> I suppose that in European casinos a half century ago they had roulette wheels that had been in service for 80 years without any real maintenance and would show massive bias without anybody noticing.

Sure. But not a contingent bias. And certainly not a contingent bias that was bigger than any concomitant non-contingent bias.

He didn't say contingent bias, you did. Nothing he's written is in any way related to prior individual rolls.

He said he knew how to place a winning bet, he never said it was because the previous results were exactly 1, 2, and 3 and no other numbers. Those results can indicate something other than a "contingent" bias; because they landed on 1, 2, and 3, the wheel is leaning more towards that half of the wheel, meaning he could therefore assume bets on that half of the wheel were more likely to hit, which would then allow him to accurately say he knew what to bet next (a complex bet) that would pay out (win more than wagered).

You're thinking about this mathematically, when the edge comes from mechanical physics.

1 2 and 3 are distributed around the wheel, there is no "that half"

Not perfectly evenly distributed, which is the point.

You could think of the statement as being about knowing the actual bias as a function of only being able to look at the previous rolls. In other words, it is “contingent” on the specific roulette wheel being observed, not “contingent” on which exact rolls were seen recently, per se. But language and thinking on these subjects has evolved since 1969, so the way to frame it to a layman would obviously evolve as well. It’s been over 50 years, after all! Trying to understand the statement in the context of 1969 makes the framing much more reasonable.

Edit: I can also see how this framing might have been important to funding ongoing research, ie, perhaps you are right that there was some degree of unethical framing here.

Recall the quoted passage:

> “If numbers 1, 2, and 3 won the last 3 rounds, [I could determine] what was most likely to win the next 3.”

That is unambiguously claiming not just a contingent bias, but a bias contingent on the previous three rounds. My math was based on a one-round contingent bias. That is already in the realm of extremely unlikely. Three-rounds is just flat-out impossible.

I don't think it's unambiguous. I think the parent's interpretation is also quite possible (even probable, given your point that the contingent interpretation is impossible).

Would a bias contingent on the previous 3 be considered Markovian in this case?

> That is unambiguously claiming not just a contingent bias, but a bias contingent on the previous three rounds.

No it isn't, only that three rounds may be enough to predict the bias. Whether the bias actually depends on previous results is irrelevant to the quote.

> three rounds may be enough to predict the bias

That is an even more absurd claim.

Wouldn't this be a very simple example (simple to the point that it's obviously exaggerated and wouldn't happen in a real casino): each spin of the wheel takes it exactly one whole rotation plus one number, which if the dealer always picks up the ball and spins starting from the previous winner would mean the next winner is always the next number around on the wheel?

It's pretty clear that you've never actually seen a roulette wheel in action. Introducing a bias intentionally (i.e. trying to control where the ball drops) would take extreme skill. Doing it unintentionally is impossible.

I've played roulette many times, and yes I agree that doing it with skill on a fair wheel would be impossible. I thought we were talking about potentially rigged games, and just as I couldn't roll fair dice to the same number 20x in a row I'm aware that dodgy dice can be made, so why couldn't a dodgy wheel (or accidentally dodgy by being so old that a noticeable pattern emerges) be made?

There was a Spanish family that took advantage of that and won big in European casinos. They recorded a large number of results, found biases big enough to overcome the house's edge, and used them. The casinos responded by moving the tables around and moving wheels between tables, but the family has spent so long observing the wheels that they could recognize individual wheels by the wear and tear they had accumulated over the years.

But this was not contingent biases. The odds of a given number were not dependent on what number had previously come up.

These people were covered in the episode "The Roulette Assault" of the old History Channel series "Breaking Vegas" [1]. You can watch this episode at the Internet Archive [2].

"Breaking Vegas" also did an episode, "Beat the Wheel", on Doyne Farmer and Norman Packard's computer-in-a-shoe that used physics to predict roulette outcomes. These are the people who were the subject of the book "The Eudaemonic Pie" which another commenter has mentioned. You can watch "Beat the Wheel" at the Internet Archive [3].

