The probability and predictability of dice.

[Conduit]

Re: The probability and predictability of dice.

 

That's just by yourself' date=' 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.[/quote']

 

I'm not sure I agree with you here. I don't doubt that there will be people who at some point roll 4 straight 18s, I very much doubt there will ever be someone who does the feat twice. The odds are just too astronomical. It is a once in a lifetime event and even then most people will not do it.

[Robyn]

Re: The probability and predictability of dice.

 

I'm not sure I agree with you here. I don't doubt that there will be people who at some point roll 4 straight 18s' date=' I very much doubt there will ever be someone who does the feat twice. The odds are just too astronomical. It is a once in a lifetime event and even then most people will not do it.[/quote']

 

Keep in mind that calculations of probability for dice-rolling have more of a theoretical than a practical application. Just because a 6-sided die has exactly 6 sides, does not mean that there is exactly a 1 in 6 chance of getting any particular result. Environmental factors such as how someone rolls the dice (shakes them in the hand, how many other dice are in the hand, topography/texture of the hand, angle dice are thrown at, velocity dice are thrown with), the surface of the table, and the distribution of weight over the die's mass will all play a role, even if those factors are not planned. Outside of perfect laboratory conditions, interference will occur, and the best hopes we can have are that the interference will balance itself out.

[Siren]

Re: The probability and predictability of dice.

 

@Conduit, no prob, **Hugz**

[prestidigitator]

Re: The probability and predictability of dice.

 

@prestidigitator' date=' 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.[/quote']

Certainly. I was responding to this comment more than anything:

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']

 

BTW, discussions do often get heated, and that is usually the source for disparaging remarks. I think it comes most often from simple spontanaity and frustration rather than outright meanness. We do a great deal of lighthearted teasing, too, so don't be too ready to take things as an insult. I'm sorry this unfortunate impression was one of your initial experiences here. The Hero Games Forums are a great place with a lot of cool and well intentioned people. Don't get discouraged yet!

[Kenn]

Re: The probability and predictability of dice.

 

Big Jules: I remember where the spots was.

[TheRavenIs]

Re: The probability and predictability of dice.

 

Ok, here is my take. Probability is the study of certian things that happen in the real world. In the real world, things don't always follow the pridicted course. Ok my small addition.

[Guyon]

Re: The probability and predictability of dice.

 

While you are at it. How many people take into account rough probability of making a roll before they actually try it?

[prestidigitator]

Re: The probability and predictability of dice.

 

While you are at it. How many people take into account rough probability of making a roll before they actually try it?

Rarely. Mostly just when the stakes of a single roll are quite high.

[Dust Raven]

Re: The probability and predictability of dice.

 

While you are at it. How many people take into account rough probability of making a roll before they actually try it?

 

I definately do. Even with a simple character with a single attack power (such as a plain EB for a simple energy projecter) it's important for me to know how likely I am to roll certain levels of damage depending on how much dice I use. It could make the different between firing normally, spreading for accuracy/area, performing a haymaker or not wasting my time trying.

[Anatosuchus]

Re: The probability and predictability of dice.

 

While you are at it. How many people take into account rough probability of making a roll before they actually try it?

I most definitely do if there are going to be significant consequences in the event of failure, which is a lot of the time.

 

Your question might explain I have found this whole debate very peculiar as some people seem* to be arguing that players aren't interested in knowing the odds of any given outcome prior to rolling, only in knowing the actual outcome after the fact. This seems a very odd way to play as it effectively removes educated guesses as to the effectiveness of any given plan of action.

 

*I say "seem" because many of the posts have been so long as to defy my ability to raise enough interest to decipher them.

[SteveZilla]

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.

 

Cool. I've thought about getting some, but the larger size tends to put me off. It'd be kinda hard to use them to roll an 18d6 pushed haymaker.

[SteveZilla]

Re: The probability and predictability of dice.

 

There is no set definition for' date=' "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.[/quote']

 

Which is what I would call average, given that the bell curve is symetrical. Which also makes it (or the nearest two values in case of a '.5') the most likely result/ I.e., the % chance of rolling that number (or either of the two closest numbers) is greater than any other single number.

