Or in other words, is the probability of hitting Red at least once at a Roulette table "1" if you play it infinitely often?

Well I guess it should be but does that mean Roulette turns into a positive expectation game then, using the Martingale system?

In answer to your title ... Wouldn't time also have to be infinite for you?

Oh God !!!

---------------
Old story.. man says, "God is a million years like a second to you?".. God answers, "Yes".
Man asks, "Can you give me $1,000,000". God answers, "Sure, give me a second".

The probability of winning at least once is 1. However, that is not necessarily what others would mean by having the martingale work. It may be more accurate to say that when you have an infinite bankroll, playing a martingale simultaneously works and doesn't work. Infinity is well-understood in mathematics, but it is not necessarily intuitive for those who haven't studied it.

Yes, in a technical and useless sense, the expectation of your results when you first win is positive. It is also the case that your expectation at any prespecified time is not positive, and your average results at time t go arbitrarily far into debt exponentially rapidly. Your expected debt after 1000 spins (trying the martingale once, not 1000 times) is over 10^22 units, for an average loss of over 10^19 units per spin, even though you are very likely to have stopped with a win of 1 unit before then.

What is relevant in reality is that the limit of the performance of the martingale, as your bankroll increases, is a catastrophic loss. It is more expensive to play a martingale when your bankroll is larger.

Half-clued people insist on repeating that the martingale "works" with an infinite bankroll, just to brag about knowing something about infinity, I guess. People who don't understand the martingale or the technical details take the idea that it somehow works for an infinite bankroll to mean that it is a reasonable strategy if your bankroll is just close enough to infinity, i.e., large. This is wrong. There is a discontinuity at infinity, and the martingale works for no finite bankroll. The martingale is not even the best strategy at roulette if you want to win a fixed amount. The main use of a martingale betting strategy for an advantage gambler is to appear stupid while counting cards, as it gives you an excuse to raise your bet sizes when you want to.

It's also a reminder for mathematicians that the optional stopping theorem is nontrivial, and some technical assumptions on a stopping time for a martingale process are needed to rule out an unbounded martingale strategy.

Does infinite wealth imply that you own the casino?
If you own the casino then you've already won.
Or at least you can't lose. Or win because you've already won.

Infinity is so confusing.

Martingale would work if you had an infinite bankroll, since you cannot go broke --> once you will win. However, infinite + x = infinite, so you cannot make anything...

Your expected value is negative no matter how much money you have or how long you have.

If you are infinitely wealthy, you can't win, because infinity + x = infinity, so you can't make any progress i.e winning does not increase your bankroll. With no motive to play, you just waste your time by playing.

Or looking at it in another way, with an infinite amount money, you have an infinite amount of inflation, so whatever amount you win isn't worth anything.

Am I going to blow everyone's mind by mentioning that just because something has a probability 1 does NOT mean it is guaranteed to happen? It is "almost surely" going to happen though.

Maybe later in this thread I will argue that the martingale will work if one has an infinite bankroll, however, casinos are very clever and have TABLE LIMITS! TABLE LIMITS make the martingale even worse for most people with less than an infinite bankroll. Assuming you are playing roulette with a table minimum wager of $1.00 and a maximum wager of $500.00. You put $1.00 on Black and lose the bet. Your plan is to keep wagering on black, each time doubling the wager, until you win a bet, giving you a total profit of $1.00. In this scenario, after losing 8 more consecutive wagers, the player will be down $511.00. The $500.00 table limit precludes him from continuing his system.

I have some questions for more experienced Mathematicians:

1 - With both an American double zero wheel, and a European single zero wheel, what is the expected value of this system. Am I safe to assume that if there is more than a 1 in 511 chance of a player losing 9 consecutive wagers, this proves the system bad? With each of the wheels, what are the chances of losing 9 consecutive wagers?

2 - Assuming I have 10 Billion dollars in chips and want to win one dollar, How bad is using the martingale (with each of the American and European Roulette tables) without a table limit, beginning with a $1.00 bet. Assuming I have 100 Billion dollars in chips and want to win a dollar (again without a table limit). I each of these cases, what is the probability that I will lose enough consecutive wagers to bankrupt me?

This is another useless and potentially misleading point.

In the usual formalism of probability theory, eliminating countably many events of probability 0 does not change the model. Choosing uniformly from (0,1) is the same as choosing uniformly from [0,1] or from [0,1] + {2} or from (0,1)\{0.5}. (Eliminating (0.1,0.2) would be eliminating uncountably many points.) So, if you want, you can actually say that it is impossible to have infinitely many losses in a row, and it doesn't change the model or any result.

This is very different from saying that the uniform distribution has no atoms, so whatever its value is, the probability it takes that value is 0.

Perhaps there would be a difference between saying that a particular event has probability 0 and saying that it is impossible if you replace standard probability theory with something else.

This protects casinos from nonexistant people with infinite bankrolls who can play arbitrarily quickly. Very clever of them.

The mirrors help to protect the casinos from vampires. Defenses against Santa Claus and leprechauns are a trade secret.

The main point of the table limits is not to protect the casinos from martingalers. They mean that inexperienced/untrusted dealers do not have to handle $5 million bets right beside $5 bets, and the casino does not need to have ridiculous amounts of chips at each table. If you say you want to play a martingale and would like the table limit raised at a private table, the casino will happily raise the limits for you (as long as your minimum bet is worth their time). If you bet enough, they will comp you a room and meals until you bust out, while laughing at you behind your back. They have seen it before, and busting out does not take long.

