" THE WEAK POINTS OF ROULETTE " ENGLISH VERSION (LES POINTS FAIBLES DE LA ROULETTE) WHIDARTE J.

English Translation

🎰 THE WEAK POINTS OF ROULETTE

---

Recently, I thought it was high time to share a large part of my knowledge and my thousands of hours of research over the last 25 years, concerning the best ways to counter the fluctuations of chance in random games such as casino roulette.

The first part will be devoted to general theory. The second part to the means of managing the random series of roulette, through ingenious systems to break untimely streaks. The third part will attempt to reverse the player's disadvantage by somewhat contradicting mathematics through a selection of bets to play in finite games.

---

📑 Table of Contents

Part One

1. Presentation of chance and its principles
   • The different betting systems
   • Martingales
   • Against martingales

Part Two

The possible systems

1. Betting techniques
2. The different progressions and staking plans
3. The V2 progression
4. Differential games

Part Three

A) The Transformers

1. Horizontal and vertical deviations
2. Trend classifiers
3. Deviation classifiers
4. Accumulators
5. Compressors

B) The Whidarte 1st martingale C) The Arcsine Law D) The brain and chance E) Virtual colors F) Reflections on personal permanence

---

📖 EXCERPTS

_______________ PART ONE _____________________

First, let us return to the basic theory of probability mathematics, according to which no martingale, no system, however ingenious, can reverse the odds in one's favor when a game has negative expectation.

In the long run, nothing works — it's the law of constant loss: whatever strategy is used, one loses proportionally to what one bets. No martingale, no bet selection, no optimal management ever reverses the outcome; unfortunately, one loses constantly, nothing helps (see Dubins and Savage).

To lose the least, one must play the least, and the more one plays, the more one loses — the only winner is the casino!

According to the Bold Play theorem, in a game with two options of probability P and 1-P, where P is less than 1/2, the best way to play consists of always betting what allows one to approach the target goal as quickly as possible.

Moreover, even a fair game remains a losing one when the fortunes of the two opponents differ. Over time, in an equiprobable game (such as roulette on simple chances excluding zero, for example), the richer party will inevitably prevail — this is the "gambler's ruin" theory.

None of this is particularly encouraging for the inveterate gamblers that we are...

A random phenomenon is by definition impossible to predict. Chance has neither memory nor consciousness; it only possesses certain characteristics of its own. We can only evaluate the probability of certain observable events.

Unlike statistics, which result from observations after experiments, probabilities are theoretical calculations that provide a prediction before the experiment.

Chance blends mathematical, physical, and philosophical models.

---

BUT, is there truly no way to work around the problem, no subtlety, no ultimate stratagem to extract even a single euro of profit?

Let us see how we might approach this seemingly insoluble problem. As Albert EINSTEIN said: "A problem without a solution is a poorly stated problem."

"Wherever chance seems to play on the surface, it is always under the grip of hidden internal laws, and it is only a matter of discovering them." — Friedrich ENGELS

---

First, what do we really know about random-character series like those found in the simple chances of casino roulette (Black, Red, Even, Odd, Low, High)?

Firstly, any perfectly random series is unpredictable. A random sequence must possess no exceptional property, no verifiable periodicity. No computer program can formulate this incompressible sequence. Perfectly random sequences cannot be described more concisely than by enumerating each term.

If chance is unpredictable by definition, repeated chance contains regularities and a certain norm. Certain laws, for example, are used to test pseudo-random draws.

We know that the average frequency of occurrence of a result in a repetition of trials tends toward the probability of observing that occurrence in a single trial (Weak Law of Large Numbers).

The Laws of Large Numbers are "asymptotic," meaning they approach the truth more closely as the number of trials increases (K=30 minimum). A sufficiently large sample is therefore needed to observe the theoretical statistical rules. The minimum size, according to specialists, is around... a minimum number of trials for a result to begin showing reliability. A smaller sample may contain particular deviations or irregularities that nonetheless cannot cast doubt on the statistical laws.