 

Which is a roundabout say of saying "the peak of the bell curve".

[Guyon]

Re: The probability and predictability of dice.

 

I've thought about getting some' date=' but the larger size tends to put me off. It'd be kinda hard to use them to roll an 18d6 pushed haymaker. [/quote']

 

Yes but it looks so impressive! I also like the weighty-ness of tossing heaver dice.

[Robyn]

Re: The probability and predictability of dice.

 

Yes but it looks so impressive! I also like the weighty-ness of tossing heaver dice.

 

I have dice the size of rice. The GM (and other players) can't see the results when I roll, and sometimes need a magnifying glass to read them even after getting up and going over to peer down at the pips, but I can roll a lot of dice from one hand

[Dust Raven]

Re: The probability and predictability of dice.

 

I have dice the size of rice. The GM (and other players) can't see the results when I roll' date=' and sometimes need a magnifying glass to read them even after getting up and going over to peer down at the pips, but I can roll a [i']lot[/i] of dice from one hand

 

I had a set of those, but they kept sticking to my hand. Too light to fall off properly. (I should probably point out I live in Phoenix, AZ... sweaty hands are a part of daily life during the summer. On the upside, I can sometimes stack my rolls by placing all the dice I roll with the numbers I want face up in my hand, then roll on a paper surface.)

[Guyon]

Re: The probability and predictability of dice.

 

On the upside' date=' I can sometimes stack my rolls by placing all the dice I roll with the numbers I want face up in my hand, then roll on a paper surface.)[/quote']

 

LOL that was so funny.

 

If you like the little dice there are so many up sides to them, such as taking up very little space. One thing that bothers me most is dice of different sizes rolled at the same time.

I have a nice leather back for by big ones.

[Robyn]

Re: The probability and predictability of dice.

 

One thing that bothers me most is dice of different sizes rolled at the same time.

 

Have you seen the semi-transparent 6-sided ones, with a smaller die inside of a hollow larger one?

 

That's rolling 2d6 at the same time

 

For large dice, I've seen a foam d6 that was at least a foot long/wide/tall

[SteveZilla]

Re: The probability and predictability of dice.

 

Mean is one of several methods of averaging (others are median and mode' date=' 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.[/quote']

 

That's the exact method I was thinking whenever I said "average". So, if we add up all the possible results of a group of dice (Xd6), and divide that total by the number of results possible we derive the mean of that group's bell curve.

 

So:

Xd6 has a range of X to 6*X, and a total number of possible results of 5X+1, and a mean result of mumblemumblemathI'mtoolazytowriteoutmumble, which simplifies down to 7*X/2, or 3.5*X, for those who don't like fractions.

 

 

The law of large numbers is' date=' 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.[/quote']

 

The term "Grand Average" springs to mind. A Grand Total divided by The Total Number of Items in that total.

 

But while the extreme results (defined loosely above) become less likely, they *also* become much further away from the mean. I believe this factor needs to be taken into account. The net effect on the bell curve of probability (not the one of numbers of combinations) as we add dice is that it grows wider (covers a greater range) and flatter (each individual result -- comparing curves peak-to-peak, not # to # -- has a smaller % chance).

 

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?

 

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 deviations of the mean and 99% fall within three standard deviations of the mean.

 

I prefer to think of the Bell Curve shape (from dice) as simply resulting from a graph of the Number of Combinations that produces each result.

 

Is there a way to determine how many standard deviations it takes to encompass *all* of the results of a group of dice? Since 99% of them are encompassed by 3 standard deviations, I would expect it to be just a little more than 3 -- like 3.1 or 3.14.

 

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

 

Ah, the chance for any single result of a roll. The Number of Combinations that make that result divided by the Total Number of Combinations, times 100.

 

I find it interesting that if we graph the chance for each number instead of the combinations that make each number, we generate a *hugely* flatter bell curve. And that as we add dice, the graph of the combinations gets taller, but the graph of the number's chances gets shorter.

 

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 irrelevant to 1Md6.

