Our dice are not actually fair [Math]

see

I've read it, but it doesn't change a thing: if a GM has a certain outcome in mind there's no point in rolling at all.

No die is perfectly fair. In theory a perfect machine which tossed a die from the same position in the same manner would achieve the same result each time. Each bounce and roll could be mathematically figured out and the machine could be adjusted to flawlessly throw any desired result with 100% accuracy.

The die is an approximation of randomness which is "good enough" for most purposes, including gaming. Even the most hardcore RPG gamers probably never roll a die enough in a lifetime of playing to properly statistically evaluate the die (which I would peg at somewhere around 20000 times the number of sides) so it's usually very difficult for them to notice if 14 is a bit more statistically likely than a 17 or whatever.

Amen.

>Even the most hardcore RPG gamers probably never roll a die enough in a lifetime of playing to properly statistically evaluate the die (which I would peg at somewhere around 20000 times the number of sides)
Sure, the more times you roll a die, the more certain you can be about its level of bias, but if one number is coming up more than twice as often as another, that's a pretty good indication the die is flawed, even if your sample size is only 1000.

...

Sometimes the illusion of the roll being a tipping point is valuable. In fact, either Gygax or Perkins (don't recall which) once said, "The only reason a DM rolls dice is for the sound it makes."

That said, Gygax was of the opinion (and I agree with him) that you should play a game straight and more-or-less by the book until you actually understand why it is written as it is.
'Fudging' well, and being able to make unshackled decisions that feel internally consistent with everything else in the game, is a learned skill and is naturally more art than science. Those reference points can be important.

>Oh boy this graph again.
>Wait...
>It's the same fucking file name too.

Abandon thread. This user uses randomness as b8 to cause havoc.

Yeah they're unfair, but it's also only a game so if people are seriously going to cheat then they're just sad fucks.

>I've read it

Sure chief.

Go ahead and keep cheating I guess, just don't be surprised when the group unanimously gives you the boot when you roll that third NAT 20 in a row.

>you can cheat your ass off!
Yes. You can. You don't need to rely on dice inaccuracies to cheat, either.

Have you considered not? And if you play with people who cheat at a tabletop roleplaying game, have you considered not playing with them or asking them to stop?

...

With all my respect to Persi Diaconis (he's a great man and one of the better professors out there), he's putting the entire thing in too simple of a fashion - he could explain the topic much better and comprehensively.
That said, I guess Numberphile is a normie channel for uneducated masses, so it's fine.

I like how he uses Gamescience dice in the video

My point is that you can see significant bias in those dice, especially given the similarity of their distributions, even though they were rolled only 200 times less than suggested. More trials would allow you to peg the extent of the bias more exactly, but it's pretty clear that bias exists.

>This user uses randomness as b8 to cause havoc.
I've no idea what that even means.

Just the other day we were talking about how many different possible fair die ROLLS were possible -- that is, including using 2d10 for 1d100. I want to write a math paper on this, but I'm a computer science person and not a math person, so I'm shit at mathematical writing.

Not sure what you mean by how many are possible. As in how many possible fair dice configurations?

You know how 2d6 is not the same thing as 1d12? There's exactly one way to roll any given result on a d12, so there's a 1-in-12 chance for any given result. But for 2d6, there's a 1-in-6 chance to get a seven (you can roll 1 and 6, 2 and 5, 3 and 4, 4 and 3, and so on for 6 out of the 36 possible combinations) but only a 1-in-36 chance of rolling a 2 or a 12 (snake eyes and boxcars, respectively).

What I want to do is to write some kind of mathematical proof that gives a rule for how many different "fair" die rolls (that is, how large of a range of results can you get while still having every result be equally likely) are possible with any given combination of dice.

For example, if you're allowed d2, you can get any die roll equivalent to 1dN, where N is any power of 2 (just do the same thing as with 1d100 from 2d10, but use binary instead of decimal). Similarly, if you're allowed to use a d6, you can get any power of 6, or any power of 2 or 3 (because you can get d2 or d3 from a d6 by dividing, taking a remainder, and adding one). You can also get 1dN for any product N of a power of 2 and a power of 3.

I want to write up some proofs about all this. I've proved it to myself in my head and convinced some other fa/tg/uys; but I haven't figured out how to get it onto paper yet, because, as I said, I'm shit at mathematical writing.

Speaking as a drunk mathsbro - statistics and random results mix perfectly, user. If you roll 1d100 1,000,000 times and get no 100's the odds of that result are 1 in 99^1,000,000. Your calculator won't do that because it's insanely long odds - I just did it on Windows calulator and it got to about 99^5,000 before it would go no further. At that point the odds of having got no 100's are about 1 in 0.[almost 10,000 zeroes]15. That is so a huge fraction of the actual probability across a million trials it's not worth mentioning.