The convergence of average frequencies in a Coin Toss game is 1/2. The speed of convergence was resolved by DE MOIVRE, then LAPLACE (Laplace-Gauss Law, 1821).

The famous GAUSS curve representing chance is the direct consequence of the Central Limit Theorem. This "bell-shaped" curve shows a peak corresponding to the number of events around the mean (top of the bell). There is a high frequency near the mean, while distant values are rare. The curve appears as a limit for many simple cases with frequent trials. A symmetry forms on either side of the curve's median. We also obtain a precise measure of data dispersion relative to the mean — the standard deviation encompassing 2/3 (68%) of the figure, and the variance being the total of the squares of all deviations.

CONCLUSION: The Law of Large Numbers and the Central Limit Theorem are the two fundamental results of probability calculus.

BUT:

According to STEIN's Law, if a phenomenon cannot continue indefinitely, it will stop. And according to MURPHY's Law, nothing guarantees that an event will go wrong precisely when we expect it to.

---

All we are sure of is that a series of "Heads or Tails" will converge after a certain time, at a certain speed, toward a frequency of 1/2. The deviations encountered throughout the sequence will approach the standard deviation, which corresponds to the square root of the number of trials in the sequence (a deviation of 10 for a series of 100).

The Glivenko-Cantelli theorem ensures the convergence of the empirical distribution function as the sample size tends toward infinity.

When we toss a coin often enough, the ratio between the number of Heads and Tails will become arbitrarily close (not equal) to 1/2, and will remain close for as long as we continue tossing.

Convergence only occurs in a "relative" sense. The absolute deviation between the number of Heads and Tails will only INCREASE...

This is why betting methods based on overly long periods are doomed to fail. Sooner or later, an ever-growing cumulative deviation will end up swallowing your entire capital. It will therefore always be essential to close out betting sequences — whether in profit or loss — in order to either start fresh from zero, or at least reduce one's stakes to preserve theoretical gains accumulated up to that point.

In the field of probability, standard deviation is a real quantity, potentially infinite, that characterizes the distribution of a random variable around its mean. Used in statistics, survey theory, meteorology, physics, and biology, standard deviation attempts to assess the dispersion of a randomly drawn sample population.

That is to say, in a random series of 100 casino roulette spins, the average difference at the end between two simple chances such as red and black will be about 10 bets (approximately 45 blacks to 55 reds). These deviations encountered throughout the permanence will follow the GAUSS bell curve.

In a binary random process such as "Heads or Tails," there is 1 chance in 1024 of drawing heads 10 times in a row, and 1 chance in a billion of drawing heads 30 times in a row (we have a greater chance of crashing in a plane).

We measure the uncertainty of events on a scale from 1 (certain events) to 0 (impossible events), and although everything is probability and absolute certainty does not exist, if the mathematical probability of an event occurring is too low relative to physical numbers, we can conclude that it is impossible...

Let us not forget one thing: the interaction between chaos and harmony is the engine of evolution in our world. Everything could theoretically be predicted by an intelligence powerful enough to integrate the totality of initial data, parameters, and factors. Nature obeys laws that govern both the macroscopic and microscopic world; chance plays a fundamental role (quantum mechanics). The "Butterfly Effect" prevents long-term weather forecasting, as infinite variations in initial conditions eventually lead to significant effects. On the other hand, insurance companies calculate their premiums based on random, covered risks and their probability of occurrence, all while remaining profitable.

---

ust like casinos, insurance companies split up their risks, not allowing a single person to destabilize their finances.

Another consequence of this phenomenon is therefore to apply this same principle when developing a betting method, by splitting up our own risks.

When we observe an authentic Roulette permanence with a naive eye, any sensible person cannot help but think that a certain constraint, a certain lack of degree of freedom, comes into play so that in the end the result between each chance is so close and homogeneous. Even if each spin is independent, one can still think that the first ball is "programmed," and that the following ones take into account a certain accounting so as not to skew the probabilities in the end...

