The Old Dice Controversy

[guilessa]

I know how annoying it can be to have to listen to so many complaints about FIBS dice. Personally, I don't have a problem with them and enjoy the games without constant paranoia.

Nevertheless, I want to report some matches I've been playing against FIBS bots, especially the stronger ones such as Monte Carlo, GammonBot_V etc.

In recent games, I put those games (not so many to be a definitive ) into GNUbg and the bot was getting about 25% of doubles, when we know that the average number for doubles is 16,66% of total rolls. Either that, or it gives me "bad dice, man", in GNUbg's own words.

This reminded me of a friend who used to contend that stronger bots roll better dice (or you roll worst dice) and that's the main reason which makes them strong. Anyway, I will continue to play matches against them, losing my precious points, and once I compile a more expressive number of games, I will post them here.

Maybe I'm wrong, but it annoyed me so much that I had to post. Sorry for the inconvenience of bringing up this old boring issue. But I just had to.

Cheers!

[diane]

There are two things you should do before continuing on with your plan to post more games here...

1 - read this thread..[all of it, no skipping  ]

http://www.fibsboard.com/fibsboard-forum-matches/how-do-'luck'-calculations-work/

2 - Ensure you analyse the games you win too - and look for unusual rolling sequences by you in matches you win. My immediate guess would be that sometimes they get lucky, sometimes you get lucky, but they play better so they do better with the luck they get.

Over all the games, the doubles rolled out will tend towards 16%, but it will take a lot of matches to see that. Some will be 22% and some will be 3 %, some will be 50% and some will be 10%...all of these are possible, and not indicative of anything unusual going on.

You can also play much longer matches, 13 points as a minimum. The longer they are, the less the rolls will seem one sided. You may well still see the bots analysing as lucky - when you read that thread, you will see why.

[dorbel]

As Diane points out, almost any number of doubles is possible in a match. The match where 1/6th of the rolls are doubles will be a rarity. However, if you want to collect stats, it's no use saying, "Hey the bots seem to have been getting a lot of doubles recently" and then when you look back, surprise, surprise, they have been getting more than average. You have to be scientific and start from scratch. You need to look at a minimum of ten thousand rolls before things start to even out.
You may like to know that the fibs dice have been tested and the server itself keeps running stats which you can see for yourself with the dicetest command. The bots get their dice from the server, just like you, so their average will be the same as yours. Interestingly, the stats show that doublets are slightly under what they should be. This is because when the server rolls a doublet on the first roll of a game, it junks it and rolls again. 5-2 is currently the most common roll.

[moonshadow]

The topic of whether FIBS dice are random or not usually brings out the crazies so its nice to see guilessa working through the issues with logic instead of paranoia and hysteria.

However, there really is no need to keep track of your matches or do any other experiments in order to determine if FIBS dice rolls are truly random, because this has already been done.

In 1997 Stephen Turner, after an impressive analysis of 10,593,121 rolls of FIBS dice, concluded that "FIBS dice appear fair" and that, "the number of doubles in the sample was 1,763,911, or 16.651%, fractionally less than expected, but well within normal statistical variation."

Keep this link handy: http://www.bkgm.com/rgb/rgb.cgi?view+262

When I'm playing a whining dimwit who claims "FIBS dice are fixed!" i just give them this link in the faint hope they will see the light, though more often than not, they lack the intelligence to be able to comprehend what they are reading and end up disagreeing with something they don't even understand.

Turner is (or at least was in 1997) a professional statistician at the University of Cambridge Statistical Laboratory and used to be an avid backgammon player (I believe he had one of the first sites devoted to backgammon in the early days of the Internet), so if he says FIBS dice are truly random after having analyzed 10,593,121 rolls, I would call that pretty conclusive.

Did that number sink in? That's ten million five hundred ninety three thousand one hundred twenty-one.

Turner sums up the randomness of FIBS dice issue thusly:

I can supply the data on request, but the chi-squared statistic gave an answer of -0.355521, showing no evidence of bias. As I said above, this test will check both whether some rolls come up more than others, and whether some rolls are more likely than others to come up after specific rolls. As a check, the number of doubles in the sample was 1763911, or16.651%, fractionally less than expected, but well within normal statistical variation (1.32 s.d.s from the mean, in fact).

