Virtual Losses and the Limits of Randomness

[Bayes]

This is a "fork" of another thread, in which I made the following statement regarding virtual losses:

The reality is that no matter how many virtual losses you use (and it would seem that more is better, for "safety"), outcomes behave like a receding horizon. It seems to be there in the distance, but never gets any closer. This is because being selective about what you "take" from a continuous random stream of numbers doesn't change the fact that you will end up with another continuous stream of random numbers which are indistinguishable from the original stream. "Removing" some elements doesn't leave any less.

I appreciate that this is very counter-intuitive and hard to get your head around, but sooner or later you have to come to this conclusion.

Here is part of Kav's reply:

Imagine this scenario. Someone is observing a number repeat 8 times (or whatever the realistic upper limit) then another player comes to the table and for the next 8 spins he sees the same number repeats itself 8 more times (the repeats limit for the second observer). Then a third person comes to the table and it is perfectly possible that he will observe the same number repeat another 8 times because he has not reached his limit. The problem is that the first person who sits on the table all this time has now seen the same number repeat 24 times!

This apparent objection to my proposed "model" of a receding horizon is that it must break down at the limits, because no one will ever see 24 consecutive hits on the same number (or 100 reds in a row, or whatever). So if it isn't true "at the limits", then that calls into question its accuracy as model, so perhaps, just being a "theory", it shouldn't be taken too seriously.

palestis seems to agree with this:

That's basically what it comes down to.
Let's take a more realistic event rather then the almost impossible 8 repeats of the same number.
A streak of 5,6,8,10 or whatever black in a row. As you walk around among many active roulette tables you will find that it's not that rare to observe a situation like this. Whether it is red or black or odd or even or 3 same DS's. If this situation is a virtual loss for me, I count on the fact that there will be at least one streak break in the next few spins.
But if my virtual losses don't count and the distant horizon becomes even more distant, simply because of my arrival at this table, ( and the roulette read my mind), then all players that happen to be playing  what I plan to bet on,  will be forced to see a new record. Or be PUNISHED, simply because  of my way of thinking as I came up to that table. Then if someone else thinking like me came up to the table, I will be forced to lose because A NEW STATISTIC has to be constructed specifically for the new player to push the horizon even further in a greater distance.  Ignoring every player that has been there earlier.
If that was the case then we would be witnessing "horror streaks" a lot more often. But we don't. Fortunately.
Virtual losses count and count heavily. And there are a lot of players using them without realizing that they use them. In fact most players moving around from table to table that's exactly what they do. Some follow the streak, the others go against it. if those who go against it are doomed, then we would be seeing streaks that we never saw before. But we don't. Something, some higher power makes sure that things remain normal, most of the time.

As I see it, there are several things wrong with this assessment. In the first place, there's no need to invoke any mysticism of a "higher power", or the idea that the roulette table knows what you're thinking and changes the outcomes accordingly. The idea of a limit to randomness is connected with the fact that we're usually interested in one particular sequence of outcomes selected from a staggering large number of possible outcomes.

In fact, each sequence of 24 spins has a probability of 1/37 x 1/37 x 1/37 ... up to 24 times, which gives a probability of 1 in 4.33 x 10³⁷, a vanishingly small number. This is why we are never likely to see such an event, but each one of these possibilities is equally likely, so in choosing any one of them and testing for its "limit", the actual limit is not inherent in the outcomes themselves, but in the computing power at our disposal.

Thus, in testing for worst case scenarios, I've discovered that there seems to be a "limit" of about 5 standard deviations, no matter what the bet is. But is this really a limit? is there really some "higher power" which constrains the outcomes to conform to these apparent limits? I think it's an illusion, and that there really aren't any limits to randomness (which, by the way, is confirmed by the bell curve).

If palestis is correct, and that "virtual losses count and count heavily", then why do all simulations show that they don't count at all?

If outcomes (streaks, or whatever) tend towards a limit, then waiting for 4 virtual losses should give slightly better results than waiting for 3 virtual losses, waiting for 5 virtual losses should be better than waiting for 4, etc, but we don't see this. The actual results conform to the "infinite horizon" model which I suggested.