---

💰 THE USEFULNESS OF MARTINGALES (Added December 2019)

Even though we know that, from a mathematical point of view, martingales in the gambler's sense provide no advantage and appear perfectly useless, it is still necessary to clarify certain points that will avoid completely dismissing this type of strategy.

As a certain Louis Bachelier, a pioneer in financial management, once said: in life, everything is but a martingale. "Whether it takes an industrial, commercial, or financial form, the Martingale is the sole cause of great fortunes"...

📐 The Mathematical Position

The position of mathematics is as follows:

No martingale, no subtle means, no manipulation, no extraction of sequences, will ever allow one to gain an advantage by betting on perfectly random series.

Sorting, selecting, waiting, choosing, distorting, or parameterizing these sequences will change nothing. In a fair and equiprobable game, it is always the richer party who wins, and in an unfavorable game — even only slightly so — it is impossible in the long run to come out ahead!

In an unfavorable game, the loss is proportional to the number of games played. In Roulette, a player betting on a single isolated spin faces almost no disadvantage, but over time, on the contrary, loss is guaranteed. The total disadvantage equals the sum of the disadvantages of the isolated games.

This is the case for Heads or Tails, studied for centuries, but also, for example, for a game on the simple chances of casino Roulette, which is an unfavorable game for the bettor.

Red and Black are equiprobable simple chances, each composed of 18 numbers, strategically distributed on a Roulette wheel containing 37 numbers (including zero). Let us recall that the distribution of numbers on the wheel — probably by the famous mathematician Pascal — cannot leave any advantage to the player either.

The phenomenon of "Gambler's Ruin" is unequivocal. This guarantees the profit of gaming organizations and casinos. Even a particularly lucky player will not shake the finances of the gaming establishment, as their winnings get diluted by the losses of all other players.

---

🎲 Beyond the Mathematical Proofs

Beyond the mathematical demonstrations of "Gambler's Ruin," we can explain the mathematicians' position on the ineffectiveness of martingales as follows:

Consider applying a martingale such as the d'Alembert progression to an infinite sequence of Black and Red in Roulette. Recall that this very good martingale is a geometric progression with a common difference of 1, increasing by one unit on a loss and decreasing by one unit on a win.

This system allows one to win half-stakes per "beat" between Black and Red, and works very well when the game is not very chaotic, when it returns to balance fairly often, and especially if sequences don't spiral out of control with long losing streaks. Because the big problem with this progression lies in the large deviations encountered on the chosen chance, and thus in stakes that become larger and larger without covering the deficit incurred. Even the winning "beats" and their gains become negligible compared to the deficit encountered. Spins of

−25 and +26, for example, on a deviation of 25, become insignificant compared to a deficit of

−1−2−3−4−5….−25. A gain of 1 unit on a deficit of 400 units becomes ridiculous. Moreover, there's the added problem of zero, which, when it appears during large bets, becomes devastating on a French roulette wheel with split-stake rules. Note that insurance on a straight-up zero bet at high stakes is mandatory, but this is still costly over the long term.

So, despite the advantages and disadvantages of this d'Alembert martingale, the mathematical position is formal:

We will never make money betting on random sequences with this type of scheme, nor indeed with any other system, however ingenious it may be.

🔍 Proof Explained

Their proof could be explained as follows:

If, over a very long equiprobable random sequence, one bets using a martingale (let's take d'Alembert here), and if we separately study all the bets by grouping them by identical amount, the bets of 1 unit will have a 50% success rate, the bets of 2 units too, the bets of 3 units too... the bets of 20 units too, etc.

So sorting the initial selection into categories by fluctuating the stakes changes nothing: sequences with 50% success will, at each staking level, reform new random sequences with exactly the same characteristics as the original sequence. And this holds true even if certain stakes become extremely rare — in the very long run, they will possess the same characteristics, thus in a Red and Black game, 50% losses!

So a game that is disadvantageous from the start will remain so, and using a martingale will not turn the disadvantage in our favor.

This applies to all types of martingales: arithmetic, geometric, loss-based progressions, gain-based progressions, staged progressions, paroli (letting winnings ride), etc.

