Math proof that roulette cannot be beaten

[The General]

link:://:.rouletteforum.cc/index.php?topic=16981.msg155759;topicseen#msg155759

[Foolwise]

We get a contradiction! All sums over the rows are negative, and all sums over columns are positive!
This can not be the case, hence my claim must be false!

There are three flaws in this proof.

1. You are assuming all sums over the column is positive with a finite 1 to M. The key word is finite. It cannot be. If it is then the law of large numbers doesnt hold good in this case.

2. You are summing over a constant bet. The betting decisions need not in all practicality point to a constant bet.

3. You are assuming that the law of large numbers will be applicable and it is making the expectation negative. What if there is a way to plot the events from roulette in a Cauchy distribution? Just like the levy flight experiment, what if the outcomes are defined in step lengths instead of outcomes themselves.

These are three flaws that I can see in your proof.

[rrbb]

Hi Foolwise,

Thanks for taking the time to read the proof

The whole idea is to find "flaws" in the proof! There is one "flaw" that almost everyone does no spot because people take it to be self-evident.

Here are my responses. Feel free to disagree!

There are three flaws in this proof.

1. You are assuming all sums over the column is positive with a finite 1 to M. The key word is finite. It cannot be. If it is then the law of large numbers doesnt hold good in this case.

A column indeed represents one session. I claim I can proof that there is a finite (albeit large) number of potential bets per strategy. For example, take a martingale with "follow the last". The number of times I can double up is either restrained by the casino, or by the physical amount of money available in this planet. If the total table bet is for example 1024 units, I can double up only 9 times. So, the total number of different bets I can place is 18: red 1 unit, red 2 units,..., red 1024 units; black 1 unit, black 2 units etc.

So for this strategy M is clearly finite.

The general idea is this: because the number of numbers is limited (37), the number of combinations I can make are limited. It is impossible to to have a infinite set of combinations with a finite set of numbers...

However, this was the first assumption I tried to circumvent long time ago. I could not as it would lead to an impossible strategy. Maybe you can!

2. You are summing over a constant bet. The betting decisions need not in all practicality point to a constant bet.

You are right about the betting decision. I used the wrong incorrect word, Actually I mean a "bet". (or, the result of the betting decision).

So please replace "betting decision" by "bet".

3. You are assuming that the law of large numbers will be applicable and it is making the expectation negative. What if there is a way to plot the events from roulette in a Cauchy distribution? Just like the levy flight experiment, what if the outcomes are defined in step lengths instead of outcomes themselves.

I never heard of this experiment. I looked it up.it is interesting stuff though.

I do not assume this, 99.999% of the people assume this (Check out the remarks of some illustrious forum members)

Even great minds like Thorpe did not deem in needed to come with a full fledged proof. As it is "self-evident that the law of large numbers, the independence of spins and the negative house-edge make roulette impossible to conquer"

When you look about the proof, there is one assumption everyone makes and takes for granted. This assumption is needed to reach the final conlusion... Play with either the formula's if you are mathematically inclined or, if you are more visually inclined, create the table  I hinted at in the post... and slowly, very consciously, simulate a strategy. Every time ask yourself: "what am I doing, why am I doing it, am I sure I can do this". In this way you might just see what the assumption is. Yes it is related to independence, and therefor to the applicability of the law of large numbers.

A strategy like Martingale with FTL is the worst choice you can make for this "simulation by hand".

grts rrbb

[Foolwise]

>>>>>>>>>>>>>>When you look about the proof, there is one assumption everyone makes and takes for granted. This assumption is needed to reach the final conlusion... Play with either the formula's if you are mathematically inclined or, if you are more visually inclined, create the table  I hinted at in the post... and slowly, very consciously, simulate a strategy. Every time ask yourself: "what am I doing, why am I doing it, am I sure I can do this". In this way you might just see what the assumption is. Yes it is related to independence, and therefor to the applicability of the law of large numbers.