Here's another way of looking at it. Many are familiar with the technique of betting the opposite of the last X decisions (in fact I think this is dobblesteen's primary system on the even chances). So the idea is that you bet the opposite of the last 10 outcomes, and (so the logic goes) this is preferable to betting randomly because it would be a rare event for the last 10 outcomes to repeat exactly in the next 10 spins.

It also seems sensible to bet on the opposite of a longer sequence rather than a shorter one; betting against the last two outcomes is better than betting against the last 1, betting against the last 3 is better than betting on the last 2, and so on. Although this technique doesn't need to wait for losses, it's still a virtual losses type system because you're in effect taking those previous outcomes as your imagined bet selection, which has just lost X times in a row, and so some losses have been "removed" from the outcomes, thus giving a better than average chance that subsequent outcomes will give a win (or so the thinking goes).

I don't have the inconvenience of having to wait for streaks of 10, 15, or 20 in a row and then betting against them to continue, I can just bet the opposite of the last 10, 15 or 20 outcomes. But why stop there? I should be able to make success even more certain by just betting the opposite of the last 30, 50 or even 100 spins. In that case, I've found a bet selection which has lost 100 bets in a row - surely success in the next spin or two is guaranteed, right?

Sadly, no. I've tested this and similar ideas and the results are no better than just betting randomly, or on red. And it turned up some losses the like of which I'd never seen, like 30 losses in a row.

And the reason why it doesn't work is that there are no limits to randomness. The illusion that there are comes from not understanding that our choice of bet selection is only one of an infinite number each of which is equally like, and so it seems that events tend to a limit. It's not that there actually is a limit, but there are limitations in confirming what the theory predicts. It's not that probability theory breaks down or becomes invalid after a certain point.

[kav]

Thanks for your reply Bayes,

Maybe we should split the discussion to more specific sub-subjects. This is a very fundamental concept we are talking about and as you showed in your post it has many implications. I think that talking about its implications (triggers, progressions, winning etc.) is very interesting from a practical angle, but it is also more open to debate. So I would like to focus only on the main question about limits and extreme conditions. Do limits exist? But instead of your question "Do limits exist in randomness?" my own question is "Do limits exists in reality?"

Let me put a problem to you and please tell me your answer.
Someone has recorded 75 continuous spins of a specific, unbiased roulette wheel. He plays the video and shows you the first 25 spins: they are all Black numbers. Then he stops the video and tells you that in the rest of the video the following 50 spins are either:
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B
or
R B R R R B R R B B B B R B R R R B B B B B R R B R B B R R B R B B B B B R B R R R B B R R B R R B

We know that both sequences have the same probability. Which one would you choose to bet your money on, as the most probable continuation of a 25 all Black sequence? I believe that limits exist in reality and therefore I'd choose the second sequence. What would be your answer? Why?

Another example. The first 5 spins in the video are 5 repeats of the number 7. Then he stops the video and tells you that in the next 10 spins one of the following sequences is correct:
7 7 7 7 7 7 7 7 7 7
6 5 19 18 17 5 7 0 1 32
On which sequence will you bet your money on?

Would seriously claim that in reality, a "75 Blacks" or a "15 repeats of 7" sequence is equally probable as any other sequence of the same length? No, I don't want your theoretical answer, I want the your-life-depends-on-it answer. I know that theoretically they are equally probable. But imagine that you had to risk everything you own on this bet. The 75 Black spin sequence or the other one?  I want to know if you would bet your fortune on a 75 Blacks  or a 15 repeats streak. Believe me there is a big difference between theoretical risk assessment and risking your own money. N.N. Taleb has written extensively about it and has described it as the "skin in the game" factor. You can read my opinion about the difference between theory and reality and theoretical use of statistics vs risking your own money

I believe that in the constrains of reality and given that I don't want to pass a math exam but I risk my own money, for all practical purposes there ARE limits to roulette extreme events.

