Asymmetrical Information

0 and 00 pay the house. The strategy is a sure-fire loser.

Well, there are two glaring problems. The first is that the strategy assumes a 50% chance of winning those bets, but there ARE two greens on the wheel, and sometimes both will sides of the "50%" bet will lose.

Secondly (and hugely), this assumes that you won't have much of a losing streak. You certainly CAN, though, and can quickly get to the point where "doubling your bet" becomes financially impossible. You can pretty easily get to the point where you have to put 10's of thousands of dollars down, and A) You might not have the stake for that and B) the house will limit your bet at some amount, and won't allow you to double.

Believe me, this strategy works great... if your goal is to become poor. :-)

There is no hole in that theory. The way casinos respond is to put an upper limit on bets.

There is no need for the odds to be 50% for the strategy to work.

By the way, that strategy is called the Martingale Strategy, and is probably the most famous roulette strategy. The other problem it runs into is table limits: it doesn't take many losses in a row before the amount you would need to bet to break even is more than the house will allow you to wager. Another famous strategy is the reverse-Martingale: double on a win, go back to a single unit on any loss. You still can't beat the house odds, but it's easy to limit your downside losses.

Winning "systems" of gambling are much like systems of winning in the stock market - anyone who finds such a "system" will eventually cause the whole (larger) system to break down, and the stakeholders will change the rules so that the winning system is impossible.

Look at card counting in black jack. Card counting is not cheating, it is just memory and playing when the odds are the most in your favor. This completely destroys the incentive of a casino to have black jack, since it puts the odds in the players favor and gaurantees a long term loss. Yet most people cannot count cards, so their are still plenty of suckers to get money from. So they changed the rules. If you are caught card counting (and they watch like a hawk), they toss you out and ban you.

Simply put - the casino industry is damned smart. If there are "systems" to win their games, those games change or go away, because they make the rules and need to make money.

If you want sure fire money in Vegas, the best thing to do is play poker. That is no gambling.

-Donut

Actually, when there are 2 greens on the wheel, bets that are made on "odd/even" "red/black" or "1-18/19-36" do not lose out right. You only lose half your bet. On Eurpoean style roulette wheels there is only 1 green and you will lose your entire bet if green comes up.

That being said: The above posters are right, the casino limits the maximum amount of the bet and it is quite easy to reach that doubling up after each loss.

Let's assume House Limits don't exist.

In that case, the strategy works fine --- as long as you have an infinite amount of money. :)

I'm too lazy to do the math, but what this works out to is no change in the expected outcome. The error is in assuming you have an infinite bankroll.

Assuming you don't have infinite money to gamble with,[*] then here's how it works.

Say you have $8. If you bet it all on one coin flip with 2x1 odds you have a 50% chance of ending up with $16, and a 50% chance of ending up with $0. Expected outcome=$8.

On the other hand, let's say you bet $1 on the first bet, and use the doubling strategy after that. (You also quit after the first win, or start the game over). Possible outcomes:

1/2 - Win ($1): Result=$9
1/4 - Lose ($1), Win ($2): Result=$9
1/8 - Lose ($1), Lose ($2), Win ($4): Result=$9
1/8 - Lose ($1), Lose ($2), Lose ($4): Result=$1

In the last outcome, you have to quit playing, because you no longer have enough money to double. Your expected outcome therefore equals 7/8 * 9 + 1/8 * 1.

x = 7/8 * 9 + 1/8 * 1
8x = 7 * 9 + 1 * 1 = 64
x = 64/8 = 8

[*] Note that even if you do have an infinite amount of money, you still can't win. There exists a possibility that you will lose every time you bet, even an infinite number of times.

Actually, even as Joshua points out the odds are against the player, theoretically, the technique will work and it will eventually every time.

Here are the two problems that make the scheme unworkable in the real world:

1) The limit of one's bank roll. Say you start out with a $10 dollar bet but you are having horrible luck and you don't win until, say, the 11th bet(something that has occurred in my Las Vegas tourist life at cards or dice several times in the past decade). That 10th bet will cost you $5,120, after having wagered and lost $5,110. Most folks looking to win 10 bucks don't have that kind of pocket change for gambling.

2) The other problem is a cousin of the first--casino house limits as you can see by the example above, if the house has a maximum bet limit of $5,000 or 10 or even $20k the most bets you will be able to make under this plan will be 9-12. You don't have to spend too much time on a reservation or in the State of Nevada to know that bad streaks of that magnatude are not nearly rare enough.

In conclusion: too much risk for way too little reward.

Sorry, forgot to sum up.

Effectively, what you've done is exchange a game where you have a 50% chance of winning or losing the same amount for a game where you have a large chance of winning a small amount and a small chance of losing a large amount. Your expected outcome hasn't changed.

(Again, in the case of an infinite bankroll and infinite amount of time to gamble, you have an infinitesimal chance of losing an infinite amount, so it's a wash.)