Now that i have looked at the conversation, I understand that all the proof here hinges around the independent nature of spins. To clarify this one should understand what is a Partially ordered set or Poset. It is a concept in which you are able to order, sequence and arrange elements of a set of spins together with a binary relation in such a way that for certain pairs of elements you can establish a dependency or precedence of one over the other. I think this is what Priyanka has shown using the cycles, using a specific arrangement and sequencing. Now when this partially ordered set comes into picture, there are inequalities in statistics and probability as against a typical i.i.d random variables. One of those inequalities is application of law of large numbers. Is this what you are driving towards rrbb?

For those who understand mathematic equations, if you consider a set of exchangeable variables (true in case of roulette spins), X1....Xn(A1.....An) with means U, variance ¬2, and correlations P2(P1). If P2>P1>=0, then Xi tend to hang together more.

I am clear on the dependent nature, but what am not clear is on the applicability of this in roulette to beat it, apart from the fact that there is no proof exists which says roulette cannot be beaten. I am a firm believer in math cannot help beat roulette and math is useless here and hence the applicability stands out for me.

[rrbb]

>>>>>>>>>>>>>>When you look about the proof, there is one assumption everyone makes and takes for granted. This assumption is needed to reach the final conlusion... Play with either the formula's if you are mathematically inclined or, if you are more visually inclined, create the table  I hinted at in the post... and slowly, very consciously, simulate a strategy. Every time ask yourself: "what am I doing, why am I doing it, am I sure I can do this". In this way you might just see what the assumption is. Yes it is related to independence, and therefor to the applicability of the law of large numbers.

Now that i have looked at the conversation, I understand that all the proof here hinges around the independent nature of spins. To clarify this one should understand what is a Partially ordered set or Poset. It is a concept in which you are able to order, sequence and arrange elements of a set of spins together with a binary relation in such a way that for certain pairs of elements you can establish a dependency or precedence of one over the other. I think this is what Priyanka has shown using the cycles, using a specific arrangement and sequencing. Now when this partially ordered set comes into picture, there are inequalities in statistics and probability as against a typical i.i.d random variables. One of those inequalities is application of law of large numbers. Is this what you are driving towards rrbb?

For those who understand mathematic equations, if you consider a set of exchangeable variables (true in case of roulette spins), X1....Xn(A1.....An) with means U, variance ¬2, and correlations P2(P1). If P2>P1>=0, then Xi tend to hang together more.

I am clear on the dependent nature, but what am not clear is on the applicability of this in roulette to beat it, apart from the fact that there is no proof exists which says roulette cannot be beaten. I am a firm believer in math cannot help beat roulette and math is useless here and hence the applicability stands out for me.

.  The dependency is not related to individual spins however. Those are per definition independent (unless of course there is something amiss with the random generator e.g. a biased wheel). 

But do not forget Foolwise: the first step was to establish that the "proof" that roulette can not be beaten hinges an a very specific assumption, which is certainly true for most of the "strategies". However, the proof does not show that other possibilities does not exist! If you accept this, math has helped thusfar: no more assuming! Whether it is possible to find a viable method is the next step...

The next step is to establish the problem definition: what do we want to circumvent (or negate as I used to call it)? And if we circumvent it, what would it imply? What "features" does it need to poses?

One forum member asked how many spins are needed to proof that a strategy works: I did not want to answer in that thread because it was not a question for me. I would have answered zero (0).

If it can be stated what we are looking for, any system, strategy or whatever that does not poses this "thing" can not be a winning system: the law of large numbers will catch up!

As concerned to whether or not mathematics can help to beat roulette: I once had an Intellectual Property workshop in an town called Rijswijk . The first thing that we were taught: most innovations are not generated by "deep experts". Afterwards these professionals are the first to state that it is "logical", but somehow it is so difficult to get rid of the "shackles of knowledge" that it (usually) are not these professionals that come up with the idea in the first place...

grts rrbb

[nottophammer]

the law of large numbers will catch up!

So in KTF where we are starting with a large group of #'s is this not a good starting point to bet that the large group must go down.

[rrbb]

So in KTF where we are starting with a large group of #'s is this not a good starting point to bet that the large group must go down.

Hi Nottophammer,

I have to admit that I did not pay too much attention to KTF. So I can not answer your question.

sorry, grts rrbb

[falkor2k15]

Hi all,

Interesting discussion!