[Reyth]

I don't have the time to digest everything in this thread but one thing has stood out and its the concept that limits in randomness actually exist.  I guess I have worked really hard at programming myself to always be conscious of and admit the POSSIBILITY of any result taking place that even though I have seen the proof repeatedly of these limits and without a single instance of failure, I still consider them "virtual" limits that I expect to possibly be surpassed on some dark day during a singularly isolated unexpected moment -- it seems like the stuff for poetry:

“Once upon a midnight dreary, while I pondered, weak and weary,
Over many a quaint and curious volume of forgotten lore,
While I nodded, nearly napping, suddenly there came a tapping,
As of some one gently rapping, rapping at my chamber door.
Tis some visitor," I muttered, "tapping at my chamber door —
Only this, and nothing more."

Ah, distinctly I remember it was in the bleak December,
And each separate dying ember wrought its ghost upon the floor.
Eagerly I wished the morrow; — vainly I had sought to borrow
From my books surcease of sorrow — sorrow for the lost Lenore —
For the rare and radiant maiden whom the angels name Lenore —
Nameless here for evermore.

And the silken sad uncertain rustling of each purple curtain
Thrilled me — filled me with fantastic terrors never felt before;
So that now, to still the beating of my heart, I stood repeating,
Tis some visitor entreating entrance at my chamber door —
Some late visitor entreating entrance at my chamber door; —
This it is, and nothing more."

Presently my soul grew stronger; hesitating then no longer,
Sir," said I, "or Madam, truly your forgiveness I implore;
But the fact is I was napping, and so gently you came rapping,
And so faintly you came tapping, tapping at my chamber door,
That I scarce was sure I heard you"— here I opened wide the door; —
Darkness there, and nothing more.

Deep into that darkness peering, long I stood there wondering, fearing,
Doubting, dreaming dreams no mortals ever dared to dream before;
But the silence was unbroken, and the stillness gave no token,
And the only word there spoken was the whispered word, "Lenore?"
This I whispered, and an echo murmured back the word, "Lenore!" —
Merely this, and nothing more.

Back into the chamber turning, all my soul within me burning,
Soon again I heard a tapping somewhat louder than before.
Surely," said I, "surely that is something at my window lattice:
Let me see, then, what thereat is, and this mystery explore —
Let my heart be still a moment and this mystery explore; —
'Tis the wind and nothing more."

Open here I flung the shutter, when, with many a flirt and flutter,
In there stepped a stately raven of the saintly days of yore;
Not the least obeisance made he; not a minute stopped or stayed he;
But, with mien of lord or lady, perched above my chamber door —
Perched upon a bust of Pallas just above my chamber door —
Perched, and sat, and nothing more.

Then this ebony bird beguiling my sad fancy into smiling,
By the grave and stern decorum of the countenance it wore.
Though thy crest be shorn and shaven, thou," I said, "art sure no craven,
Ghastly grim and ancient raven wandering from the Nightly shore —
Tell me what thy lordly name is on the Night's Plutonian shore!"
Quoth the Raven, "Nevermore."

Much I marveled this ungainly fowl to hear discourse so plainly,
Though its answer little meaning— little relevancy bore;
For we cannot help agreeing that no living human being
Ever yet was blest with seeing bird above his chamber door —
Bird or beast upon the sculptured bust above his chamber door,
With such name as "Nevermore.”

Although, I will ask Bayes, you used to think "like I do" which to me sounds quite agreeable, but what exactly was/is it that made you give up this admirable mode of thought?

[scepticus]

Interesting discussion.
I think that if randomness means that we cannot know what the following winning numbers will be then there can be no limit to randomness.
 Leaving aside the zero then any one of the first 3  possibilities  of an EC recurr on each and every 3 subsequent spins. But which one on the NEXT  3 spins ? Or any subsequent 3 spins.? I think we need to make an ASSUMPTION here.
Not only that I think no matter which Method you choose -even AP- you need  to make an Assumption  in  ALL  cases and I think we all need to be aware of that.