I see, so you're looking to prove the maximum number of equally likely outcomes from any given pool of dice.

Although, it seems pretty trivial to number the possible outcomes and maximize uniform outcomes that way. E.g., snake eyes is interpreted as "1" 1 and 2 is interpreted as "2", 2 and 1 is interpreted as 3...etc. up to 36 (we'd have to specify which die is which, but we have that same problem in the 2d10 -> d100 case). So presumably you'll want some well-specified constraint on the dice output -> "value" function. I'm not sure what that constraint would be but it probably exists.

Not really.

You know how you can use 2d10 to produce 1d100? When you do that, d10 is the die and d100 is the roll. I want to see just how may possible rolls you can get out of a given group of "types" of dice. (By "type" I mean the shape -- d6 is a type, d10 is a type, etc.)

I have come up with (not myself; these are old as the hills) two methods for using a die to get a roll other than for the number of sides on that die (when I say a roll is "for" a number, I mean that it can produce results between 1 and that number. 1d6 is a roll for 6, 1d10 is a roll for 10, etc.). These are the methods:
The remainder method, where you roll for a lower number than the number of sides on the die by dividing the result of the die by an integer divisor v of number of sides on the die, and add 1 to the remainder. This is how you get d3 from d6 and d5 from d10, and d2 from either. Mathematically expressed, this is modulo(1dS, v) + 1, where S is the number you're rolling for and S is divisible by v.
The multiplication method, where you roll for a higher number than the number of sides on the die by taking another roll and adding to the result of the first die the result of the second die minus 1 times the number of sides on the first die. This is how you get 1d100 from 2d10. Mathematically expressed, this is 1dS + (1dT - 1)S.

Fuck; I can't do this; let me just write it all in Scheme:

(define (remainder-method faces divisor) (1+ (remainder (+ (random faces) 1) divisor)))
(define (multiplication-method first-faces second-faces) (+ (1+ (random first-faces)) (* first-faces (random second-faces))))

If you're allowed to use a given set of dice, you can compose these functions to produce a roll for the product of any of the powers of any of the factors of the numbers of sides on the types of dice you're given, and that's what I want to prove.

If any of that doesn't make sense, that just proves my point that I'm shit at mathematical writing. I'll get my IRL mathbro friends to help me with this and post results eventually. I'll deliver, I promise!

These.

With people I trust I don't need 100% reliable dice, I just need any inherent bias they have to be small enough to not be obvious.

>ITT: the rest of the autism

What even is this reaction image.

why dont you just git gud

meaningless if this only sampled two dice...

>1 in 99^1,000,000

Not true - 1 in 99^1,000,000 would be the probability of rolling one specific result on a d99, one million times in a row.

Rolling zero 100's over one million rolls is P=(0.99^1,000,000), which I agree gets pretty damn infinitesimal, but not to the order of the number you listed.

It's not meaningless because it shows that at least some dice are broken (the only two Chessex dice I tested, which were different types and bought years apart). It certainly tells you more than nothing. And what's the alternative? Roll 2000 dice 2000 times each? I'll let you fund that particular scientific study.

Random.org based dice rollers, for true, white - noise based randomness?

More and more in not seeing a need for physical dice.

A roller could automatically include modifiers and simply display the results rather than display a roll and have the pc perform repeated arithmetic to determine the results. You can use this that might be impossible or impractical with physical dice (d7?), and you could set it up via networking so the gm can set the DC, the pc rolls, and the program spits out the result, for maximum fairness.

Have you not seen the threads with dozens of people losing their shit over gms that cheat their players by fudging the dice?

>if people would be upset if they knew about you lying about your dice rolls, maybe you shouldn't be doing it. Maybe it's breaking the social contrast, cheating even.

I don't care if you're the gm or a player, cheating is cheating. And i say that whether I'm gming or playing - most of the time I'm gming. Many others clearly feel the same.

Your montecarlo results aren't even vaguely accurate unless they include a *minimum* of 10k rolls.

If you have 10k+ rolls and still show bias, that bias has some degree of statistical significance.

>cheating is cheating

What faggots and other retards don't get is that "cheating" means breaking the rules. As GM, players are in my house, at my table, eating my popcorn and adventuring in my realm. So - GUESS WHO GETS TO MAKE THE RULES FAGGOT.

Protip: Players only roll to select from outcomes I have already prepared for the situation - they don't roll to make-up new rules! I can only surmise that posts like these are made by aspergers victims who have never played and cannot therefore see how dumb their opinions sound.

Even the OP assertion that players must think cheap, mass-produced blobs of plastic are perfect, laboratory-grade probability generators is fucking retarded. No one thinks that. No one cares - except compulsive bean-counters and other autists. tl;dr:

Troll better.