I think it has been proven again and again that spins are independent. Also roulette is a negative expectation game.

i think that the question should be made more specific: "proof that there is no strategy that can overcome the house edge".


I will show my "proof" in words. For anyone versed in the mathematical language: just translate it.



Now lets assume i claim i have a winning strategy. This would mean that i would see a steady increase in my bankroll.

because of this i could define "sessions". A session ends when i'm in the plus. A new session starts after an ended session.

So: a strategy consists of sessions, and sessions consist of "betting decisions"

Now, lets assume that we can proof that any strategy has a finite amount of possible betting decisions. Either by rules, physical boundaries, or inherent features of that strategy.

For example: if the strategy were a simple maringale + FTL, the betting decisions are very limited.

Let's call these possible betting decisions the "template". It just a name i chose.

Let's number these betting decisions from 1 till M. M betting the total amount of possible bets. And lets make a list of this template: we fixed the order. This comes in handy later.

Each session will consist of playing at least one of bets from the template one or multiple times.


Now lets play my "winning" strategy ad infinitum that is, till the end of times,

Also, lets keep track, per session, what the result is per betting decision in the template.

If we would put the template in a row on the lefthand site of a piece of paper, and the make columns to the right of it, we could eadily keep track of what a session does.

Per session we can add the result in the appropriate row.

Now we do this for an infinite amount of sessions.

Because each sessions ends in the plus, we could sum all the results in one session (we sum over a column) and write this down in a row below the sessions (say row M+2).

On the other hand, we could also sum the results per bet in the template!

Well, i used a fixed template, so when we som over a row, i sum over a "constant" bet.

Now, due to the law of large numbers, we know that these sums must be negative.

We get a contradiction! All sums over the rows are negative, and all sums over columns are positive!

This can not be the case, hence my claim must be false!




Now, i claim, there is one HIDDEN assumption that needs to be made to reach this conclusion.
If you can negate this assumption, the "proof" is not valid any more!

Good luck!

This is the big secret I believe that Priyanka has deliberately kept hidden from us - the key to gaining edge on the very next spin.

[praline]

What is the hidden assumption?

[Scarface]

I think roulette can be proven to win mathematically.  No one would disagree, given an infinite number of spins, all things are possible.  If betting 1 unit a spin on red, and parlay our bets, there will eventually be a series that will put us in the profit.  10 wins in a row off a 1 unit bet is 1024 units...11 wins is 2048...12 is 4096.  At some point, given enough time, a parlay bet will put us in the profit.

There is actually a math paradox that explains this.  Parrondos, I think is the name.  The paradox says that a player should be willing to play any amount of money, no matter how large, to play this game because mathematically the reward is infinite.

[praline]

If the reward is infinite, you are right.
But unfortunatly there are always a table limits, you could win even with marty if it was infinite.

Hope i understand your post to make this comment.

[Scarface]

If the reward is infinite, you are right.
But unfortunatly there are always a table limits, you could win even with marty if it was infinite.

Hope i understand your post to make this comment.

Yes, that's the problem...table limits.   

[praline]

If i remember, the Parondos paradox is about a dependency beetwen games that can potentialy lead to a positive edge for a player.

[Scarface]

I read this somewhere and found it interesting.  Suppose you are playing a single number for 36 spins.  A stranger walks up and bets you $20 that you will not be in the profit after 36 spins.  The stranger is confident because he knows that roulette is a negative expectation game.  So, you take his wager for 36 spins.  In fact, you play many sessions of 36 spins.  Guess what?  You are mathematically guaranteed to win money in the long run.  Seems like a paradox, BUT this has been proven by mathematicians.

[Scarface]

If i remember, the Parondos paradox is about a dependency beetwen games that can potentialy lead to a positive edge for a player.

Can be interpreted in different ways.  Basically, if you take an infinite amount of spins then all possibilities WILL happen at some point.  Ex..20 reds in a row, or 5 same numbers in a row.  At some point, there will be a parlay that will put you in the profit.  But like you said, it has to be confined to table limits....also, be practical to play in a playable numerous of spins.  Really, only works in theory