[palestis]

Thanks for your reply Bayes,

Let me put a problem to you and please tell me your answer.
Someone has recorded 75 continuous spins of a specific, unbiased roulette wheel. He plays the video and shows you the first 25 spins: they are all Black numbers. Then he stops the video and tells you that in the rest of the video the following 50 spins are either:
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B
or
R B R R R B R R B B B B R B R R R B B B B B R R B R B B R R B R B B B B B R B R R R B B R R B R R B

We know that both sequences have the same probability. Which one would you choose to bet your money on, as the most probable continuation of a 25 all Black sequence? I believe that limits exist in reality and therefore I'd choose the second sequence. What would be your answer? Why?

Another example. The first 5 spins in the video are 5 repeats of the number 7. Then he stops the video and tells you that in the next 10 spins one of the following sequences is correct:
7 7 7 7 7 7 7 7 7 7
6 5 19 18 17 5 7 0 1 32
On which sequence will you bet your money on?

Again that's an extreme case, but the good news is that the same logic can be applied on a more realistic scenario. Something we see every time we go to the casino and we can take advantage of it to win. It doesn't have to be 75 B in a row because this is never gonna happen. It can be 8 or 10 B in a row, or whatever sequence beyond 10 that we happen to come across as we go around observing score boards. With the stipulation that we are only going to bet 5 spins, or stop anytime there is a hit, It doesn't have to be 5 bets. The goal is at least one hit.
If my money is on the line I would certainly bet Red. Aside theory, the main reason is that I rarely see 15 B in a row. I may be in the casino all day and I never see it. But seeing a red after 10B  in one of  the next 5 spins, I see all day long.
Y would I choose the rare and risk my money and not chose the most likely?
One can say that you can bet either B or R for the next 5 spins and still have the same chances.
But the rarity of an image developing into 15 B on the score board, compels  me to bet red.  Just to be on the safe side.
One can also claim that I don't have to wait for 10 B to bet R. I'll get the same results if I start betting R without looking at the board for the past results.
But doesn't that lead to betting every spin?

[kav]

Palestis,

Your approach is of a much more practical value, and you may very well be correct, but there can be much more debate about your everyday example. One could dismiss your observation as confirmation bias and gambler's fallacy. 

The reason I'm focusing on extreme events and limits is that they are harder to dispute.
Therefore I would kindly ask Bayes or anyone else to give a specific separate answer about the extreme limit cases I gave as examples.

[palestis]

This is "fork" of another thread, in which I made the following statement regarding virtual losses:

I don't have the inconvenience of having to wait for streaks of 10, 15, or 20 in a row and then betting against them to continue, I can just bet the opposite of the last 10, 15 or 20 outcomes. But why stop there? I should be able to make success even more certain by just betting the opposite of the last 30, 50 or even 100 spins. In that case, I've found a bet selection which has lost 100 bets in a row - surely success in the next spin or two is guaranteed, right?

Bayes this was a long post, though a very important one.
But I will only comment on the betting against a long streak to discontinue in the next long stretch.
Sure betting against a long streak like 20-30 numbers not to repeat itself, most likely will succeed.
The problems is that it is only practical if you succeed in the first spins, not the latest. Betting against a 30 numbers sequence and doubling each time (even starting with just one penny), will require more money than 10 Bill Gates's  combined. Do the math. If the starting chip is $5 you might need more money than the entire US population put together.
You can't possibly be serious recommending a system like this.
The second comment is on the value of virtual losses.
You said you have done simulations that prove that virtual losses have no value. 
I'd like to know what "command wording" did you use to code a simulation like this.
If you tested  millions of spins thru simulation, and saw 30 or more consecutive losses it doesn't count for this particular issue. It may prove that long losing streaks can happen but it has nothing to do with the use of virtual losses.
You may have seen another 20 red after an already streak of 10 occurred ( to make it 30), but this is not how virtual losses is used.
Most likely the command instruction should be only 5 bets on B after 10 R, with the stipulation that the bets stop as early as when a hit occurs.  Did you program the simulation with those conditions?
Secondly does your simulation involve randomly varying $ amounts on those bets?
Also does it take into account that after a streak of successful triggers, the player can change at will the starting chip to the lowest value, to protect his profits? (Believe it or not, after a few consecutive successful attempts, the next attempts will have a little bit harder time winning regardless of system. And that's from my own experience and other who have been playing for a  long time).
I doubt very much that you used those and other constrains to program the simulation.
Because simulation is impossible if it involves "spur of the moment" decisions, and also money values that change randomly according to the player's will.
However your idea of 10,20,30 numbers streaks not to repeat, gave me an idea that is based on that, but with a unique  twist.
Pattern breakers involve betting too many numbers.
EC pattern breaker require 18 numbers bet.
Dozen pattern breaker require 2 dozens bet (24 numbers).
Single DS pattern breaker requires 5 DS's bet (30 numbers).
However the opposite (PATTERN REPEATER) requires the same bet as the group you anticipate to see repeated. I will come back on that as soon as I have some test results.