"... the casino industry is damned smart. If there are 'systems' to win their games, those games change or go away, because they make the rules and need to make money. If you want sure fire money in Vegas, the best thing to do is play poker."

Richard Feynman, the Nobel Prize physicist, spent a lot of time in Vegas and became friends with the famous gambler Nick the Greek (not to be confused with TV football's Jimmy).

In his book Feynman tells that when they met he, Feynman, said: I know a fair bit about probabilities but still don't understand how you can make a living like you do here taking money from the house.

The Greek replied: I don't make take money from the house, I take it from the people who come here thinking they can take money from the house.

BTW, in the same book Feynman tells how he learned to get women in Vegas pick him up by getting them to buy him drinks. Great book. I wish I'd read it when I was young enough to use it.

Following up on J Mann's statement about how the expected value remains the same, but the probabilities shift around -- what happens is the variance, or standard deviation (sqrt of variance), keeps getting larger and larger as the house limit increases. In the infinite limit (no house limit and limitless bankroll), the variance is infinity. Expected value =is= zero, but one has an infinite spread.

By the way, this martingale betting can be done one better: bet the amount lost so far and add 1. This is an even quicker way to run out of money, but a "sure" way to win 1 buck. People who do these doubling schemes must warm the heart of the casinos -- every half-assed system means that a bunch of suckers pay the casino quickly rather than dribs and drabs.

I wrote a little simulation in Java just to test out what most have said, and indeed, it's borne out. The house always has the better of this deal. Any individual gambler has a good shot of making a little cash (if your stake is a small enough percentage of your total available gambling money), but at the risk of being the sucker who loses everything (the only two options in this strategy being winning your original stake or losing absolutely everything). A reverse lottery of sorts: it's like buying a $150 dollar lottery ticket with 12 out of (just under) 13 odds of winning, with a winning ticket worth $160 dollars. Doesn't sound so great now, eh? You can only further minimize your loss risk by increasing the ticket price or decreasing the payout.

It's not blatently obvious from the strategy rules, but keep in mind that by choosing to enter the game with this strategy, you are in fact risking all the money you've got to gamble with.

To echo the previous posters, the Martingale Betting Strategy works perfectly fine, assuming an infinite bankroll and no betting limits.

Assuming no betting limits and a finite bankroll, to win n dollars has a chance of success of 1 - 2^(-k), where the initial bankroll is n * (2^k - 1), in the fair game (50% chance of winning) case. It's pretty easy to show that. In a non-fair case, with probability of winning p, the chance of success is 1 - (1-p)^k.

Probabilists say that one will win eventually, with an infinite bankroll and no limits, almost surely in the probability space. Analysts would say that it happens almost everywhere in the measure space. The chance that one fails an infinite number of times is an event of measure zero.

The expected value as time goes to infinity is NOT the same as the expected value originally or at any point. The reason that the martingale convergence theorem doesn't hold for either the fair game or the biased against the player game, is that the expected value of the absolute value of the martingale is not bounded. (Another sufficient condition that doesn't hold is that the martingale can be uniformly integrable.)

"In the infinite limit (no house limit and limitless bankroll), the variance is infinity. Expected value =is= zero, but one has an infinite spread."

No, it's not. In the infinite limit, the expected value is 1, or the amount you're trying to win. The Martingale Convergance Theorem doesn't hold, as this martingale's absolute value is not bounded, and it is not uniformly integrable. It's a moot point, as one won't have an infinite bankroll, and with anything short of an infinite bankroll, and at any finite number of plays, expected value is zero.

For refence, see Billingsley, Patrick. Probability and Measure. New York: University of Chicago Press, 1995, p. 468-469. Or, for a really good basic explanation that understandable without measure theory, Lawler, Gregory F. Introduction to Stochastic Processes. New York: Chapman & Hall, 1995, pp. 90-102 for a discussion of the martingale betting strategy and the martingale convergence theorem.

Dr. Lawler is my advisor in my mathematics Ph.D. program in probability at Cornell that I'm in right now, so I think I qualify as a "budding mathematician."

I dunno... my dad used to do something similar at the racetrack. He always bet on the third horse, and he doubled his bet every time he lost. Mom says he consistently brought home more than he left with. It drove her nuts worrying though.

Odds are not 50/50. They are 36/37. Odds of all games at least slightly favor the casino. The longer you play, the closer your results should look like the strict mathematical probality. But hey, you could win the first time.

A friend tells a story, perhaps an urban legend, about his cousin. He didn't have a enough money for a down payment on a house. Took 10k to Vegas and put it all down on black. He won and left the casino immediately. The casino gave him a free limo ride to the airport. I don't recall if he bought a house or not.

Go Bears!