A quick upper estimate for the median time to bust out is the time it would take to double your bankroll. You will double up less than 1/2 of the time. In fact, it will be much less than half if your bankroll is large in comparison with your minimum bet. Decreasing the size of your minimum bet decreases the probability that you will eventually double up.


It's that there is a more than 1/512 probability, not 1/511, since you only gain $1 on the complement.

(20/38)^9 ~ 1/323.
(19/37)^9 ~ 1/403.

You are selling lottery tickets for $1 with a prize of $512 which pay off too frequently.

This won't prove that the martingale is bad -- for people who hate money. If you like money, the martingale is bad, except as I mentioned, cover play designed to make you look like a harmless casual gambler, when you have really identified bets you would like to exploit without a martingale.

That's underspecified. What do you do when you are not bankrupt, but don't have enough to win back your losses? If you quit at that point, you can figure it out from the above formula. If you keep betting smaller amounts, you lose even more, while increasing your probability of winning slightly.

By the way, the logistical costs of moving $1 billion to a casino exceed $1.

There is no way to add negative numbers to get a positive number. If you want to win on average, and you aren't a vampire or Santa Claus or someone with an infinite bankroll, you need to find some wager with a positive expected value. Don't try to get creative with a really complicated sum of negative value wagers.

Technically, a probability space is some triple (Omega, F, P) where Omega is
a universal set, F is some sigma-algebra of Omega (a "nice" subset of the
power set of Omega) and P is a probability measure (a "nice" function) that
maps elements of F to [0,1]. Sometimes the unit interval (0,1] is used as
Omega. [ Also, by the axiom of choice, there are some "inadmissible" subsets
of Omega that can't belong to F, because probabiliists want the "usual nice
properties" for a probability space. ]

As pzhon mentioned, any countable subset can be removed/added; it also
turns out you can remove an uncountable set such as the Cantor ternary
set which has Lebesgue measure zero on the real line and wouldn't "change
the model or any result". Removing any nonempty interval will change the
result since intervals have positive Lebesgue measure in (0,1]. Lebesgue
measure is the generalization of length on the real line and it turns out to
generalize integration (because Riemann integration isn't "powerful" enough).

"Almost surely" is used probabilistically in the sense that the complement of
the event in the Omega space has probability measure of zero ( as seen
above, it can even be uncountable but not contain an interval if Omega is
(0,1] ). "Impossible" could be taken as the empty set, and a "null set" is
pretty standard for a set with measure zero.

A: "I know that roulette is unbeatable, but lets just say, for a moment, that you have a magic leprechaun with you that makes sure that the ball never lands on red. Then, theoretically, wouldn't you have a huge edge?"

B: "I guess so - but what's your point?"

A: "Of course it's impossible to take a leprechaun to the table with you, but I have a mint condition Yoda hand puppet, which is as close to a leprechaun as you'll get in the real world. If I took that with me to the table, I'd have enough of an edge to make it profitable."

B: "You're going to look ridiculous with a Yoda hand puppet in the casino."

A: "They laughed at the Wright Brothers too."

Hi. I read this thread last night before trying the ebook that I got. Although I agree with a lot of posts here, there's an enhancement to the martingale that I tried and it worked for me. Last night, after reading the stuff here are seeing that it's a lot different from the betting strategy I read, I tried to invest $200. Then, after observing how it works, I started to bet $1 at a time, but I did it after the probabilities reached my standard. My total reached $549. But I found out that when you get a bit tired, you tend to think that winning is so easy that you have to bet a lot more. That is where the problem lies. I can safely say that for me, the system worked, until I wanted more. So you too can try it but you have to guard yourself. By using the system that I found, I was able to get an average of $20 an hour. If you need information, just let me know. I could send you the info and the link to the site so you can try it for free. Then when you see that my stats are logical, then you may try the real thing.

Statsboy, I'm sure that a system which works the same way no matter what stakes you play at, but is a positiv expectation in $1 bets but a negative when you go "too high" is a great system.

And a serious reply to the thread:
If you play for an infinite ammount of time, wouldn't that make you have an almost infinite streak of losses and an almost infinite streak of wins, too? So it all comes down to when you quit gambling? And if you then play for a fixed time, the casino has the edge? But then again, if you Can play for an infinite ammount of time, you can also expect to sometime reach your "goal of play" (in example, $500 plus), then turning the game into a positive expectation? But, since again you have set a time of play, but that is not fixed, the casino can still expect to make more $$ of you then you of them?
I'm confusing myself but you get the picture... and the technicalities that you've put up is way to advanced for me!

I CAN VOUCH FOR THIS GUY. HE IS TOTALLY LEGIT.

...

The way I like to think about it is this.

Suppose you found a casino that would extend infinite credit and had no table limits. To make it even nicer, it's willing to play infinitely fast if necessary. It will let you bet $1 on red, doubling the bet each spin until you win, taking 1 second for the first spin and cutting the time in half for each subsequent spin. You will win $1. The casino could save trouble by handing it to you and not investing in an infinitely fast roulette wheel and risking getting caught for violating credit rules.

However, you never pay anything in this game. You're not gambling, your "bets" and "losses" are only theoretical. It's the same game if the casino says, "we'll spin the wheel until red comes up and then pay you $1." No one would be confused about that game (either that you could find a casino to play it, or that it has a $1 expected value to the player).

So yes, under perfect conditions (infinite credit, no bet limits, infinitely fast wheel), the martingale system has a positive expected value. But "expected value" is not meaningful because you never pay your losses.