[palestis]

Palestis,

Your approach is of a much more practical value, and you may very well be correct, but there can be much more debate about your everyday example. One could dismiss your observation as confirmation bias and gambler's fallacy. 

The reason I'm focusing on extreme events and limits is that they are harder to dispute.
Therefore I would kindly ask Bayes or anyone else to give a specific separate answer about the extreme limit cases I gave as examples.

I see. I am curious too. You got my answer anyway.
If the gambler's fallacy applies in every day ordinary sequences, and not is extreme cases like you mentioned, then the gambler's fallacy is a myth. 

[Reyth]

If you program your range technique:

1) wait for trigger
2) bet range
3) repeat 1 & 2 if no hit in range until hit is established in the range

You will find that the max loss (say on a EC) will drop significantly from its normal level.  It doesn't matter how many spins you try, the max loss is always less which means you have a genuine edge. 

The reason for this is exactly as you explain:

Rare events will not repeat themselves successively very often or for very long.

This "successive frequency rarity" is not achieved when betting an EC in one long stretch because that is only one event (not frequent nor successive); the stacked ranges subsequent to the trigger act as a statistical sieve to improve results.

It is interesting to note that the rarer the intial event, the less likely it is for it to repeat; i.e. the longer the triggers & the bigger the ranges, the greater the max loss reduction.

This is a statistical breakthrough in roulette that I don't think has been fully explored...

[Reyth]

Believe me there is a big difference between theoretical risk assessment and risking your own money. N.N. Taleb has written extensively about it and has described it as the "skin in the game" factor. You can read my opinion about the difference between theory and reality and theoretical use of statistics vs risking your own money

I believe that in the constrains of reality and given that I don't want to pass a math exam but I risk my own money, for all practical purposes there ARE limits to roulette extreme events.

In thinking about this I think there is a certain "life or death", "fight or flight" instinct involved that could come into play in order for one to "intuit" the truth.

I think the real problem here is that roulette is so difficult as it is, that it is far too easy to believe GF related theory than it is to fight the variance with contrary theory; its too easy to lose in roulette and its too hard to prepare a proper statistical defense to deal with losses. 

This conceptual environment may only get worse if one programs simulations because they tend towards simplicity whereas statistics in roulette and the means required to deal with those statistics, tend towards complexity.

[Bayes]

Let me put a problem to you and please tell me your answer.
Someone has recorded 75 continuous spins of a specific, unbiased roulette wheel. He plays the video and shows you the first 25 spins: they are all Black numbers. Then he stops the video and tells you that in the rest of the video the following 50 spins are either:
B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B
or
R B R R R B R R B B B B R B R R R B B B B B R R B R B B R R B R B B B B B R B R R R B B R R B R R B

We know that both sequences have the same probability. Which one would you choose to bet your money on, as the most probable continuation of a 25 all Black sequence? I believe that limits exist in reality and therefore I'd choose the second sequence. What would be your answer? Why?