Jim English
Chicago, IL

John wrote:

"Or, for a really good basic explanation that understandable without measure theory, Lawler, Gregory F. Introduction to Stochastic Processes. New York: Chapman & Hall, 1995, pp. 90-102 for a discussion of the martingale betting strategy and the martingale convergence theorem."

Martin Gardner also discussed the martingale system in his Sci. Am. column, which is collected in his book: "The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems"

That might be more accessible to a layperson, umm, like me.

Several years ago a friend of mine suggested that we should have the President use this strategy, using the National Debt as the stakes.

It works, mathematically, if and only if you have:

1) Infinite money to bet.
2) No betting limits, either imposed by the house or by the financial reserves of the house -- the house must have infinite money to pay you off with.

Otherwise it's just a way to lose a lot of money with mathematical certainty. Since #1 and #2 never apply, it's utterly useless as a real-world strategy.

Sorry, I meant 18/37.

Go Bears!

Jim English
Chicago, IL

There's also a discussion of Martingale systems in Thomas Bass's book "The Eudaemonic Pie." The "probability of ruin" graph in the book is most alarming and instructive. These systems fail for exactly the reasons given above: the inability to play with an infinite bankroll, with an infinitely high betting limit. As a bonus, there's the psychological pressure of putting down thousands of dollars in an attempt to win back your original $2 chip.

The book chronicles an early attempt to use computational aids in a casino environment - specifically, using a shoe-mounted computer in a search for non-randomness in roulette. They took a lot of readings on different tables, croupiers, etc., looking for wobbles and dead spots. The plan was that with a very bored (and machine-like) croupier, the information about where on the wheel the ball was released might be enough to raise the odds of it landing in a specific quadrant of the wheel.

It worked, but the equipment wasn't too reliable (and the whole system was prone to operator error, as you'd imagine.) Things are presumably run a bit more tightly these days. I need hardly add that the casinos will treat you in a manner you will regret if they catch you wired up with this kind of equipment.

The main flaw is this: if you have the infinite amount of money needed to guarantee success you probably don't need to gamble.

I'm probably doing something wrong, but here's my take:

There are 38 slots on the wheel: 18 ways to win and 20 ways to lose. On any spin, your odds of winning are 47.4%. You'll only end up ahead by the amount of your first bet: if you bet $10 at first, you'd end up $10 ahead, whether you went 1 spin or 15.

As the number of spins approaches infinity, the expected return approaches 100% of your bet, so a million monkeys would be even-steven.

The problem is that the casino won't let your bets approach infinity, and they won't let you start with $1 bets. The squeezes on your minimum and maximum bets create the house's margin. For example, if there's a minimum of $10 and a maximum of $5000, the expected return on your $10 is $9.97. The odds of ending ahead $10 are greater than 99%, but if you're the one loser who hits the limit you can really get hurt.

I got bored and set up a simulation of this in Excel, if anyone wants a copy.

J Mann and Johnathan are exactly correct. Given 50/50 odds (which I know do not apply in roulette), you have an equal chance of doubling your initial bankroll with this method as you do of losing your initial bankroll.

If your goal is to win one unit, there is a great likelyhood that you will succeed, but with minimal reward. A 255/256 chance of winning 1 with a 1/256 chance of losing 255 is no winning gambling system.

Katherine - your Dad came home consistently with more than he left with, but your mother was right to be sick with worry as your father may have come home one day having to remortgage the house to continue his system.

I used to work in a liquor store with lottery terminals. One guy thought he came up with a system for playing the daily numbers game using the law of averages (Silly, as each drawing is independent of prior drawings, but anyway). This fellow played thousands of tickets, increasing his stake after he was in the hole. He eventually had to tell his wife so she could sign the mortgage papers. He bailed out with a win on his last chance, after about 7 or 8 days, clearing a few thousand dollars.

So, umm, I guess the moral of the story is systems work and we should all try the Martingale system!

I understand that you can get almost 2 and 1/2 percent on a 36 month CD these days.

Jim English
Go Bulls!

When in Vegas, I didn't play to win, I played to lose as slowly as possible. That way, I got to nurse free drinks for the longest time.

Over time, the house takes 1/37 of every roulette dollar bet. Yes, one must deal with long losing streaks; that said, the problem with this 'Martingale' strategy is that it forces one to bet more than necessary, thus losing faster and getting fewer 'free' drinks.

If I may be so bold, why do you ask?

The only way to win in Las Vegas consistently is to play games against the rest of the public, not the house. As mentioned above, poker allows this, but I think the sports books are a better avenue, in that they allow a much larger swath of the public to play against at any given time, which increases the likelihood of competing against incompetent or poorly informed players, which is the key. Yes, you may get lucky and sit down at a poker table with a patsy with a fat bankroll, but there isn't anything like the Super Bowl to bring out players with just enough knowledge to be dangerous to their cash. Add in the emotionally attached fan's penchant to suspend analysis, and there are real opportunities to spot situations where the overwhelming public opinion is at odds with reality, and the more widely publicized the event, the better the chance the phenomena will occur. I haven't checked the money line on tonight's baseball game, although I suspect that this game is sufficiently evenly matched, both in reality and public sentiment, to make it not worth risking money on. The sixth game between the Marlins and the Cubs probably would have been attractive, though. I try to stick to football when I pursue wagering, however, because I believe there are more football fans who overestimate their knowledge than baseball fans.