 

I don't know how to calculate standard deviations, or what purpose standard deviations have in the mix. How are they calculated?

 

The standard deviation gets larger as you add dice but becomes less significant to the total faster.

 

I understand the first part of that statement, but not the last part (the "but become less significant to the total" as well as the "faster" part). No matter how many dice are used (more than one, I'd presume), 3 standard deviations still cover 99% of the results. Both the range of results and the size of the standard deviation seem to be in exact proportion to each other.

 

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).

 

Okay, a graph of the combinations per result of the dice is a "distribution curve". But it's a graph of the combination distribution, right? Doesn't this differ from a graph of the chance of each result? Which could still (AFAIK) be called a distribution graph, just a distribution of %chance?

 

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

When you say "more predictable" here' date=' I presume you mean "more predictable than a greater number of dice"?[/quote']

I mistyped' date=' I mean to say 2d6+28[/quote']

 

Okay, so if I understand you correctly, you're saying that 2d6 will be more predictable than 2d6+28 because 2d6+28 will/can generate a greater possible variance relative to the mean?

 

Defining predictability like you have above, 2d6 and 2d6+28 would seem to be equally predictable. Everything has been "shoved right" on the graph. All the results and also the mean they generate. Rolling 4+5 to get 9, and rolling 4+5+28 to get 37 each have a % chance equal to each other, and both are equally distant from their own means (9 is 2 away from the mean of 7, and 37 is 2 away from the mean of 35). 2d6 can vary at most +/-5 from it's mean, and 2d6+28 can vary at most +/-5 from it's mean.

 

Right' date=' it is a different curve because it is shifted to the right. It doesn't vary in any other way.[/quote']

 

Why would this shifting make it more predictable? I would think that it would be (IMO) more accurate to say "It's the same probability/combinations curve, just shifted right". The chance of rolling 7 on 2D6 is the same as rolling 7+28 on 2d6+28. The same 1:1 relationship can be made for each result. How does the "end result number" affect the whole thing's predictability?

[Cancer]

Re: The probability and predictability of dice.

 

I picked up three big wooden dice, 3 inches along the edge. (I think they're supposed to be paperweights, actually.) Can't roll them on the table; they just knock over everything, figures, drinks, etc. So I throw 'em into an empty laundry basket on the floor.

[Conduit]

Re: The probability and predictability of dice.

 

First off, SteveZilla, great questions.

 

That's the exact method I was thinking whenever I said "average". So, if we add up all the possible results of a group of dice (Xd6), and divide that total by the number of results possible we derive the mean of that group's bell curve.

 

So:

Xd6 has a range of X to 6*X, and a total number of possible results of 5X+1, and a mean result of mumblemumblemathI'mtoolazytowriteoutmumble, which simplifies down to 7*X/2, or 3.5*X, for those who don't like fractions.

 

 

 

 

The term "Grand Average" springs to mind. A Grand Total divided by The Total Number of Items in that total.

 

But while the extreme results (defined loosely above) become less likely, they *also* become much further away from the mean. I believe this factor needs to be taken into account. The net effect on the bell curve of probability (not the one of numbers of combinations) as we add dice is that it grows wider (covers a greater range) and flatter (each individual result -- comparing curves peak-to-peak, not # to # -- has a smaller % chance).

 

Actually, while the bell curve does grow wider, it grows taller as well (speaking of the curve with possible results on the x-axis and number of possible combinations on the y-axis).

 

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

Correct

 

I prefer to think of the Bell Curve shape (from dice) as simply resulting from a graph of the Number of Combinations that produces each result.

Agreed

 

Is there a way to determine how many standard deviations it takes to encompass *all* of the results of a group of dice? Since 99% of them are encompassed by 3 standard deviations, I would expect it to be just a little more than 3 -- like 3.1 or 3.14.

 

It's not a fixed number. 68.2% are within 1, 95% are within 2, 99% are within 3 but that is all that can be said. For example it takes slightly more than 4 to encompass all possible results from 10d6.