As scep points out, there are always assumptions. If you hadn't explicitly said that the wheel was unbiased, I would have chosen the first sequence (all B's). This is because the empirical evidence suggests (according to Bayes' theorem) that there is a bias towards black. However, since we know the wheel is definitely not biased, I would choose the second sequence. But (and this is crucial), not because red is "due", but because a mix of red and black is more likely than all one colour, regardless of what has come before.

This doesn't contradict the fact that all sequences of the same length have the same probability, because in the case of a sequence the order of outcomes is taken into account. There are permutations and combinations; order matters for the former but not for the latter. To clarify this, consider the following sequences:

RRR
RRB
RBR
BRR

Ignoring the zero, each sequence has the same probability of 1/2 x 1/2 x 1/2 = 1/8 but if we ask, what is the probability that there will be exactly one black in 3 spins? The answer is the sum of each of the sequences which contain one black, which is 3 x 1/8 =  3/8. We can add them up because the sequences are mutually exclusive. So since 3/8 is a larger probability than 1/8, it makes sense to bet on there being 1 black in the 3 trials, rather than none. Notice that I don't need any information about what went before in order to come to this conclusion.

So in the light of this, I have a question for you, Kav. Suppose the conditions in your example are the same, but in this case you're given a choice not of just the two sequences

B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B

and

R B R R R B R R B B B B R B R R R B B B B B R R B R B B R R B R B B B B B R B R R R B B R R B R R B

but also a third, which is

B R R R R B B R B R B B R B B R R B B B B B R B B R B B R B B R B B B B B R B R B R B R R R B R R R

This last sequence has the same number of reds as the second, but the sequence (order) is different.

Which would you choose?

Another example. The first 5 spins in the video are 5 repeats of the number 7. Then he stops the video and tells you that in the next 10 spins one of the following sequences is correct:
7 7 7 7 7 7 7 7 7 7
6 5 19 18 17 5 7 0 1 32
On which sequence will you bet your money on?

The second sequence, for the same reasons given above.

@Reyth, you asked:

Although, I will ask Bayes, you used to think "like I do" which to me sounds quite agreeable, but what exactly was/is it that made you give up this admirable mode of thought?

Good question. I've never been satisfied with just theory, no matter how plausible is it. The way to know is to actually put it to the test. This attitude is beautifully summed up by the great physicist Richard Feynman in a one minute video:

https://www.youtube.com/watch?v=b240PGCMwV0

I believed there was a limit. And if there is a limit, it just seemed like common sense to wait for prior losses because in that way you're at least preserving your bankroll - losses cannot continue indefinitely so by "using up" some of those losses without having lost any bankroll, then losses should be fewer. It seems you have gained an advantage, not necessarily overcoming the house edge, but at least you should have reduced the length of the losing streaks. All very logical as far as it goes, but I was wrong because I didn't question the assumption that there was a limit. When I tested the assumption thoroughly I found that there wasn't a limit at all.

After doing the empirical tests I realized they only confirmed the simple conditions and nature of the game of roulette. On each spin each number has just as much chance to hit as any other. There are no limits implied, but players create their own based on what they see at the table, e.g. we never see 100 reds in a row, etc, therefore there must be limits and by extension outcomes can't be independent after all, therefore virtual betting works!

[Bayes]

I will reply to palestis later, but for now I have other things to do.

[palestis]

As scep points out, there are always assumptions. If you hadn't explicitly said that the wheel was unbiased, I would have chosen the first sequence (all B's). This is because the empirical evidence suggests (according to Bayes' theorem) that there is a bias towards black. However, since we know the wheel is definitely not biased, I would choose the second sequence. But (and this is crucial), not because red is "due", but because a mix of red and black is more likely than all one colour, regardless of what has come before.

How can a wheel be biased towards a specific color? That's impossible.
Even if the wheel was heavily tilted, there are still 18 numbers alternating between black and red in the lower part of the incline. The only way that could happen would be if  the bottom of all the pockets of one color was raised, then and only then you would have color bias.