For those talking about the player with infinite money in a no-limit game: Does he also have unlimited time?

Nothing wrong with it. Would you please lend me 10^264th dollars?

To inject some actual calculations here, but unrealistically assuming 50-50 odds so I can do the calculations in my head:

You bet $1. If that loses, bet $2, then $4, etc. Suppose you brought along $1023. You will be able to handle anything up to 9 losses in a row, but 10 will wipe you out. (2^10 = 1024.) This is the same as betting $1023 against $1 at 1023:1 odds. In the long run, it's a wash - but you could get wiped out on your first night out.

In the real world where the odds are tilted slightly against you, the odds after 10 turns are considerably worse than in a single turn.

So, you're a millionaire and want to keep going? OK, if you can find a cooperative house[1]. But if you keep losing, you'll lose somewhat more than a million after 10 more turns. And what will you do for the rest of the night?

[1] One thing the house won't be worried about is losing too much money - they can always cover your bet with just $1 more than they already took from you. If they won't waive the limits, it's probably because they are afraid of the effect witnessing house-mortgage size losses will have on their other customers. And seeing you ultimately walk out $1 richer, as you probably will, is hardly going to bring the crowds back.

If you've got a million dollars and are gambling for fun, you want to make it last, and the Martingale is almost the opposite of the strategy you ought to be playing. If you've got $1 million and want to make it $2 million, your best (but still not so good) chance is making the biggest bets possible, so the house odds are against you are getting compounded as few times as possible.

If you're trying to make your lunch money every night with casino gambling, either you're a casino or you're a fool, plain and simple. If you must try this, get very, very good at a game where you have a real influence on the odds, such as poker (against live players, not computers, and not under the roof of a casino). Horse racing is another venue where I've heard a smart person can come out ahead on the average, if he really works at collecting information and figuring the odds - but in the book I read long ago about this, the gambler eventually realized that a regular job would be less work and pay better.

If you want to gamble in a game with a better than even chance to win, try playing the stock market. As an added bonus, you will be financing what you believe to be the best ideas.

I don't know anything about roulette, but the obvious economic problem is that most people are risk averse. Most people are not, for example, indifferent between 1) no action, and 2) placing a bet wherein they can gain or lose $10,000 with probability .5. Even if the certainty monetary equivalent works out in their favor, the expected value as adjusted for some degree of risk aversion will not.

I would like to hear comments on a slightly diferent game:
you toss a coin untill it turns head up. If head turns up in the first flip, you receive $1, $2 if head turns up in the second flip, $4 in the third and so on.
How much do you pay to the casino for tossing the coin in this game ?

There is a good discussion of the martingale strategy at Mathworld:
http://mathworld.wolfram.com/Martingale.html
http://mathworld.wolfram.com/GamblersRuin.html

And, a related topic, since since another poster asked about it:
http://mathworld.wolfram.com/SaintPetersburgParadox.html

ITS TIME WE STARTED TO LISTEN TO ARAFAT

Arafat Invokes 1974 Phased Plan Calling for Israel's Destruction
Compares Oslo to Temporary Truce
PA Chairman: "I Envy the Martyrs and Hope to Be One"

Arafat's Secret Agenda Is to Wear Israelis Out

Arafat estimates that the final-stage agreements between the Palestinians and Israel will ultimately bring about Israel's collapse. He reportedly told the diplomats that a migration of Arabs to "the West Bank and Jerusalem" and the psychological warfare the Palestinians would wage against the Israelis would cause a massive emigration of Jews to the United States.

Arafat: Jihad to
Liberate Jerusalem

The Jihad [Islamic holy war] will continue, and Jerusalem is not [only] for the Palestinian people, it is for all the Muslim nation.
You are responsible for Palestine and for Jerusalem before me [applause], the land which had been blessed for the whole world.

Now after this agreement you have to understand our main battle.

Our main battle is Jerusalem. Jerusalem. The first shrine of the Moslems.

This has to be understood for everybody and for this I was insisting before signing to have a letter from them, the Israelis, that Jerusalem is one of the items which has to be under discussion and not the state, the permanent State of Israel! No! It is the permanent State of Palestine [applause]. Yes, it is the permanent State of Palestine.

"When we stopped the Intifada we did not stop the Jihad [Islamic holy war] to establish Palestine with Jerusalem as our capital.... We know only one word: Jihad, Jihad, Jihad.... We are in a conflict with the Zionist movement, the Balfour Declaration, and all imperialist activity...."

