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A **martingale** is any of a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.

Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler possess infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who chose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses occurs more often than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

A martingale is any of a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet.

The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close to 50%.

## Intuitive analysis[edit]

The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative.

The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).^{[1]} It is only with unbounded wealth, bets *and* time that it could be argued that the martingale becomes a winning strategy.

## Mathematical analysis[edit]

The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem.^{[1]}

However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity.

## Mathematical analysis of a single round[edit]

Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler 'resets' and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value of one round.

Let *q* be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). Let *B* be the amount of the initial bet. Let *n* be the finite number of bets the gambler can afford to lose.

The probability that the gambler will lose all *n* bets is *q*^{n}. When all bets lose, the total loss is

- $\sum _{i=1}^{n}B\cdot {2}^{i-1}=B({2}^{n}-1)$

The probability the gambler does not lose all *n* bets is 1 − *q*^{n}. In all other cases, the gambler wins the initial bet (*B*.) Thus, the expected profit per round is

- $(1-{q}^{n})\cdot B-{q}^{n}\cdot B({2}^{n}-1)=B(1-(2q{)}^{n})$

Whenever *q* > 1/2, the expression 1 − (2*q*)^{n} < 0 for all *n* > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average loss.

Suppose a gambler has a 63 unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking *k* as the number of preceding consecutive losses, the player will always bet 2^{k} units.

With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet.

With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued.

In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19)^{6} = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19)^{6} = 97.8744%.

The expected amount won is (1 × 0.978744) = 0.978744.

The expected amount lost is (63 × 0.021256)= 1.339118.

Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 .

In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64. Assuming *q* > 1/2 (it is a real casino) and he may only place bets at even odds, his best strategy is **bold play**: at each spin, he should bet the smallest amount such that if he wins he reaches his target immediately, and if he doesn't have enough for this, he should simply bet everything. Eventually he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible in this circumstance.^{[2]} However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds (but still with the same expected loss of 10/19 of the stake at each bet), and can only place one bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely bold play.^{[3]}

## Alternative mathematical analysis[edit]

The previous analysis calculates *expected value*, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll.

As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll.

In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown that since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely.^{[4]} This intuitive belief is sometimes referred to as the representativeness heuristic.

## Anti-martingale[edit]

In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach, also known as the reverse martingale, instead increases bets after wins, while reducing them after a loss. Jokers wild blackjack. The perception is that the gambler will benefit from a winning streak or a 'hot hand', while reducing losses while 'cold' or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning 'streaks' is merely an example of gambler's fallacy, and the anti-martingale strategy fails to make any money. If on the other hand, real-life stock returns are serially correlated (for instance due to economic cycles and delayed reaction to news of larger market participants), 'streaks' of wins or losses do happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and can be used in trading systems (as trend-following or 'doubling up'). (But see also dollar cost averaging.)

## See also[edit]

## References[edit]

- ^
^{a}^{b}Michael Mitzenmacher; Eli Upfal (2005),*Probability and computing: randomized algorithms and probabilistic analysis*, Cambridge University Press, p. 298, ISBN978-0-521-83540-4, archived from the original on October 13, 2015 **^**Lester E. Dubins; Leonard J. Savage (1965),*How to gamble if you must: inequalities for stochastic processes*, McGraw Hill**^**Larry Shepp (2006),*Bold play and the optimal policy for Vardi's casino, pp 150–156 in: Random Walk, Sequential Analysis and Related Topics*, World Scientific**^**Martin, Frank A. (February 2009). 'What were the Odds of Having Such a Terrible Streak at the Casino?'(PDF). WizardOfOdds.com. Retrieved 31 March 2012.

Roulette is one of the most popular table games in modern casinos. Although variations on the game have been around for several hundred years, there are now only 3 variations in American casinos.

You’re likely already familiar with American roulette and European roulette. The most recent addition to the table game inventory is Sands Roulette.

Which of these games should you play?

How should you bet on them?

What’s the smartest strategy for roulette betting?

I’ll explain all that in this post:

## What Are the Differences between American, European, and Sands Roulette?

Although these games have a few other differences, **the most significant distinction between the 3 versions of roulette are the number of green slots the wheels contain.**

Every roulette wheel has at least 37 slots.

36 of those slots are always numbered 1 to 36, and they’re alternately colored RED or BLACK.