🛡️ Casino Safeguards

Moreover, to prevent any unforeseen circumstances, the casino has put in place a few safeguards: a zero, or an even more penalizing double zero, and above all, betting limits. This cap prevents any clever player from using a "grande martingale" by systematically doubling their bets after a loss until achieving a win. With the cap set at 200 times the base bet on simple chances, the grande martingale (1, 2, 4, 8, 16...) breaks down after a series of 11 losses — and this is far from impossible. Such streaks are encountered almost every day at a Roulette table, on Red and Black, Low and High, or Even and Odd!

Let us add that this hugely popular grande martingale, even though it relies on the Shadock maxim ("by continually trying, one eventually succeeds"), is particularly perilous, because when the game spirals out of control, one is ready to risk 2,000 euros just to win a single starting stake of 1 euro! And some people sell this archaic method on the internet...

This safeguard also protects the casino from a lucky player letting their winnings ride indefinitely, starting from a trivial stake and multiplying their gains exponentially.

In summary: an unfavorable game remains unfavorable regardless of the system used, no matter how ingenious. The martingale only accentuates the disadvantage, even if it appears to delay the bankruptcy phenomenon, which will eventually become inevitable in the long run.

---

🎯 2. The Systemic Player's Point of View

We will now return to a few nuances that mathematicians, perfectly confident in their proofs, may have overlooked.

Firstly, Heads or Tails — seemingly so simple — is, when studied in depth, a cutting-edge random experiment in mathematics. It's the game that gave birth to probability theory.

Torn between balance, the Laws of Large Numbers (Weak and Strong), the Central Limit Theorem, Brownian motion, the Gauss curve, recurrent random walks, and the Arcsine Law.

I won't even mention "wild randomness" and the out-of-the-ordinary fluctuations that could occur, but as we can see, numerous proofs attempt to categorize this balanced 50/50 Heads or Tails game. And above all, let us not forget that even in mathematics, absolute certainty does not exist!

Probability is the antithesis of exact science.

👑 The "King's Choice"

First, a very important point is what's called the choice of the "King."

Nothing obliges us to play! In fact, in a game, if it is disadvantageous for us, we should logically abstain. But of course, we are always ready to take on challenges, to try to be better than our opponents, or it's simply human nature to be a "gambler"!

So, assuming we are obliged to play, if there are indeed many ways to "play badly," there surely must exist some that are better than others.

Knowing already that the longer we play, the more we expose ourselves to the Law of Large Numbers, which guarantees certain loss in an unfavorable game, common sense would tell us to stay at a Roulette table for as short a time as possible.

Secondly, and a major advantage: we are not obliged to play every spin! Even though, once again, mathematical laws tell us that it changes nothing to choose when to play, since each spin, even taken separately, is disadvantageous.

The example that tells us that when, for instance, we observe a series of 10 Blacks in a row, the eleventh spin still has exactly the same probabilities of landing on Red or Black — this is indeed true. But we also know that we shouldn't focus on looking at a single spin, but rather at a group of spins... And in that case, if we compare the probabilities of getting 10 Black results out of 10 spins versus 11 Black results out of 11 spins, the result is markedly different!

For 10 spins, we have a 1 in 1,024 chance, and for 11 spins, a 1 in 2,048 chance.

Let us not forget that a long streak will always have a greater tendency to stop than to continue.

🧮 Thirdly: The Question of "True" Randomness

Let us not forget that mathematicians have been performing sophisticated calculations for a few hundred years on series of Heads or Tails that are purely and perfectly random.

But where, in an earthly system, do we ever find something "perfectly random"? We don't even know how to manufacture it!

Everything we produce resembles pseudo-randomness. Random number generators, online poker games, programming software, etc.— but also horse racing results, sports betting, stock market values, and even casino roulette wheels.

This is precisely why a somewhat "clever" system could very well draw a slight advantage from the tiny distortions present in these random sequences.

---

<div align="center">

WHIDARTE Joël

This is an excerpt from my book "The Weak Points of Roulette," written in 2012.