IF WE LISTEN, WE WILL UNDERSTAND

There are better ways to guarantee either a very small win or a very large loss.

jebus, how disappointing.

You're some sort of faux economist with an mba, and a) you've never heard of this, and b) you have to ask your readers to explain it to you?

jebus, the idiot right will have to do better than you. you gonna strip to extend your fifteen minutes?

Just a footnote to John Thacker's excellent explanation; convergence "almost surely" always trips me up so I'll assume it does others. Important to remember that "almost surely" in context doesn't "maybe, kinda, almost surely". It's actually about as strong a form of convergence as you can get; it means that you will win your dollar with probability 1 in finite time.

The way to win at gambling is to be the house. Side bets on craps are good as the odds are easy to reember

The way to win at gambling is to be the house. Side bets on craps are good as the odds are easy to remember

The way to win at gambling is to be the house. Side bets on craps are good as the odds are easy to remember

rpl,

thanks for the source on St Peterburg Paradox.

For excellent reading on gambling, and Vegas in particular, try and get hold of a copy of "Inside Las Vegas", by Mario Puzo.

In it, he recounts most of the tricks and techniques used by the Casinos to attract, hold and gently fleece the punters. By my memory, much of the stuff in the movie "Casino" was based on Puzo's book.

Also, on the roulette question: You can increase the chances of winning by two methods:
1. don't restrict yourself to red/black; bet all of the "50/50" side bets;
2. work with a mate on the table. When you hit the table limit, you friend steps up and bets next to you to keep the system running.

Some of the comments in this thread are scary in their lack of comprehension. :-)

Regardless of the odds, sometimes you win, sometimes you lose. If you graph your bankroll over time, you'll get a really random looking graph that wanders up and down. If you play long enough, you'll come out ahead. Increasing the size of your bet increases the variability, but doesn't change the expected outcome.

So far, so simple - and it is, in fact, simple. The only other factor to consider is that casinos do not accept credit, so you're out of the game if your bankroll ever hits zero (or if you are otherwise unable to bet the amount you'd like to bet, for example due to "table stakes" limits). Much like coming out ahead, this too WILL happen if you play long enough - and is, sadly, quite likely to happen before you come out ahead.

Example: Assume that you follon the "size of bet after nth loss is equal to 2^n" forumula. Assume you lose 40 times in a row (not impossible). If you cannot bet a further $1 trillion, despite having already lost $1 trillion, then you are out of the running, and must eat the $1 trillion loss. No casino in the world would play that game (and if you lost much more than 40 times in a row, the global economy doesn't have that much liquidity).

I heard of another Roulette system once while attempting to teach a young woman (the fact that it was a woman is obviously irrelavent to the story) about options trading. She said her father would wait until the wheel had come up red 5 or 6 times in a row, then he would bet on black.

I asked her if the previous spins of the wheel had any effect on the current spin and she admitted that no, they did not. "Well, then what difference does it make?" I asked.

"But he waits until it has been 5 or 6 times in a row," she repied.

"OK, is the next spin like a coin toss, completely random, unaffected by previous spins?"

"Sure," she said.

"Well, then isn't the next spin just random, so there is no increased chance of black this time?" I asked, again.

"Sure, yeah..., but he waits until it has been 5, or even 6, in a row."

"Oooh, OK, got it."

Just for your mathematical enjoyment.

Cody,

Reread the post. The premise is that the first time you win, you quit.

"Some of the comments in this thread are scary in their lack of comprehension. :-)"

Yeah, reading comprension.

Go NIU Huskies!!

Jim English
Chicago, IL

Jim English,

Learn how to spell or at least proof read your work.

Jim English
Chicago, IL

The game in Vegas with the best house odds is Blackjack if you can count cards without the house catching you. The next best game is Craps, if you play conservatively and are willing to leave the table after a "big roll". Roulette is a horrible game. The worst is Keno.

I've spent a lot of time investigating gambling of all sorts, mostly to increase my chances of winning. Never figured out a 100% foolproof winning strategy, but currently I favor craps and horseracing. Slots and video poker are for when I'm not rolling the dice or watching the ponies. Keno is for when you're having lunch or dinner.

The Martingale strategy has also been advanced as a possible winning Blackjack strategy, but it breaks down on table minimums/maximums.

After years of study, I think craps is the way to go.

My $.02.

dsquared:

Yes, but remember that "finite time" is not the same as "bounded time". It'll happen eventually, but there's no length of time in which it's sure to happen.

The problem with this theory is that it involves betting on roulette.

I am told that Vegas casinos post roulette outcome history on a big board, so that gamblers can look for trends.

You know, the fellow who first thought of serving drinks to gamblers was smart. But whoever was the first one to realize that greed can incite even the smartest person to think he can detect hidden patterns in random noise is a genius.