The additional slots are green.

**In European roulette there is only one green slot, the “0”. **

**In American roulette there are two green slots: “0” and “00”. **

**In Sands roulette a third green slot, “S”, has been added to the wheel.**

The green slots are there for one reason:

They make the game’s statistical probabilities uneven.

This is because of the way roulette bets are paid off. You can win anywhere from 35-to-1 (for betting on a single number) down to 1-to-1 (for betting on 18 slots at a time).

The payoffs, called “odds”, are not as fair to you as the actual estimated probabilities of the roulette ball landing on any given slot. This is how the casino makes its money.

In a game of roulette the house should keep at least 2.70% of all the bets players make over time. The casino has no need to cheat the players. In fact, the players often make really bad bets that improve the “house edge”, as that casino profit is called.

One of the other differences between European roulette and both American and Sands roulette is that the European roulette table has an additional betting area. This secondary betting area is used to place specially designed bets. They are more complicated than the normal bets made in American and Sands roulette. I’m going to ignore this section of the table, because I’m going to show you how to place bets that have the best chances of paying off.

## Is There a Winning System for Roulette?

Everyone who gets into roulette sooner or later starts to think about how they can “beat the system”.

I’m going to be honest here:

**There is no way to do that.**

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The green slots on the wheel make it impossible for anyone, anywhere, to ever design a betting system that is guaranteed to win. If you really want to guarantee yourself a win every time, then put a chip on each of the 2-to-1 outside bets and on each of the green number bets.

That’s the only way you’ll be paid money every time the wheel spins.

**You’ll also go broke.**

You may have heard about a system called the Martingale System. It’s a popular betting system with new roulette players.

Experienced roulette players just turn their heads and roll their eyes when someone mentions the Martingale System. **The only way you can make money with the Martingale System is to write a book about it and get people to buy your book.**

Even that’s a gamble, though, because most people now know that the Martingale System promises more than it delivers.

Here’s how this system works:

You start out betting the minimum. If you lose, you double your bet. If you win on your doubled bet, you go back to betting the table minimum. If you lose again, you double the size of your bet again.

This sounds great to inexperienced bettors but the problem is that you’ll either run out of money or hit the table limit before you can recoup your losses as they add up.

**The Martingale System is a sucker bet, plain and simple.**

Every betting system in every form of gambling tries to leverage probability theory. The Martingale System and other roulette betting strategies also rely on probability estimates.

But there’s a flaw in the thinking behind these systems. If you account for the flaw you’ll be okay. You won’t always win but your expectations will be more reasonable.

The secret to not going broke when you gamble is to set reasonable expectations and maintain your self-discipline. You should never drink or take drugs when you gamble. They lower your inhibitions and impair your judgment.

You might as well just hand your money over to the casino at the cashier window and say “keep it” if you’re going to drink or do drugs when you gamble.

## How Do Probabilities Work in Roulette?

Probability theory came out of statistics. It tries to give us rules by which to guess what happens next in any situation. The guesses are seldom accurate predictions. Sometimes the guesses work out, and sometimes they don’t. Gamblers love probability theory because they think it helps them pick the best betting strategies.

You’re actually more likely to double your money during a roulette session if you put all your money on a single bet. The more bets you place, the less likely it becomes to double your money.

That’s because every bet brings you close to the long term expectations. The closer you are to the short term, the more likely you are to get better than expected results.

**In roulette, the probabilities are simple.** The dealer spins the wheel and releases a ball that whirls around the outside of the wheel and finally settles in a slot. With only 37 slots on a European roulette wheel you have a 1-in-37 probability of the ball landing on a specific slot.

This probability never changes.

This probability is calculated on the basis of all the known possibilities.

What probability theory cannot do, however, is predict where the ball will stop.

Nor can it predict whether the ball will land on red, black, or green any number of times over the next 100 spins.

Nonetheless, a lot of gambling guides tell you that you have the best chances of winning if you do this because of such-and-such probabilities. And many of these guides warn you that there is no way to predict the future, but by setting the expectation that the ball will land on red about 47% of the time, these guides are making predictions and promises they cannot keep.

They’ll even back up their claims by talking about how to run computer simulations for 1 million spins of the wheel so that you see how often the ball lands on red, black, or green.