[Jesper]

If somebody was showing me 12 of the same number in a row, like   12,12,12,12,12,12,12,12,12,12,12,12.
I would agree it is a very  rare event. If he ask me if I think it will repeat next 12 spins, I will say NO, it is extreme unlikely, and if it is a odds bet against it happens I may bet all I own.

But if I have to chose between a repeat and a certain other mixed frequency like  1,2,34,12,19,21,1,0,17,21, I would think it does not matter which I chose, as all (exactly) outcomes are the same probability.

If somebody use a system with a negative progression, there are more chances to win, when ever we try, but that has a cost when unwanted rare events happen.  There are 37^10 ways to get 10 numbers, some  methods will make it on most of them, and it may be so just one of the 37^10 is a loss.

Regarding bias of a color, it is possible if it is done by somebody can fiddle with the slots, by lack of maintenance it is very very unlikley.

[Bayes]

How can a wheel be biased towards a specific color? That's impossible.

Actually it's not. Real posted this on another forum some time ago:

Oddly enough several years back we found a casino that had a few wheels with a red bias.  The casino had replaced the red inserts, but not the black inserts..I guess because they didn't have the correct material.  The red inserts were deadening the ball more, I guess because of how they were glued.  Regardless, it was one of the few times where a player could get an advantage playing on the outside.

But in any case, whether you know the cause of a bias is irrelevant. If the empirical data is suggesting that some outcomes are dominant then Bayes rule will always tell you to bet on whatever the dominating outcomes are. As you collect more data the dominance may decrease, in which case you would adjust your bets accordingly. If there is no dominance either way then the data says you shouldn't have a preference. Red is just as likely as black. You take account of the data you actually have, not some theoretical model of an ideal wheel which says that all outcomes are equally likely and that probabilities are fixed. The GF comes in when you say "this is a fair wheel and therefore when I see 10 blacks in a row I'll bet red because it's more likely than black". But that contradicts your assumption that the wheel is fair (which implies outcomes are equally likely and independent), hence the fallacy.

Regarding your reply to my earlier post:

Sure betting against a long streak like 20-30 numbers not to repeat itself, most likely will succeed.
The problems is that it is only practical if you succeed in the first spins, not the latest. Betting against a 30 numbers sequence and doubling each time (even starting with just one penny), will require more money than 10 Bill Gates's  combined. Do the math. If the starting chip is $5 you might need more money than the entire US population put together.
You can't possibly be serious recommending a system like this.

I'm not recommending any system at all. The point I was trying to make was that randomness doesn't have any limits, and that waiting for a longer streak of losses doesn't increase the likelihood that wins will come sooner rather than later. Isn't that what you're trying to achieve by using virtual bets?

Most likely the command instruction should be only 5 bets on B after 10 R, with the stipulation that the bets stop as early as when a hit occurs.  Did you program the simulation with those conditions?

No, but I'll write another one with the conditions you specify. In an earlier post you said you weren't interested in comparing the results of tests done when betting randomly or just on one side. But this means you must be just assuming that the method works. Without a "control" group how do you know this? You can't. The point of the simulation is to discover whether virtual betting results in reduced losses or variance compared to just betting continuously on one side, or randomly. So there should be two simulations done and the same number of bets should be placed in each, otherwise the comparison isn't a fair one. Conditions should be the same in both simulations, except that in one you're using virtual bets and in the other you're not. That way you know that the results tell you whether there is any difference with regard to virtual betting, and only virtual betting, not some other criteria.

Secondly does your simulation involve randomly varying $ amounts on those bets?
Also does it take into account that after a streak of successful triggers, the player can change at will the starting chip to the lowest value, to protect his profits?

As I just explained, including these factors in the simulation would only muddy the waters and make it more complicated. It isn't necessary because if you include all this extra stuff in both simulations, the only difference would be in the fact that one strategy uses virtual bets and the other doesn't, so any differences in results will be attributed to that. All other factors such as money management etc will cancel each other out.

Because simulation is impossible if it involves "spur of the moment" decisions, and also money values that change randomly according to the player's will.

I agree that this can't be simulated, but how can a "spur of the moment" decision make any difference to your results?