God Bless America.

Slight aside - is this called martingale after the tack? Anyone know?

There are people who make their living gambling...

None of them do it playing roulette.

They do it playing poker - against people who are not very smart.

regards,

vtrtl

The approach I use is to increase my bets when I win and go back to my base bet when I lose. A sequence I've often used in blackjack

1-2-3-3-6-9-10-20-30-30-60-90.

If I lose at any point in the sequence, I go back to the first bet (1 unit) in the sequence and start over. The last numbers in the sequence may not be available in some casinos, as some casinos have max bets only 25 to 50 times the minimum bet.

The obvious (and common) problem that can arise is if you start a win-lose-win-lose sequence (commonly known as "chopping"). If this happens, you will be losing money faster than if you were making straight one unit bets, as you would have broken even with the same sequence (and made a one unit profit per hand won with the Martingale system). At blackjack, another problem that can arise is that if you've won several hands in a row, you'll have a fairly large bet out there, and may have to double (or even triple or quadruple it) to take advantage of a correct splitting or double down situation. These situations favor you in the long run, but can result in a devastating short term loss if you're unlucky at the wrong moment (this shows why even a Martingale true believer should avoid blackjack--imagine the poor SOB who has lost seven hands in a row and has a 128 unit bet out--probably near the house maximum--and is dealt a pair of eights, and has to split and re-split them when eights keep showing up? Not fun when you're already down over 100 units). On the plus side, if a blackjack shows up for you with a big bet on the table, you pick up a nice extra chunk of change (always insure a multi-unit bet if the dealer shows an ace when you have BJ--you're giving up a bit of percentage, but it's a big guaranteed payout and avoids the risk of a push).

If you want to try Martingale in a lower risk environment, try making Bank bets in baccarat. The payoff is only 19 to 20, so you'll have to add a unit or several to the doubled wagers every so often to keep the end profit for the sequence around a unit, but the Bank bet wins more than half the time (unlike most "even money" bets), making it far less likely that you will lose enough hands in a row to get to the house maximum and get wiped out. Still a dumb bet, but a slightly less dumb bet.

The basic problem with any such "odds on your side / brute-force" strategy is that it pretty much requires substantial financial resources in order to be allowed to pay off. If you can't continue to make bets then you lose. Moreover, house rules almost always make such simple minded betting strategies even more diffcult by putting in place various limits, etc. Ultimately, it comes down to house odds turned around, except the house has the money, usually, and the house makes the rules, so they can favor their house odds and disfavor your quasi-house-odds. Otherwise one casino, or any suitable financial backer, could simply bankroll a player to go to another casino and clean out its bankvault.

Finally, precisely because such simple betting strategies could be used to drain all the money from an unwary casino given proper financial backing, it stands to reason that any casino of that sort would be run out of business quickly and thus all established casinos currently in existence have rules set up to prevent that.

Theoretically as long as you have a big enough bankroll (infinite), this strategy will always work as sooner or later you will win and upon winning, you will instantly earn back all you've lost plus your original bet. I suppose this is one reason why every table has a minimum and maximum bet.

Jim English,

I guess I'll pass up the obvious snipe about your reading comprehension. I *WAS* trying to be clear after all, but obviously wasn't clear enough. My bad.

The "Martingale strategy" is designed so that a single win will always "get you ahead", therefore my assumption that you would stop when you were ahead, and the strategy's assumption that you stop after a single win, are not so much contradictory as logically equivalent. Of course, you'll be ahead by a whole dollar (or whatever your starting stake was), which is why the normal Martingale system (it's pretty common) has a step 5 that reads "go to step 1". That's neither here nor there, however, and doesn't change anything I said.

Incidentally, the math is painfully obvious if you've counted in binary much. 2^n is exactly one more than sum(2^1..2^(n-1)). Example: The 8th bit of a byte represents 128, the sum of bits 1 through 7 = 127. I suppose if you haven't messed with binary much, the behaviour of 2^n numbers wouldn't be as obvious. :-)

If you have access to a very large sum of money, and to tables with very low and very high minimum and maximum bets, you could extend the Martingale (or the Martingale derived N, 2N+1, 4N+3 . . . sequence)by starting at the cheapest table and playing there until a sequence forces you to make a bet too large for that table; at that point, you can either leave the table or simply make a note that you've got one ongoing sequence and start over--the latter is probably better for time management purposes. After you've lost five or six sequences, leave the low-stakes table and find a table where the bet you were unable to make is slightly over the minimum bet, and begin the interrupted sequences again, ending them when you win. If possible, if you lose enough times in one of the interrupted sequences to hit the table maximum at the second table, find a table with a higher limit and start again.