In the real world the Probability Fairy is always on vacation. She’ll never be there to wave her magic wand to make things happen the way experts say they should. The ball could land on red over the next 20 spins. Or it could land on black or green or some random mix of color combinations.

You have no way of knowing how many of the next [X] spins will turn out a certain way. Talking about probabilities in this way is just dishonest.

What you *can* do is look at the wheel and ask yourself how much it costs to bet on the largest possible set of numbers. The idea here is to get as much coverage as you can without losing money too fast.

**But even if you cover every number on the wheel you’ll lose money.**

So the only way to win in roulette–and this is completely random, never guaranteed–is to bet on less than all the numbers on the wheel.

You also want to play bets that pay better than even money. You can place a variety of bets, but most of them aren’t worthwhile.

Betting on single numbers is a bad idea. You can place bets on the lines between the numbers (these are called “street bets”) and on lines at the corners of numbers (these are called “corner bets”).

But even though you get pretty good odds (payoff) you’re still covering too few numbers.

## How Bets Work in Roulette

Divide the bets into two groups:

- Inside bets
- Outside bets

**Inside bets are based on individual numbers or small groups of numbers.** When you see players betting on the lines, corners, and individual numbers on the table they are making inside bets.

**Outside bets are based on pre-selected groups of numbers on the wheel.** The “2-to-1” bets cover 12 numbers each: 1 to 12, 13 to 24, and 25 to 36. The “1-to-1” or “even money” bets cover 18 numbers each:

- Odd
- Even
- Black
- Red
- 1 to 18
- 19 to 36

**The bets more likely to pay are the even money bets.**

But unless you can win 5 times out of 9 on even money bets you’ll lose your stake. That’s the problem with roulette. You always have to win at least 1 more time than you lose no matter how you place your bets.

The “2 to 1” bets pay better than the “1 to 1” bets because they cover fewer numbers. You have less of a chance of winning.

There are 6 types of “2 to 1” bets:

**3 kinds of dozens bets:**(1 to 12, 13 to 24, and 25 to 36)**3 kinds of columns bets:**([1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34], [2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35], [3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36]).

You can make a bet by betting on any two of the “2 to 1” groups. That means that instead of covering only 18 numbers you’ll be covering 24 numbers.

**This type of bet is often called the “double dozen” bet.** It’s popular among gamblers who like to hedge their bets. They have a better chance (all other things considered) of scoring a win with a “double dozen” than with one of the standard even money bets. If you’re playing it safe and going for even money odds, you should always play a double dozen bet.

If you want to bet more aggressively, then instead of betting more money on your double dozen, you can cover all 36 of the red and black numbers. Leave the green numbers alone. Yes, they’ll come in every now and then, and you’ll lose money.

But there’s a way to keep your losses low.

## How to Bet on Columns or Dozens Aggressively

Take 6 chips and distribute them across EITHER the three dozen bets or the three column bets.

Place 3 chips on 1, 2 chips on the 2nd, and 1 chip on the 3rd. If the ball lands on a green number you’ll lose your entire bet, so always play the table minimum with this aggressive style.

If the ball lands on any number with your single chip bet, you’ll win 2 chips and lose 5–for a net loss of 3 chips (half your bet).

That’s the safest way to bet aggressively on the table.

If the ball lands on any number in your 2 chip bet you’ll win 4 chips and lose 4 for no loss. This keeps you in the game.

If the ball lands on any number in your 3chip bet, you’ll win 6 chips and lose 3 for a net gain of 3 chips. This will offset 1 single chip win.

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The way this betting strategy works out, your money can grow substantially and still take some big hits. Where the strategy will fail you is when the ball lands on green or if the ball lands on the single chip bet more often than it lands on the 3 chip bet.

Sorry, but there’s no way to prevent that from happening.

## There Is No Guaranteed Way to Win in Roulette

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I can’t say this often enough:

**You can’t win at roulette in the long run.**

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I think roulette is a fun game to play. It’s exciting because you don’t know where the ball will land. You take an active role in making your wagers.

And you’ll find there are a lot of different betting systems to experiment with. The only thing that is guaranteed in roulette is that the casino will make a profit. What you hope for is that they make their profit at someone else’s expense.

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Players who try to improve their luck by making big bets do sometimes win, but most often the people who come out ahead are the patient players who use conservative betting strategies and take money off the table. If you only walk away with your beginning stake you’ll be luckier than most gamblers.

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And you can take *that* to the bank.