I disagree. If the big villain of the game, who was being hyped up since session 1, does not provide the challenge that the players were promised, what do you think the players will say after the fight?

>Man, that was totally anti-climactic
>The miniboss was easier than that
>What a letdown
>I thought he was supposed to be tough

All because of a few rolls of the dice.
I admit that I power-game to some degree and I am always let down when the villain of our games can't take more than one hit from my character. Especially if they're supposed to be big, strong and tough as nails.

There are times where it's appropriate to fudge dice for or against players, to make the game more tense and enjoyable.

Wasn't trolling.

If you didn't spell out your houserule that you don't do fairly in advance, you're cheating.

If you do have the good grace spell it out in advance, it's not cheating, as anyone who objects to you saying "there are no rules for the gm and the dice mean whatever i want them to mean at that particular moment" has fair warning and can either deal with it or opt to avoid your campaign and find a new gm, or run a game themselves and find new players.

But if I'm a player in such a game, I'm hosting (seems I'm always hosting) and you're one of like, 5 people in the group taking turns gming, 3 of us would give you a big ol' nope, rather than play under those conditions, and the game wouldn't even get off the ground.

And if you're one of those pricks who pretends theyre gming fairly until we catch on 4 months in that we thought we were playing shadowrun (or whatever) but really you've just been lying through your teeth and it's been a bait and switch the whole time - this is really just joe's storytime, we're going to be understandably perturbed.

>Your montecarlo results aren't even vaguely accurate unless they include a *minimum* of 10k rolls.
Is that Monte Carlo testing? I always thought Monte Carlo was performing random trials in place of mathematical calculation, in that it's a substitute when crunching the numbers is too difficult, time consuming, etc. But in this case, it's just testing. You can easily crunch the numbers on a fair d20, but those calculations will not indicate whether bias exists in a die or dice.

>If you have 10k+ rolls and still show bias, that bias has some degree of statistical significance.
The larger your sample, the narrower your confidence interval, but there is no magical line where results become valid. Large discrepancies in smaller samples are as indicative of bias as small discrepancies in larger samples. Pinpointing the near-exact level of bias is only going to be possible with a huge sample size, but detecting significant bias can be done with fairly small one. If you think that two thousand rolls tell you absolutely nothing, you're crazy.

I was under the impression montecarlo testing comes specifically from the casino testing its dice for bias.

As for the 10k minimum, when I was going to school and taking psychology as part of a double major, they drilled it in pretty clearly that if the sample size is small like this chart, all it can tell you is "there might be something here, we should do a proper study to see if we have anything".

The reason i picked 10k is iirc that's what the casinos use for their montecarlo style dice testing.

It indicates there's probably something there with your smaller sample, but you can't know how accurate or how severe.

But the fact that normal game dice are not fair is not news.

If you want fair, use random.org.

>Is that Monte Carlo testing? I always thought Monte Carlo was
>I was under the impression montecarlo testing comes specifically from
>As for the 10k minimum, when I was going to school
>The reason i picked 10k is iirc

wew, lads it's getting pretty scientific in here.

Its late af, and I'm posting from my phone because insomnia. No double-checking or research before posts is going to happen at this hour. That's just how it is. You get posts from memory. Double-checking was out like 5 hours ago.

>I was under the impression montecarlo testing comes specifically from the casino testing its dice for bias.
I don't think so. I think it's testing for probabilities in games of chance like dice and cards. Sort of like seeing how likely you are to draw a straight flush (only something harder to calculate than that).

>It indicates there's probably something there with your smaller sample, but you can't know how accurate or how severe.
Not precisely, but on that bottom die here , I'd put down a thousand dollars to your hundred that if we rolled the thing ten thousand times, we'd get more 6s than 20s. Would you take that bet against me?

meh, who cares.
I use online dice roller anyway.

Oh it certainly seems likely we would get more 6s than 20s.I'm simply saying the degree of difference could be rather exaggerated .

>If a GM fudges a roll he shouldn't have rolled in the first place.
>I've read it, but it doesn't change a thing: if a GM has a certain outcome in mind there's no point in rolling at all.
There are at least two reasons to roll that you are not considering:
1. The illusion of chance: When the PCs attempt to accomplish something that, for reasons their character is not aware of, will never be able to be successful, an ignored roll does not convey the same meta-information that would be revealed by the GM simply stating that the player fails, without rolling.
Fake rolls help hide info the players shouldn't have.

2. Unwanted Outcomes: You suggested that the GM has an outcome in mind, but sometimes there aren't planned outcomes that the GM wants, but unexpected twists of fate that they would like to avoid. My classic example was when several extreme dice rolls would have resulted in a beginning session TPK of a party that did nothing wrong, just by chance failed to spot the snipers that all got crits.
When the dice were rolled, no outcome was in mind, but the implausible result was unwanted.