If you carefully read Turner's analysis you will see that these 10,593,121 are all rolls generated by the FIBS server, including the first rolls of a match which happen to be doublets. Turner evidently used the undocumented and mysterious "matrix" command (go ahead, try it!) to compile this data. dorbel's concern that the data would be skewed due to first roll doublets doesn't apply here--perhaps he is alluding to the skewed data derived from the old Big Brother matches, which did not include first roll doublets.
 
In any case, next time someone is looking for hard data on FIBS dice, point them to Turner's study.

[diane]

In any case, next time someone is looking for hard data on FIBS dice, point them to Turner's study.

I have that page bookmarked for such moments 

[guilessa]

Ok, guys. Thanks for all the prompt responses. I'll take all you said into account.

Diane, I read the thread you recommended, but not all the 5+ pages  The first one was enough for me.

Thanks, moonshadow, for acknowledging that I'm not trying to make a point out of nothing.   

If I have any data that I think should be considered, I'll post it.

Thanks a lot for the attention.

Cheers

[diane]

Diane, I read the thread you recommended, but not all the 5+ pages  The first one was enough for me.

The conclusion is the important part...the better you play, the more you will tend to appear lucky in a match analysis - so the observations that bots are lucky more often on analysis is a function of how well they play, rather than that they get more of the random luck on offer...

[pck]

In 1997 Stephen Turner, after an impressive analysis of 10,593,121 rolls of FIBS dice, concluded that "FIBS dice appear fair" and that, "the number of doubles in the sample was 1,763,911, or 16.651%, fractionally less than expected, but well within normal statistical variation."
[...]
Turner is (or at least was in 1997) a professional statistician at the University of Cambridge Statistical Laboratory and used to be an avid backgammon player (I believe he had one of the first sites devoted to backgammon in the early days of the Internet), so if he says FIBS dice are truly random after having analyzed 10,593,121 rolls, I would call that pretty conclusive.

Did that number sink in? That's ten million five hundred ninety three thousand one hundred twenty-one.

Pointing to serverwide statistics isn't as strong an argument against allegations of manipulation as it may seem.

Imagine the following scenario: Unbiased dice generator (DG), exactly two players are logged into fibs. Fibs is programmed to behave in the following way: If on player 1's turn the DG rolls a 66, the roll is put on hold and instead the player is given the next roll produced by the DG (if that happens to be a 66 as well, the process is iterated until one of the other 35 rolls comes up). The 66(s) is(are) given to player 2 on her next turn(s). Result: Non-random dice sequences for both players, while any serverwide stats like dicetest will detect no bias, no matter how large the the number of rolls.

This scenario is modifiable in many ways and extensible to an arbitrary number of players.

[moonshadow]

Pointing to serverwide statistics isn't as strong an argument against allegations of manipulation as it may seem.

What you are really suggesting--and all without any evidence or motive--is that in 1992 Andreas Schneider (aka marvin) intentionally programmed the FIBS software to diabolically "manipulate dice" and then foisted that off on the public on July 19th of that year.

What happened is that in your rush to brush aside and casually dismiss what you thought to be a sophomoric argument presented by statistician Turner's analysis, you perhaps hadn't fully read the article which I had linked but not quoted, or perhaps you read it, but not truly comprehended it.

Imagine the following scenario: Unbiased dice generator (DG), exactly two players are logged into fibs. Fibs is programmed to behave in the following way: If on player 1's turn the DG rolls a 66, the roll is put on hold and instead the player is given the next roll produced by the DG (if that happens to be a 66 as well, the process is iterated until one of the other 35 rolls comes up). The 66(s) is(are) given to player 2 on her next turn(s). Result: Non-random dice sequences for both players, while any serverwide stats like dicetest will detect no bias, no matter how large the the number of rolls.

Though intellectually stimulating, there is no need to tax our brains with this scenario.

I don't think you understand the matrix command that Turner used to collect his 10,593,121 FIBS dice rolls and to which I also alluded in my previous post. In his analysis of FIBS dice Turner says of the matrix command which was used to collect the data:

You probably don't know about this command because it's not on the list of commands, but it produces data not only on the frequency of rolls, but on the frequency of each of the possible 1296 possible pairs of consecutive rolls.