There's one more "Breaking Vegas" roulette episode that is very interesting. It is called "Ultimate Cheat".

The Spanish family was definitely not cheating. It is not against the rules to gather data--in fact many casinos encourage it because 99.99% of the time someone gathering data thinks they have some foolproof system for winning when in fact they have completely botched it and casinos love these people. They blow a ton of money before they figure out that their system is completely worthless (at best).

The "Eudaemonic Pie" people may or may not have been cheating depending exactly on how the rules of the casinos were written back then.

The guy in "The Ultimate Cheat" was definitely cheating. At first he was cheating using a particular method. He was successful at it, but then casinos installed video cameras that recorded everything at the tables. When he would win, they would detain him while they checked the tape, and the tape showed the cheating. This seemingly put an end to his career as a roulette cheater.

But then he started winning again, and when the casinos checked the tapes they showed the win was clean. They were certain he was cheating and they were scrutinizing his wins in excruciating detail, but they could not find any problems with them.

There was a good reason for that. His wins were in fact entirely legitimate. He legitimately put a stack of his chips on a number, did not influence the wheel or ball or croupier in any way, and if his number came up by chance (which happened just as often as you would expect it to) he collected his winnings.

What he did was take what he had been doing originally, the method he had to abandon due to the cameras, and apply a brilliant twist to it. That method is post roll bet modification. You put down a stack of low value chips on a number, and if that number wins you try to swap that stack for a stack that has low value chips on top and a high value chip on bottom displaced a little so it is not visible to the croupier.

The cameras catch you doing the stack swap and the jig is up.

His twist was to apply it to losing bets instead of winning bets. He would bet a stack of low value chips on top with a high value chip on the bottom. If he won he did nothing. But if he lost he tried to swap the stack for one that only had low value chips.

The casino was only scrutinizing his wins so he got away with cheating on his losing bets for an embarrassingly long time. You can watch "The Ultimate Cheat" at the Internet Archive here [4].

If you are checking out "Breaking Vegas", there is one more I will recommend even though it has nothing to do with roulette. That's the "Counterfeit King" episode, on the Internet Archive here [5].

That's about a guy who was making counterfeit casino tokens. They were good enough that the casinos only knew they were there because their token counts were unexpectedly going up. When the casinos sent tokens back to the manufacturer to check, the manufacturer said they were all legit.

The downfall of this counterfeiter is instructive. One day he was using his tokens in slot machines. He was playing a fairly expensive machine and it jammed. He just moved over to the next machine and continued. A guard saw this on live surveillance and became suspicious because when a machine eats a high value token the gambler is always very annoyed and comes complaining to casino staff to get their token back. The gambler just going "meh" and moving over a machine was just not believable. That got him on their radar as someone up to something which led to them discovering he was the counterfeiter.

[1] https://en.wikipedia.org/wiki/Breaking_Vegas

[2] https://archive.org/details/breaking-vegas-s-1-e-10-roulette...

[3] https://archive.org/details/breaking-vegas-s-1-e-08-beat-the...

[4] https://archive.org/details/breaking-vegas-s-1-e-01-ultimate...

[5] https://archive.org/details/breaking-vegas-s-1-e-06-counterf...

Breaking Vegas was a fantastic series. My favorite episode was the one where a programmer working for the Nevada Gaming Commission hacked the ROMs of slot machines and was able to activate a jackpot with a secret combination of coins.

For those who want to see it, that episode was "Slotbuster" and is on the Internet Archive here [1].

It looks like all 14 episodes of "Breaking Vegas" are there [2]. I haven't seen them since around the time they were originally broadcast, but my recollection is that most of them were quite worth watching.

[1] https://archive.org/details/breaking-vegas-s-1-e-02-slotbust...

[2] https://archive.org/details/@vegasfan

> It is not against the rules to gather data--in fact many casinos encourage it

Except when it's blackjack and you're counting cards.

you are allowed to count cards at most (all?) casinos in the usa.

It's legal, but if you're too successful, a lot of casinos will show you the door, sometimes preventing you from converting your chips back to money. See https://youtu.be/rjfLuM-Pqr8