That said, threads on Veeky Forums have actually dissuaded me from fudging as much.
Take the "Cinematic Battle" example:
>Fighter crits the enemy with a climactic critical strike, pulling off an impressive move and bringing it down to 1hp.
>Bard anticlimactically hits it for 3 damage with a slingshot and it dies.
Now, you could make the argument that it would be more cinematic for the impressive blow to finish the enemy off.
However, I consider that a failure of narration:
>Staggering from the fighter’s blow, the enemy wavers, but bears down on the group.
>The bard quips a clever line and shoots straight and true, striking it between the eyes, with an audible crack, it reels from the strike against its skull and collapses.
Unless there’s something like the enemy happened to be the fighter’s personal sworn nemesis that he vowed to slay himself, I see no reason to fudge.

Fudging rolls is like eating fudge.
Eating only fudge all the time is sickening and disgusting.
Eating fudge with every meal is too much is to be avoided.
Declaring that nobody should ever eat fudge for any reason is categorically stupid.
Sometimes fudge is damn tasty and if you add a little, it can make a person’s dessert amazing even if they don’t know it’s there.
Just don’t go putting it the damn tuna casserole.

Honestly, the guy lost me about halfway through, but I kept watching because I never get bored of passionate people talking about things they are passionate about.

Newfriend here, can someone explain what "fudging" rolls means?

>I'm simply saying the degree of difference could be rather exaggerated .
Or understated, of course. But I'm in no way claiming that the results in paint a precise picture of exactly how the dice are balanced. The comparative results for the 13 and the 14 in the bottom table are only 6 points apart, which doesn't really tell us much. I could easily see the results reversing themselves with additional testing. The 7 and the 8, which are 28 points apart? Probably not, but it wouldn't be shocking. The 6 and the 20, which are 79 points apart? I *really* don't think so.

Anyway, here are that bottom die's results compared to random trials in Excel.

Cheating. Changing the results. Saying "I got a 7" when you got a 3. Generally, fudging is a softer, more forgiving term, and is applied to GMs who nudge results for reasons of game play and so forth. It's kind of like the difference between stealing and "borrowing without asking".

Well put

No. You could only manage that with a significantly flawed die. Far too many variables. Me and my billiards robot proved this during my grad work at Stanford.

t. Stanford Phd candidate

Fudging is when the GM changes a number, either a star or a roll, to something other than what it was, typically to achieve a different result.
Such as changing a damage roll or hp total to save or kill an enemy.

Fudging is not necessarily cheating.
Lying is not necessarily cheating.

That's a hell of a lot of noise for the excel baseline. Gives you some idea of just how inaccurate only using 2000 rolls is.

You could use those excel baselines to get a decent estimate of just how unreliable the actual rolled results are worth that sample size.

See Don't listen to the first guy, he doesn't like others having fun in a way he does not.

If the sample size is big enough that your control varies by

Stat not star, obviously.

It's interesting how similar the results for the two dice were though. For both, 20 had the least results, 8 had the second least results, 1 had the third least results, and 14 had the fourth least results. For both, 6 had the most results (though for the top die, it was a three-way tie).

Anyone who has stepped on a d4 can tell you: ain't nuthin' fair about it.

You're right. Having sobered up what I wrote does not make much sense. I was clearly too drunk to be attempting probability.

Probably.

Which is worse, stepping on a d4 or stepping on a lego?

In my experience, that depends. If you're barefoot, a Lego is worse. If you're stepping on it through a sock, d4 is much worse.

Having a smaller surface area of my foot where it would really hurt to step on either has to be one of the few advantages of having really high arches

Maybe because there's no such thing as randomness. All we can do is generate results that are based on environmental factors, and that for all intents and purposes appear to be random in nature.

Even computers do this when generating "random" outcomes. They might look at things like RAM states, and deterministically choose a number based on that. Since the observer doesn't necessarily know what's going on in the computer's memory, this outcome appears to be random.

With dice, it's all based on the trajectory and momentum of the throw, distance, etc. But of course you already know this.

Well, that actually makes sense in the context of the game:
Given that there are hidden variables preventing true random outcomes, that holds in the game world as well: It's simply a matter of determining what hidden variables happen to be at work in the situation presented, by creating a relationship between those in person and those in game.
The usefulness of the dice is its unpredictability- we don't know whether someone truly succeeds in a situation or not until they do, though we have a good idea what their rate of success in multiple trials is.
In short, dice are an artificial tension generator representing unknown factors that can't be precisely determined but fall within a particular spectrum we standardize for general use.
Overthinking ho!

>:p
Get out.