(Bold added for emphasis)

fergy, graphic designer of the old MacFibs interface and the GamesGrid (now Grid Gammon) graphics and who still occasionally plays here puts it thusly:

   

•   Syntax: matrix
   

Display matrix of conditional roll frequencies. The matrix command shows you for each of the 36 possible combinations of numbers on the dice, the distribution of the next roll. For each of the 36 possible outcomes you'll see a matrix showing you how often each pair of numbers was rolled after that outcome. The numbers in the matrix represent the following rolls:


   •     1-1 1-2 1-3 1-4 1-5 1-6
   •     2-1 2-2 2-3 2-4 2-5 2-6
   •     3-1 3-2 3-3 3-4 3-5 3-6
   •     4-1 4-2 4-3 4-4 4-5 4-6
   •     5-1 5-2 5-3 5-4 5-5 5-6
   •     6-1 6-2 6-3 6-4 6-5 6-6 (X-X)


The last number shows how often a double was rolled after the given outcome. At the end of the list you'll also be told how often it has happened that a double was immediately followed by another double, together with Theoretical value for that number.
The values used for this test are all rolls that FIBS has made since it was last restarted. If the number of rolls is very small (i.e. smaller than 250000) the output of this command might lead you to wrong conclusions.

http://www.fibs.com/~fergy/help/fibsCommands/pages/matrix.html

So I think its quite reasonable to say that pointing to server wide statistics, especially those generated by the matrix command, provide a much stronger argument against dice manipulation than pck would suggest and that Turner's analysis of 10,593,121 rolls of FIBS dice, though not conclusive, is certainly far more compelling by an order of several magnitudes than that of any utterance by a caterwauling dice whiner.

For me, the most compelling argument against FIBs being programed to manipulate dice is that there is absolutely no evidence that Marvin did this and there is absolutely no motivation for him to have done so.

Fifteen years ago in 1996, several months before Turner completed his analysis of 10,593,121 FIBS dice rolls, Kit Woolsey modestly gives the reasons why his opinion should carry some weight:

There has been much discussion on r.g.b. about the possibility of the FIBS dice being skewed, so I thought I should put in my two cents worth.

I have been playing on FIBS for about three years, and have logged plenty of experience points.  It is my personal judgment from my playing experience that the dice are not biased in any way.

How qualified am I to make such a judgment?  I have been playing backgammon for over 30 years, both money play and tournaments.  In addition I have been active in many other gaming activities which involve chance all my life.

When one is involved in gambling, there is always the danger of cheating.  It is important to be on the lookout for irregular occurrences, and to distinguish them from the normal lucky and unlucky sequences which will always happen.  On more than one occasion I have sensed that things weren't right, and walked away from a situation which could have cost me a lot of money had I stuck around.  Thus, from my experiences in the gaming world, I believe my judgment in this area is better than most people's judgment.

Kit goes on to address the point I made earlier that there is absolutely no reason for Marvin to have rigged the dice:

The claim that the dice are intentionally programmed to be rigged in some way is patently absurd.  What would marvin's motivation be?  He has no profit from FIBS.  He created the program because he enjoys backgammon and wanted a free backgammon server to be available on the net.  In addition, programming the dice to be rigged is much more difficult than programming random dice.  Any such assumption about the dice being intentionally programmed to be rigged is completely contrary to logic.

(For Kit's full post from October 16, 1996, go here: http://www.bkgm.com/rgb/rgb.cgi?view+52)

As I understand it, it was Kit and his wife Sally that initially paid the FIBS server costs.

Overall, I suspect that pck regards this discussion of FIBS dice to be more of an intellectual puzzle than he is convinced there is outright dice manipulation and conspiracy, but in the event he gets too carried away with attributing sinister motives to Marvin and dice conspiracy theories,  I have Ockham's chainsaw ready.

[diane]

Overall, I suspect that pck regards this discussion of FIBS dice to be more of an intellectual puzzle than he is convinced there is outright dice manipulation and conspiracy, but in the event he gets too carried away with attributing sinister motives to Marvin and dice conspiracy theories,  I have Ockham's chainsaw ready.

I actually don't think for one moment that pck thinks there is any bias, or that anything untoward going on with regard to dice at fibs. I think his/her only issues are with the strength and nature of the arguments used to prove that they aren't. For each 'proof' offered that all is well, pck offers some rationale for that proof to be flawed. I have never heard an allegation that the proof is flawed because the dice are biased/skewed.

I am definitely going with the 'intellectual puzzle' theory

[Slipshod]

Moonshadow,

Pck is not suggesting Marvin intentionally programmed the dice in a manner that would introduce bias. Anytime someone codes a software program there is the possibility of unintended consequences. Actually, we might not be incorrect in saying everytime someone develops complex code without rigorous code reviews and extensive testing, unintended consequences are virtually assured.

