Anyway, see post #5.

[Conduit]

Re: The probability and predictability of dice.

 

@Conduit That is not nearly enough samples to make an experimental determination.

Though in my Discreet math class in collage they had said the more samples you take the greater the odd behaviors and straight runs you will have ex: With 3d6 getting the sum 18, 4 times in a row.

 

?

 

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.

[prestidigitator]

Re: The probability and predictability of dice.

 

Well, trying to roll 18 four times in a row isn't likely to be successful anytime soon, but rolling a bunch of times and finding some, "highly unlikely," pattern is quite common. While any one particular result may be highly unlikely, when summed up all the, "highly unlikely," patterns you could imagine offer a surprisingly fertile field, as it were.

[Siren]

Re: The probability and predictability of dice.

 

?

Not nearly enough samples? Go ahead and work out a couple more examples for yourself then.

 

Wow easy Conduit, I was not being mean. I was just adding to the discussion on probabilities. I didn't mean to offend you. I am sorry! trhough i do knwo this borad can get a little mean. hugz.

 

 

@prestidigitator, My point about rolling 4 18's in a row was that when you get very large samples, strange things happen. And the chances of that happing grow as the trials are increased. Just a fact from my college math class, nothing more.

[Robyn]

Re: The probability and predictability of dice.

 

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

 

That's just by yourself, though. Taking into account all the people who have rolled dice before (and will in the future), everywhere, it becomes nearly impossible to impose the rules for distribution over larger sets onto any one person's dice rolls.

[Conduit]

Re: The probability and predictability of dice.

 

Wow easy Conduit, I was not being mean. I was just adding to the discussion on probabilities. I didn't mean to offend you. I am sorry! trhough i do knwo this borad can get a little mean. hugz.

 

 

@prestidigitator, My point about rolling 4 18's in a row was that when you get very large samples, strange things happen. And the chances of that happing grow as the trials are increased. Just a fact from my college math class, nothing more.

 

Stupid internet not conveying tone:mad: . Not offended at all, I just didn't really understand you comment. I didn't sample anything I actually ran the numbers on all possible results of 2d6 (and 3d6 I just didn't bother to post them)

 

As far as strange things happening, you are right in as far as the more tests you perform the more likely you are to see a random strange thing. Strange things as a group are not all that rare, it's the specific strange things that are rare. For example I would not be terrible surprised if someone came on the board and said they rolled 4 straight 18s, they would be increadibly unlucky but surely we are as Herodom collected approaching 2 billion rolls it will happen sooner or later. But if you told me you started the night with 4 straight and ended with 4 straight with other results inbetween, I wouldn't believe you.