 

Ah, the chance for any single result of a roll. The Number of Combinations that make that result divided by the Total Number of Combinations, times 100.

 

I find it interesting that if we graph the chance for each number instead of the combinations that make each number, we generate a *hugely* flatter bell curve. And that as we add dice, the graph of the combinations gets taller, but the graph of the number's chances gets shorter.

 

I'm not sure what you mean here.

 

I don't know how to calculate standard deviations, or what purpose standard deviations have in the mix. How are they calculated?

 

PhilFleischmann gave us a generalized formula for this. The standard deviation of Nd6 is (35N/12)^0.5.

 

I understand the first part of that statement, but not the last part (the "but become less significant to the total" as well as the "faster" part). No matter how many dice are used (more than one, I'd presume), 3 standard deviations still cover 99% of the results.

 

Let me say it this way then, the mean of Xd6 increases significantly faster then the standard deviation.

 

Both the range of results and the size of the standard deviation seem to be in exact proportion to each other.

 

This is where you error in the change of shape of the bell curve fails you. When X becomes large, and for our purposes this happens at 3 but becomes "more true" as the number increases. The number of combinations that form results near the mean increases drastically faster than the number of combinations that form results near the extremes. There will only ever be one combination at each extreme while 4d6 has somewhere in the neighborhood of 44 ways to reach the exact mean. 10d6 while still only having one way to hit each extereme has somewhere north of 100 ways to reach the exact mean. Meanwhile the standard deviation has only increased from 3.4 to 5.4. The statistic that is used to measure this is the coefficient of variation which divides the standard deviation by the mean to give an idea of how large the deviation is The CoV of 4d6 is .24 while 10d6 is .15. The smaller number shows that 10d6 is grouped tighter around the mean than 4d6 is.

 

Okay, a graph of the combinations per result of the dice is a "distribution curve". But it's a graph of the combination distribution, right? Doesn't this differ from a graph of the chance of each result? Which could still (AFAIK) be called a distribution graph, just a distribution of %chance?

 

Same thing. The % chance of rolling a given result is the number of combinations that form that result divided by the total number of possible results. Graphing the number of combinations just saves you the dividing.

 

Okay, so if I understand you correctly, you're saying that 2d6 will be more predictable than 2d6+28 because 2d6+28 will/can generate a greater possible variance relative to the mean?

 

No, I mean the opposite. 2d6 has a mean of 7, a standard deviation of 2.4 and a coefficient of variation of .34. 2d6+28 has a mean of 35 a standard deviation of 2.4 and a coefficient of variation of .07. Xd6+Y is more predictible than Xd6 because it transfers an absolutely smaller Standard Deviation to a larger mean, making the deviation less significant (a smaller percentage of the result). Xd6+Y is also more predicitble than the number of dice normally thrown at that damage class for the same reason, a smaller than normal standard deviation for that mean

 

Defining predictability like you have above, 2d6 and 2d6+28 would seem to be equally predictable. Everything has been "shoved right" on the graph. All the results and also the mean they generate. Rolling 4+5 to get 9, and rolling 4+5+28 to get 37 each have a % chance equal to each other, and both are equally distant from their own means (9 is 2 away from the mean of 7, and 37 is 2 away from the mean of 35). 2d6 can vary at most +/-5 from it's mean, and 2d6+28 can vary at most +/-5 from it's mean.

 

 

 

Why would this shifting make it more predictable? I would think that it would be (IMO) more accurate to say "It's the same probability/combinations curve, just shifted right". The chance of rolling 7 on 2D6 is the same as rolling 7+28 on 2d6+28. The same 1:1 relationship can be made for each result. How does the "end result number" affect the whole thing's predictability?

 

The reason is that 5 is a smaller percentage of 35 than it is of 7. This is the point where you have to start adding application. If a result of 7 would be sufficient for what ever I want (maybe an attack against 5 defense) 5 is going to make a big difference. If 35 is sufficient, 5 makes less of a difference.

[SteveZilla]

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.

 

So, if the number one is adding up are consecutive integers (like the possible results from a group of dice), their average will be their mean, right?