Turner's statistics are accurate and Pck knows that; Pck is pointing out that a Chi square test for all rolls is not necessarily a perfectly accurate test to eliminate the possibility of bias of all types. That is, a bias which is not solely attributable to random number generation. In this instance the Chi test is asking the question is the observed distribution of numbers consistent with a random distribution of numbers? The answer is clearly yes (although I will deny this the next time those cheating bots out roll me).

It is, of course, highly probable that there is no bias whatsoever in FIBS dice. However, that is an entirely different issue as to whether or not there is a rigorous proof of the same. I haven't read Turner's analysis so I can't comment on precisely what it was he set out to prove.

Pck's statment is accurate; Turner's data and statistical calculations are accurate; an accurate and careful use of language is difficult.

[pck]

What you are really suggesting--and all without any evidence or motive--is that in 1992 Andreas Schneider (aka marvin) intentionally programmed the FIBS software to diabolically "manipulate dice" and then foisted that off on the public on July 19th of that year.

What happened is that in your rush to brush aside and casually dismiss what you thought to be a sophomoric argument presented by statistician Turner's analysis, you perhaps hadn't fully read the article which I had linked but not quoted, or perhaps you read it, but not truly comprehended it.

Looks like I might have to reconsider my belief that vic is the fibster with the highest bp.

Though intellectually stimulating, there is no need to tax our brains with this scenario.

Most of what I need to reply here has already been said by diane and Slip. My motivation was indeed to show that both sides of the dice manipulation debate have their weaknesses. Nothing more, nothing less. While it can get mathematically complex, the hardest part of statistics is interpretation. In the case of FIBS dice it amounts to considering what constitutes a proper test for manipulation and what doesn't.

Overall, I suspect that pck regards this discussion of FIBS dice to be more of an intellectual puzzle than he is convinced there is outright dice manipulation and conspiracy, but in the event he gets too carried away with attributing sinister motives to Marvin and dice conspiracy theories,  I have Ockham's chainsaw ready.

While on the subject of attributing sinister motives, look at my ficticious manipulation scenario as a napkin that wipes the smug smile off the faces of those whose arguments carry insufficient clout. You don't always need a chainsaw to restore logical balance. (And btw, the logical weight of Ockham's principle is zero.)

I've never been much interested in intellectual puzzles. What gets my interest are conceptual problems and analyses, and randomness and probabilty are perhaps the hardest ideas a mathematician gets to wrestle with conceptually.

I've attached a pdf and two jpegs with an eXtremeGammon (XG) analysis of 120.000 of my and my opponents' rolls on FIBS (about 1100 matches, carried out by gumpi about two years ago; I do not have XG). On the last page of the pdf (which is also shown in pck_X2_dist.jpg) you'll find the result of XG's chi squared (X2) test. My opponents' dice can be viewed as representing the server since I played many different people (and bots). You'll note that my opponents' X2 turned out fine, while mine sucks, indicating that my rolls were rather far away from what can be expected from a random dice generator. What does this mean? pck_luck_skill_stats.jpg shows that my total luck over all these matches was pretty much exactly zero. This rules out the interpretation that the server "had it in for me" and in fact is a much better counter against accusations of fixed matches than the serverwide "dicetest" or "matrix" commands. So what I got were "crazy dice", but not "fixed dice". This is a distinction that hadn't even occured to me until I started thinking about randomness as a concept and gumpi being kind enough to provide me with this analysis. I have certainly never seen it even casually mentioned in any debate about rigged dice.

Asking myself how it is even possible to have serverwide stats which are fine but nonetheless skewed for an individual player is part of the covert and sinister history behind my posting above. The simple manipulation scenario I described is consistent with my X2 results, but what clears FIBS of suspicions concerning fixed dice (with respect to backgammon) is my luck.

[dorbel]

You don't need all the maths and computer tests do you? I think that all you need to ask is, "Are there any numbers on my next roll that are more likely to appear than probability suggests?" If the answer is no, then you have random dice. If on the other hand the answer is yes then you don't. I invite anybody who thinks that fibs dice are not random in any particular scenario (dice for a bot in a race for example) to bet real money on the outcome. I won't hold my breath while I wait for a response.

[pck]

You don't need all the maths and computer tests do you? I think that all you need to ask is, "Are there any numbers on my next roll that are more likely to appear than probability suggests?"

