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Why can’t I win at roulette in the long term?

Why can’t I win at roulette in the long term?In this article, we will talk about the possibilities of winning at roulette in the long term and why mathematics does not support these games. So, why can’t I win at roulette in the long run? The main thing is that the layout has 37 (European) or 38 (American) numbers. In this article, we will consider the European one because it's more popular. So, even if you bet on all of the numbers (1 CAD), your final bet will be 37 CAD, the payout will be 36 CAD. Thus, according to the statistic, you lose 2.7% of the stake each time you bet. So, even if you win 10 times in a row, in the long term, a day, a week or a month, you will lose all of the money. If you don't know whether to believe us or not, you can easily check the roulette; just read the article until the end.

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Why are strategies useless?

Roulette gameLet's consider all of the possible bets at roulette and see the final results of the mathematical expectation. 

In the picture below, you will be able to see a formula that calculates each separate bet in roulette. 

How to beat the roulette, (1)

  • Where is xi– event I,
  • Pi– The probability of event I,
  • К – The total number of events forming a complete group.

European roulette mathematical expectation (ME)

RatesThe formula (1) ME, unlike any made bets on "European roulette," can be converted taking the full probability, not collaborative events, that is Pwin.+ Ploss.=1, to the view of the game in the N sectors (rooms), is equal to:

Why can not you win in roulette, (2)

Since the probability of winning Pwin in "European roulette" while you are playing for N sectors (rooms) is equal to

Win at the roulette so, we finally get ME for any bet at European roulette when you are betting on N sectors (rooms), is equal to:

Formula, (3)
Let's calculate ME for every single bet in European roulette. Then, the result of ME can be seen in table 1.

The mathematical expectation for the bets

Table 1. Calculation of ME for "simple" bet

«Simple» bet




МО, у.е.









=35×1/37-36/37= -1/37






=17×2/37-35/37= -1/37






=11×3/37-34/37= -1/37






=8×4/37-33/37= -1/37


Six Line




=5×6/37-31/37= -1/37


Column & Dozens




=2×12/37-25/37= -1/37


Even Chance




=1×18/37-19/37= -1/37

What does the table show? ME is exactly equal to the value obtained by the formula (3). Let's look at the results.

The player will lose at the end, even if he wins

ZeroWhen the punter places a bet (any bet) on the European roulette, he loses 1/37 of the made bets in the end. By the way, the results for American roulette are even worse. That’s why owners of casinos don’t care if the gamblers win 1, 2 or even 10 times in a row. As long as they return to play roulette, the owners know that they just “borrowed” the won money but with huge interest. So, no matter how many times the player wins, he will return all of the money and will even spend more than he won at the casino. Only 1-2% of the people will succeed in stopping the game on time, and they won't return the money for a long time.

Using bets strategies

So, no matter what kind of strategy you use, when the ME is -1/37 of each bet, the ME will always be negative for the player in the long term. We should not forget that casinos are made for the owners to become wealthy. That’s why they add restrictions to the maximum bets; thus, the strategies are useless in the long term.


Let's calculate the volatility (dispersion) of any bets made at roulette depending on how many sectors N (rooms) we bet on. Then, use the variance (dispersion) to determine the optimal Bank Kelly criterion when playing European roulette.


Bank on Kelly's criterion shows what a gambler's bank to the total balance should look like aspired to infinity.

We can calculate the dispersion of the bets like this:

How to beat the roulette, (4)

The formula (4) for the variance "D" of any bets on the European roulette can be converted to take the full probability, not joint events, Pwin.+ Ploss.=1, to the mind:

Is it possible to win in roulette?, (5)

Since the probability of winning Pwin in the European Roulette while you're playing in the N sectors (rooms) is equal to:

Formula, finally, we get the expression for the dispersion" D "of any bet at the European roulette when you are playing in the N sectors (rooms), is:

Impossible to win in roulette, (6)

The magnitude of dispersion in"D" has a positive value throughout the range of games in the N sectors. That is an important detail.

Places where you should calculate the required bank for the game at "European roulette" using Kelly's criterion:

Formula, (7)

Using the expressions (3) and (6), we finally obtain the expression:

Mathematical justification, (8)

According to Kelly's criterion for playing at European Roulette, the formula (8) shows that the optimal amount of the bank is a negative value.


If the optimal value of the gambler's bank following Kelly's criterion is negative, then playing roulette isn't recommended because the balance will always be close to zero. In the long term, he will lose all of his money. 

If you want to get the total picture of gambling and how “good” it is to gamble, you can follow Kelly’s criterion for a mathematical expectation of the game result that is:

How to win, (9)

This picture shows that a small bank balance is actually better for players. 

The optimal criteria of the Кoptima game can only be used to evaluate games with a positive expectation! If the game has a negative ME, it will not attract the attention of the gamblers. For European roulette, it's ME -1/37, which is less than zero, so this game is not profitable for the punter. This is the reason why roulette is not mathematically good for a player, regardless of the used strategies, tips, bets, etc.; in the end, you will be the one who loses all of the money.

Note: the formula (3), (5) and (8) can be obtained for "American roulette" with two sectors of zero: 0 and 00.

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Written by Kamen Valev
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