Again, I don't recommend this, and there are far more productive ways to "invest" the hundreds of thousands of dollars needed to make it a plausible strategy, but it does reduce the chance of short-term disaster, as if you can find one table with a minimum bet of $1 and another with a maximum bet of $100,000, you would have to lose 16 consecutive bets in the sequence 1, 3, 7. . .65,535 to hit the final table max and be officially screwed. Very unlikely to happen in the short run--certain to happen in the long run (I've lost as many as ten hands in a row [in a win-based progressive system], in relatively limited live play).

Cody,

Thanks for the lesson on Binary math. My confusion is with the following statement:

"If you graph your bankroll over time, you'll get a really random looking graph that wanders up and down."

As I understand the rules, your bankroll goes down until you win. Then as you say, you are up one bet unit. I don't see this as randomly wandering up and down. I guess you are referring to multiple applications of the system.

Jim English

But there are limits. The HOUSE doesn't allow this because it 'limits' its payouts, by setting limits on what you can bet.

Of course, if you've never gambled. And, treat this as a one up shot. Win and walk. Or lose and walk. You get the 'experience' with the least cost.

Remember, when you bet on a lottery ticket, you LOSE a dollar in most cases. So, if you want to 'win' at a lottery, you just don't bet. You keep your buck in your pocket.

Gambling is a poor choice to riches.

John and Dan:

Is the probability of losing a 50% bet every time for an infinite number of times actually zero or immeasurably close to zero?

J Mann:

Not to dodge your question, but the short answer is probably that your question as asked doesn't quite make sense.

You're asking: "If I flip a coin an infinite number of times, might it always come up heads?"

To which the answer is, "You can't flip a coin an infinite number of times."

To "Is there any finite number of coin flips that give a 100% chance of at least one head", the answer is no. However, the answer is yes for any number less than 100%.

We tend to sling infinities around as if they were any other number, but they're really a shorthand for something quite different than the numbers we're used to.

I didn't see anyone say it outright, so let me put in the phrase "gambler's ruin problem". Basically, for any finite amount of money, with a series of independent trials, there is always a non-zero probability that you will eventually lose everything. Consider, without loss of generality, a series of coin flips (or betting red or black in routlette with no 0 or 00 for the house) and a bet of $1 paying 2-for-1 if you win. You go bust when the number of losses - number of wins exceeds your stake, which may happen in as few flips as you have a stake. That is, if you go into a routlette game with $50 and play $1 bets, you can go bust in as few as 50 rolls -- but the probability of that happening is 2^-50. After you exceed 50 rolls, for any number of rolls there's a finite probability you'll lose everything.

I don't know if this is really any help, but it's always interested me so what the hell.

See also "random walks".

"1-2-3-3-6-9-10-20-30-30-60-90"

To maximize that approach, you would want to use the Fibonacci sequence (1-2-3-5-8-13-21-34-55-89-144..., where each number, after the first two, is the sum of the two preceding it). Just don't let the casino catch you doing it.

to recap:

0 and 00

Actually, that's the winning method for strip poker.

Scott,

Insurance is a lousy bet, unless you've been counting and know you have a favorable deck. The business about being sure of winning something is a logical trap that only helps the casino.

Look at it this way. Suppose you have a $10 bet and get a blackjack with the dealer showing an ace. if you insure you win $10. If you don't you win $15 when the dealer does not have a blackjack, nothing otherwise. So you are slightly better than 36/52 (depending on number of decks in use) to win $15 by NOT insuring.

$15 X (36/52) = $10.38.

Alternatively, and correctly, you should consider the insurance bet as a separate wager, totally apart form the original bet. You're betting the dealer has a 10 in the hole. If you win you collect 2-1, but the chance is slightly worse than 16/52, much less than the 1/3 you need to make it a fair bet. You're giving up 8%.

Clayton,

Just tell the house you're using a Fibonacci sequence and they will treat you like a king.

Every roulette bet is a negative expectation bet. You can not string together any combination of negative expectation bets to get a positive expectation outcome.

As many have said, Martingale just moves risk around so that there's a large probability of winning a small amount and a small probability of winning a large amount. But the overall expectation has to be negative.

That is, a small probability of *losing* a large amount.

I made practical use of this at a casino night, once - tickets were used in place of money, and to enter a raffle I needed 8 tickets, but had only 7.

I realized that if I bet one ticket on black and won, I'd have the tickets I needed. If I lost, I could then bet 2 tickets, and then 4 if I lost again. The 7/8 chance of winning worked out for me, and I entered the raffle and won.

Practical downside risk was zero, as 7 tickets had no value to me, but 8 did.

steve, you would be right if there was not an implicit infinity involved. If you have a genuinely infinite stake and infinite amount of time then you will almost surely win.
The scheme ensures that if you ever win you come out on top and this is almost sure to happen but the size of the loss up to the point of winning grows very large.

"Insurance is a lousy bet, unless you've been counting and know you have a favorable deck. The business about being sure of winning something is a logical trap that only helps the casino."