That, in fact, is part of what a chi squared test (and the matrix command) does. It's of course possible, but tedious, to do this without a computer if you're looking at a couple of thousand or more rolls.

Do we need the math? Yes. Consider a dice generator which puts out this sequence (spaces and linebreaks are for ease of reading, only one die is being rolled, not two):

11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
<repeat from beginning>

This passes the test your question asks for, but is as non-random as it gets. (The equivalent sequence for two dice is a 36x36 matrix with all possible combinations of two rolls, you get the idea.)

What makes a sequence of dice random is the absence of patterns. Patterns equal predictability, and predictability is the opposite of randomness. Since there are infinitely many different patterns which might occur in a dice sequence, it is impossible to have a perfect test for randomness.

Even putting it like that is a bit sloppy. Strictly speaking, any finite sequence of rolls is a pattern, which means that the idea of "patternlessness" can really only be applied to infinite sequences. Since actual infinite sequences of rolls are not available, but most dice generators use algorithms to create their output (*), the question of predictability becomes one of analysing a given algorithm for patterns. This can be attempted directly by mathematical argument or empirically by performing tests such as chi squared on samples of the algorithm's output. This is technically and conceptually challenging territory on which there is no unanimous agreement even among professionals.

---

(*) The alternative would be additional hardware which conducts actual experiments with random outcome, or software using other (hopefully) non-predictable data, for example from the internet.

[inim]

My motivation was indeed to show that both sides of the dice manipulation debate have their weaknesses.

What two sides of what debate? So far not a single piece of evidence was presented that the dice are manipulated. What is presented is a lot of conspiracy theory how they MIGHT be manipulated and how that may be hidden by a malevolent admin. How bizarre, how bizarre.

This is just another example of the reverse scientific method, where you start with a hypothesis and then desperately look for evidence supporting it - while you safely ignore anything you find which doesn't support it. And there is plenty of facts supporting the hypothesis that fibs dice are just ramdom.

Let me know when evidence was found that fibs dice are manipulated. Before that, there is no debate and I am on no "side" of it.

[pck]

What two sides of what debate? So far not a single piece of evidence was presented that the dice are manipulated. What is presented is a lot of conspiracy theory how they MIGHT be manipulated and how that may be hidden by a malevolent admin. How bizarre, how bizarre.

No claim was made about having evidence for dice manipulation. Quite the contrary: In #11 I wrote: "[...] what clears FIBS of suspicions concerning fixed dice [...] is my luck."

Thoughts about how dice might be manipulated are not conspiracy theory but necessary steps in the process of clarifying what constitutes a proper test for manipulation. "Malevolent" or "diabolical" admins were never brought up until the blood started rushing to the heads.

The two sides of the debate are: Arguments in favour of dice manipulation and arguments against it. I believe that the arguments in favour of manipulation are weak. But I also believe that many of the arguments against it have been weak or confused as well.

This is just another example of the reverse scientific method, where you start with a hypothesis and then desperately look for evidence supporting it - while you safely ignore anything you find which doesn't support it. And there is plenty of facts supporting the hypothesis that fibs dice are just ramdom.

This is both logically and factually inconsistent. If you accept that there are facts in support of the hypothesis of randomness then you must accept all facts against it too. Whether these were gathered starting from a hypothesis of non-randomness or not is immaterial. The only thing that matters is whether they confirm the hypothesis or not. In matters of logic, there is no principle of "when in doubt, for the accused".

What are the facts in support of randomness? The results of the dicetest and matrix commands. These were not - as you claim - ignored, but put in their proper place: They do constitute evidence in favour of randomness and against manipulation. All I was concerned with was showing that that evidence is not as strong as one might think. To support this I constructed two very simple scenarios which can fool dicetest (more precisely, the first half of dicetest - see reply #7) and matrix (reply #13) respectively. In addition I presented XG stats, which show that my dice in 1100 FIBS matches were on the extreme end of the spectrum of randomness.

According to XG, the probability of getting even less random dice than I did in those 60000 rolls is 3.119%. That is not the same as winning the lottery, but enough to raise an eyebrow. To put it in your language: Which part of "worrisome" do you not understand?

Even if it were shown beyond doubt that FIBSdice are non-random, this would be no good reason yet to say that FIBS is rigged. As I already pointed out in #11, to successfully be able to conclude that, luck would have to be skewed as well.

Let me know when evidence was found that fibs dice are manipulated. Before that, there is no debate and I am on no "side" of it.