 

The Law of Large Numbers' date=' 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).[/quote']

 

So, if:

 

"a whole series of values" = all possible results from a group of dice

 

and...

 

"randomly pick out a group of them" = roll the group of dice several times

 

Then The Law of Large Numbers seems to be saying that the average of the random picks (dice rolls) will be very close to the mean of the "whole series of values"? That seems significantly different than saying that TLoLN says that any particular roll will be close to the mean.

 

Also, I *do* remember that one cannot make accurate conclusions from an inadequately small sampling. For instance, if I roll 12d6 only two times (two random samples of the series) and get 12 and 38, I might inaccurately conclude that the mean should be (12+38)/2=25.

 

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' date=' if you pull out any truly random sampling and average them the result should be near 3.5.[/quote']

 

Presuming you have a large enough sampling. The larger the sampling, the greater the accuracy of the average. But I don't understand where you got the numbers 500 and 501 from.

 

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.

 

Which is the same as saying "there is a 16.66666... percent chance of rolling any number on a d6". But what kind of predictions are we talking about? To me, the "predictability" of a group of dice is essentially the average of each results' chance of being rolled. Is there a better way of quantifying the predictability of a group of dice?

 

Normal distribution is a term to say that if you roll 6000d6' date=' you would have about 1000 1s, 1000 2s, 1000 3s etc. That would be a normal distribution.[/quote']

 

Okay, I can see that, and it makes logical sense to me. But almost no roll is going to come out exactly like that. There is always going to be some variation. But I think that's not what we're discussing.

 

If one of your dice was off and tended to have odd results and you rolled 737 1s' date=' 645 2s, 997 3s, 756 4s, 2859 5s...that would NOT be a normal result.[/quote']

 

I agree that it wouldn't be a "normal distribution" of the dice faces showing, yet their total (737*1, 645*2, 997*3, 756*4, 2859*5, 6*6 = 22,373) is relatively close to the mean of 21,000. Yes, it's a far 1,373 points away in absolute terms, but relatively close (I think) in terms of standard deviation.

 

So while the faces that are showing are way off of the "norm" (only six 6s), the roll itself isn't that far off of the norm. I'm not quite sure what this means/implies. Though in games, it's usually the total and not the distribution of die faces that matter. I don't often see a graph of face distributions.

[SteveZilla]

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.

 

Which is the same odds as rolling either a 12 or a 72 on 12d6.

 

If someone games for 72 years, for 23 hours each day, and rolls once every second, they will just make it. Jolt Cola, anyone?

[SteveZilla]

Re: The probability and predictability of dice.

 

For example I would not be terrible surprised if someone came on the board and said they rolled 4 straight 18s' date=' 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.[/quote']

 

I have rolled damage on a d10 and gotten a 10 five times in a row. While the odds of that happening by itself is 1 in 100,000, it's vastly diluted by the sheet number of other results I've gotten with that character. So, my average damage roll wasn't affected all that much by that string of good luck.

 

Oh, and there were 4 other witnesses.

[SteveZilla]

Re: The probability and predictability of dice.

 

While you are at it. How many people take into account rough probability of making a roll before they actually try it?

 

In Battletech, which uses 2d6 for to-hit rolls, I *always* take it into account on which target (if I have a choice) I shoot at. The only exception is if I'm really trying to take down an already badly damaged mech. Then I'll target it to just about the exclusion of all else.

 

In Hero System, it has made a difference on which target I shoot at (the speedster/martial artist vs the brick/mentalist/energy projetor).

[SteveZilla]

Re: The probability and predictability of dice.

 

I had a set of those' date=' but they kept sticking to my hand. Too light to fall off properly. (I should probably point out I live in Phoenix, AZ... sweaty hands are a part of daily life during the summer. On the upside, I can sometimes stack my rolls by placing all the dice I roll with the numbers I want face up in my hand, then roll on a paper surface.)[/quote']

 

Try putting them in a 25 count clear card box, like here.

 

Just shake the box to roll the dice! And if you put in the proper mix of various colors, it will take care of just about all of your die rolling needs in Hero System.