The “HEC Ball” was held last weekend and the event’s first part was at Casino Montreux, where each student received CHF15 at the entry to gamble. Many of my friends lost their initial bet while I was lucky enough to cash back CHF60 after a few games with roulette. This unexpected success made me wonder about the nature of luck in games of chance and if any strategies could be exploited to make some profits.

Indeed, it is stated on several forums that it is possible to make money by gambling by using some “secret method” named *martingale*. In this article, I will explore the classical version of this method using the example of roulette and explain the probabilistic intuition behind it.

#### The Martingale strategy

To simplify our analysis, let’s assume the gambler plays European roulette and only bets on odd/even or only black/red type of bets. Thus, at each game, the probability of winning is around 18/37=48.65% and the payoff is always double our bet. If you bet $1, you receive $2 back with a probability of 48.65% and $0 with a probability of 51.35%. Therefore, the expected loss for each dollar and each game is $0.027. Although the expected profit is negative, some websites affirm that it is possible to beat the croupier and make long-term gains by using a strategy called martingale.

The classical version of the martingale is when the gambler always bets the same amount after a win and doubles their bets after a loss. This method allows the gambler to win the initial bet or win back their accumulated loss after a successive set of losses.

To illustrate this, let’s take a numerical example where you go to the casino with $1,000 in your wallet with an initial bet of $1:

- In the first game, you bet $1, you win, and you now have $1,001.
- In the second game, you still bet 1$, but you lose it, so now you have $1000 in your wallet.
- In the third game, since you have lost $1 in your last game, you bet ($1*2=) $2, and you lose again. So, you have $998 in your wallet.
- In the fourth game, you bet ($2*2=) $4, and you lose again, so you still have $994 in your wallet.
- In the fifth game, you bet ($4*2=) $8, and you win. So now in your wallet, you have $1,002.
- In the sixth game, you need to bet $1 again, since you won the previous game.

This method looks miraculously great, but the gains are linear while the losses are exponential. In other words, if your initial bet is $1, after successive losses of 10 bets, you will need to gamble 2^10=$1,024 for the 11^{th} bet.

#### Practical Testing: Martingale Strategy Through Python Simulations

*(My code for Python is available in the Sources at the bottom of the article)*

To test this method, I developed a loop in Python that simulates color betting in roulette and incorporated the martingale within it. I assumed that the player has $1,000 to gamble in each simulation, and the initial bet is always $1. Therefore, each graphic starts with $1,000 on the Y axis, and then the available funds will fluctuate according to the success or failure of the strategy. Another important assumption here is that the player can “borrow” unlimited amounts of money for free when the balance dips into negative numbers.

#### Early Results: Analyzing the Outcome of 100 Simulated Games

After simulating 100 games, this strategy appears effective. The gambler starts with $1,000 and ends with over $1,050. The lowest point on the graph is at 990 and the gambler always had sufficient funds to finance subsequent bets. Nevertheless, it is observable that the gains are linear while the losses are exponential, which might be an issue in the case of successive losses.

#### Scaling Up: Insights from 1,000 Game Simulations

With an increased number of games in the simulation, the probability of encountering successive losses rises. The graphic below clearly shows that, although the gambler’s wealth reaches $1,489 after 1,000 games, successive losses almost made the available funds reach negative numbers.

#### Long-Term Analysis: 10,000 Games Under the Microscope

This graphic shows the danger of the martingale strategy. While the gambler’s wealth nearly reaches $6,000 by the end of the simulation, this outcome depends on the accessibility to infinite financial resources. This is necessary to constantly double the bet when successive losses occur, which is not realistic in a real-world gambling scenario.

#### Real-World Constraints: Testing the Martingale with Limited Resources

In this simulation of 10,000 games, I tried to make things closer to real-world conditions by not allowing the gambler to borrow money. If the player cannot double the bet after a loss because of insufficient funds, the betting cycle will start again at $1. In this simulation, the gambler mostly runs out of money even though the strategy is the same as before.

#### Statistical Risk: Estimating Bankruptcy Rates with Monte Carlo Simulations

Since it would be too hard to calculate the probabilities of ending with $0 in the wallet, I used the Monte Carlo approach to estimate them. An important assumption here is the possibility of going bankrupt, thus, here the gambler does not have access to infinite financial resources. Other assumptions are still valid (initial bet is $1 and available funds when starting is $1,000). The graphic below shows that the probability of running out of money is around 40% after 2,000 games.

#### “The last 10 spins have been red; I should bet on the black for the next spin”

One of the most common misconceptions about roulette strategies is the emphasis on using previous numbers to predict the next one. For example, some players might bet on black after seeing 5 consecutive reds, believing the probability of having 6 successive reds is minimal. The proponents of this strategy state that the statistics of each color should converge to its mean which is around 48.65%. They claim that if one of the colors is underrepresented in the previous games, we should bet on it for the next games because roulette needs to overrepresent this color to balance the ratio and reach the long-term statistic of 48.65%.

Nevertheless, each spin of the roulette is independent of previous ones, meaning that past results do not influence the next spin. The probability of the ball landing on black or red remains at 48.65% for each spin. Furthermore, roulette does not “need” to overrepresent a color that was underrepresented during previous games.

For example, if over 10 games, 9 were red and only 1 was black, representing 10% of previous games, the probability for each color will still be 48.65% for the next games. Roulette does not need to overrepresent blacks to rebalance the statistics. Indeed, with an equal portion of both colors for the next 90 spins, there would be 54 reds (9 from before plus 45 new) and 46 blacks (1 from before plus 45 new). Thus, the proportion of blacks went from 10% to 46% without underrepresenting red numbers for the 90 new spins, making the statistics converge to the true mean of 48.65%.

#### Final Thoughts

After testing the martingale, it seems theoretically possible to make a profit by gambling with roulette under two assumptions:

- The gambler has access to an infinite amount of money,
- There are no limits on the maximum bet size.

However, since these assumptions do not hold, we conclude that expecting to be profitable by gambling at roulette with this method is unrealistic. Indeed, every casino has a maximum bet limit, and no one can access to infinite amount of money. Furthermore, betting by looking at past numbers or colors is ineffective since each spin of the roulette is independent of the previous one.

Lastly, if there were a method to make long-term gains from roulette, casinos would adjust their rules to ensure their profitability. For now, the most profitable strategy with casinos is to avoid them.

##### SOURCES (cliquez sur les titres pour en savoir plus)

An academic study on martingale

Python code available on my GitHub repository