In general, I agree with you, though I routinely do it, on the theory that the irritation when I push with the house is more debilitating to me in a gambling session (particularly if I've been having a bad session anyway) than the extra boost in morale I'd get from winning an extra fifty percent on my bet if I gutted it out. Effective gambling is partly a function of concentration and energy, and I'm willing to give up that small edge to help keep both as high as possible.

Speaking specifically of my advice regarding multi-unit bets, suppose I've won five hands in a row in the sequence I described above--on the sixth hand I have a nine unit bet on the table. I get a blackjack, and the dealer shows an ace. If I take insurance, I'm reducing my average expected take on the hand by about four-tenths of a unit (9.4 without taking insurance, 9 with insurance). However, that opportunity to win nine units is rather rare--it probably occurs about once every sequence of forty hands on the average--isn't it worth the small surrender of theoretical advantage to keep the winning streak going for certain, as well as to obtain more capital to allow the exploitation of lucrative double down and/or splitting possibilities with an added cushion in the event of misfortune? In the long run, probably not. However, the successful gambler is going to play hit and run in any event, barring a truly epic run of luck. The strategy is to get ahead and start putting away money, walking when your remaining table stake is gone. Doing what you can to keep that one big run going and keeping your energy and spirits high so as to maximize your chances is the way to go--unless you're counting, in which case this betting strategy wouldn't be what you were using anyway.

Here's another example--say you're playing a video poker machine with a 2% house edge, and that it has an option that allows you to go double or nothing on a winning hand at true fifty fifty odds. You hit a royal flush, which pays eight hundred for one. Do you take the double or nothing bet? Technically, the percentages are better than the one you just made in drawing for the royal flush--0% house edge on the double or nothing bet vs. 2% on playing the game in general. Of course, this ignores the fact that unless you've been playing for many hours, you've been rather fortunate in drawing that royal, and risking the fruits of that good fortune would be rather foolhardy, unless you're in it for the really long haul.

"Insurance is a lousy bet, unless you've been counting and know you have a favorable deck. The business about being sure of winning something is a logical trap that only helps the casino."

In general, I agree with you, though I routinely do it, on the theory that the irritation when I push with the house is more debilitating to me in a gambling session (particularly if I've been having a bad session anyway) than the extra boost in morale I'd get from winning an extra fifty percent on my bet if I gutted it out. Effective gambling is partly a function of concentration and energy, and I'm willing to give up that small edge to help keep both as high as possible.

Speaking specifically of my advice regarding multi-unit bets, suppose I've won five hands in a row in the sequence I described above--on the sixth hand I have a nine unit bet on the table. I get a blackjack, and the dealer shows an ace. If I take insurance, I'm reducing my average expected take on the hand by about four-tenths of a unit (9.4 without taking insurance, 9 with insurance). However, that opportunity to win nine units is rather rare--it probably occurs about once every sequence of forty hands on the average--isn't it worth the small surrender of theoretical advantage to keep the winning streak going for certain, as well as to obtain more capital to allow the exploitation of lucrative double down and/or splitting possibilities with an added cushion in the event of misfortune? In the long run, probably not. However, the successful gambler is going to play hit and run in any event, barring a truly epic run of luck. The strategy is to get ahead and start putting away money, walking when your remaining table stake is gone. Doing what you can to keep that one big run going and keeping your energy and spirits high so as to maximize your chances is the way to go--unless you're counting, in which case this betting strategy wouldn't be what you were using anyway.

Here's another example--say you're playing a video poker machine with a 2% house edge, and that it has an option that allows you to go double or nothing on a winning hand at true fifty fifty odds. You hit a royal flush, which pays eight hundred for one. Do you take the double or nothing bet? Technically, the percentages are better than the one you just made in drawing for the royal flush--0% house edge on the double or nothing bet vs. 2% on playing the game in general. Of course, this ignores the fact that unless you've been playing for many hours, you've been rather fortunate in drawing that royal, and risking the fruits of that good fortune would be rather foolhardy, unless you're in it for the really long haul.

Scott,

You seem to be saying that what you consider a "good" strategy is a mixture of psychological ("keep the streak going") and mathematical factors. That's subjective and unarguable. After all, the best long-run strategy is not to play at all, so by sitting down at the table we are automatically adopting an inferior long-run strategy.

Still, it is important to understand that, mathematically, insurance is a bad bet, and really is a separate bet from the original one. After all, the insurance bet is the same proposition at the same odds - does the dealer have a ten in the hole? - whether you have blackjack or not.

The poker case is just a straightforward example of risk aversion, but again note that the second bet is totally unrelated to the first. The question is, are you willing to bet $800 at even money. It doesn't really matter how you got the $800. It's the same decision whether you won it or simply pulled it out of your wallet.

The royal flush is, from a logical point of view, irrelevant, just like the fact that you have blackjack is logically irrelevant to your decision on insurance.

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