I never looked for evidence of manipulation and I never accused anyone of it. I looked at the data available to me and tried to make sense of how both sides of the debate were defending their positions. The logical and conceptual sloppiness inherent in "arguments" such as you present here is part of what prompted me to write about these issues in the first place. I believe that much of the dice manipulation debate is, if not plainly confused, talk at cross purposes.

On a personal note, while moonshadow's "burn in hell, heretic" approach was funny (yet effective, I got extra insurance for my house against hail and frogs after that), your persistently condescending tone is merely annoying and diminishes your credibility.

[inim]

Which part of "worrisome" do you not understand?

The part where we switch from verbose panicking to concrete scientific working. Where is the data, what was the method used in the experiment, and what prediction is made for the outcome of an indepenent repetition of the experiment.

A screenshot of a closed source program based on unknown data is not a reproduceable experiment and means nothing so far. In the past, the process of deriving the data was flawed (subsets were counted twice). So deriving the input data for the statistical analysis is part of the experiment and must be reproduceable as well.

[garp_02]

Can someone explain to me, without being nasty , how the Fibs DG distributes the rolls?

I am no maths genius but am reasonably intelligent - so keep it simple.

It is my understanding that the DG's normally used in these situations are pseudo-random number generators which apply an algorithm to a particular seed number, which then produces a predictable sequence of rolls based on that seed. The same seed will always produce the same sequence. This is NOT random number generation, but, to all intents and purposes, will appear random enough for our purposes.

If this is correct then how are the seed numbers generated - as in the old 'who created the creator' argument. Not that I think this matters too much as it will still appear random, but not be random.

So if I startd a game with pck, say, does the DG start a new sequence of rolls based on a new seed which applies to our game only? Or is the number sequence running in the background and allocates the next roll in the sequence every time any player rolls the dice. In many ways, this would be more random in the true sense of the word.

Be nice with your replies

[pck]

The part where we switch from verbose panicking to concrete scientific working. Where is the data, what was the method used in the experiment, and what prediction is made for the outcome of an indepenent repetition of the experiment.

I've attached the data from which the analysis was generated. The method used is a chi squared test on the distribution of all possible rolls of two dice (as is obvious from the screenshot). There are numerous other evaluations such as race doubles etc. in the pdf I uploaded with the screenshots. Since this is a single "experiment" ("evaluation" would be a less misleading term) there can be no predictions.

A screenshot of a closed source program based on unknown data is not a reproduceable experiment and means nothing so far. In the past, the process of deriving the data was flawed (subsets were counted twice). So deriving the input data for the statistical analysis is part of the experiment and must be reproduceable as well.

This is confused on several levels but I can guess what you mean.

1) It's true that in gumpi's first attempt to have XG analyse my matches, the derivation of the results from the input data was flawed. Gumpi was new to XG then and did not operate it correctly when it came to telling the program that it needed to treat all of my opponents as a single player. Accordingly, the resulting statistics looked suspicious from the start: http://www.fibsboard.com/using-backgammon-software/extremegammon%27s-chi-squared-tests-on-fibs-dice/msg22180/#msg22180
He subsequently corrected this, but I never posted the proper results until you challenged me to in shouts.

2) As for the input data, these were of course never "derived", they are dice rolls from a little more than 1100 fibs matches. I assume that by "reproducing the experiment" you mean running the chi squared test again. "Deriving" or "Reproducing the input data" makes no sense.

The closed source argument will of course remain available as long as XG is used. I trust gumpi's ability to work XG correctly (esp. because he got it wrong the first time) and the author of XG to be capable of coding the correct formulas. But with the data I attached, you can run your own chi squared test. If you do, you need to make sure the number of degrees of freedom is the same as XG used (which is 21 - 1 = 20), otherwise the results will not be comparable, as Chi Squared depends on "binning".

[pck]

So if I startd a game with pck, say, does the DG start a new sequence of rolls based on a new seed which applies to our game only? Or is the number sequence running in the background and allocates the next roll in the sequence every time any player rolls the dice. In many ways, this would be more random in the true sense of the word.

My guess would be the latter. I don't see any need to re-seed the DG every time a match is started, unless the algorithm is known to run into a not-so-random streak after a while, producing, for example, not enough 6s anymore.

I deliberately refrained from any speculation about how FIBS works internally, software or hardware. It doesn't change the core of the debate. What I want to look at are the statistics (rolls, luck, etc.) as the players get to see them, because that is what the criticisms are about.