Last week, I went to the casino with a few friends, solely playing Roulette. As you may have guessed, we lost more than we won. One friend wanted to make the classic casino mistake: after losing all the money he entered with, he wanted to get some more money to play again. Being the good friends that we are, we ridiculed him and discouraged him from doing so. However, we thought about going to the casino every month and how much that would cost. We quickly drew the conclusion that we would lose money proportionally to the amount wagered, irrespective of betting strategy. Nonetheless, different betting strategies lead to different distributions of profit! In this article, four betting strategies will be compared and the corresponding distribution of profit will be assessed.

We use a European Roulette wheel, which has only the green 0 as house edge. Players can bet on a variety of outcomes, after which a ball is rolled and ends up in one of the 37 pockets. Every bet, if successful, returns 36/n times the wager, where n is the number of pockets that would complete the bet. As the probability of winning any bet is n/37, the return on any bet is 36/37 and the house edge is 1/37. As all bets in Roulette have this same negative expected value, we lose money proportionally to the amount wagered, whatever betting strategy we use.
We will look at the distribution of profit corresponding to different betting strategies played once per month for a year, which means they are played 12 times.
Strategy 1: Play 10 games per night betting on red/black, waging €5 every bet.
Strategy 2: Play 10 games per night betting on a single number, waging €5 every bet.
Strategy 3: Bet on red/black, double when you lose and leave when you win. The first bet is equal to €2.
Strategy 4: Bet on red/black, double when you win and leave when you lose. The first bet is equal to €2.

Strategy 1: constant bet size on a colour

Placing a bet on red or black gives a probability of 18/37 to win, as there are 18 pockets of each colour. Wagering €5, we end up with €10 after a successful bet. Per casino night, we play 10 games which do not influence each other. Hence, the probability of any number of wins per night can be found by taking a binomial distribution with 10 trials and probability of success of 18/37. Per year, we get 12*10=120 games, each again independent. Hence, we use a binomial distribution with n=120 and p=18/37. In total, we wager €5*120=€600. The total profit is then given by

It is difficult to assess this distribution directly. However, we can use the normal approximation to the binomial distribution, since n is large and p is close to 0.5. We approximate the binomial distribution for the number of won bets and then transform this to a normal distribution for the profit. For the profit, we obtain a normal distribution with a mean of -€16 and a variance of €2998. This gives us a 35% probability to make profit over a year, and a 93% probability to end up between -€116 and €84 of profit. Pretty tame for a casino year, right?

Strategy 2: constant bet size on a number

Placing a bet on a number gives a probability of 1/37 to win. Wagering €5, we end up with €36*5=€180 after a successful bet. Analogously to strategy 1, we can use a binomial distribution with n=120 and p=1/37. In total, we wager €5*120=€600. The total profit is then given by

The normal approximation unfortunately does not work because p is so low. However, using the formulas

 
We find that the profit is a random variable with expected value -€16 and variance €102241. We have a 41% probability of turning profit, 37% probability to lose €240 or more, and a 23% probability to win over €300!

Strategy 3: bet on a colour, double if you lose, leave when you win

The Martingale strategy is a well-known betting scheme, claiming to ensure small, steady wins. By doubling your bet on a colour if you lose, you are guaranteed to turn any losing streak into a small profit by a single win. However, the amount of money needed to sustain a long losing streak grows exponentially with its duration. After 9 losses in a row, you need to place a €1024 bet to recoup your losses with a win! Fortunately, the probability of losing 10 times in a row is very small.
To simplify the calculations, we leave when we have a profit. This means that scoring a single win in 10 games is enough to turn a profit. As the probability of winning a single game is 18/37, losing all 10 games has probability with associated profit -€2046 and winning at least one game has probability with associated profit €2.
We repeat this 12 times, hence we can use the binomial distribution, with n=12 and p=. The profit is given by

 
Hence, we win €24 with probability 0.9848, we lose €2024 with probability 0.0151 and we lose €4072 with probability 0.0001.

Strategy 4: bet on a colour, double if you win, leave when you lose

The Reverse Martingale strategy hopes to take advantage of a long winning streak, which a lot of people think they are likely to have when they feel lucky. By doubling your bet on a colour if you lose, you either lose very small amounts or make a huge profit, which sounds attractive.
Similarly to the Martingale strategy, we can use the binomial distribution, with n=12 and p=. The profit is given by

 
Hence, we lose €24 with probability 0.9911, we win €2024 with probability 0.0088 and we win €4072 with probability 0.00004. Comparing these probabilities with the third strategy shows the dramatic effect of the house edge!
As strategies three and four have an expected total bet size of €271 and €213, respectively, we scale these up to €600 as in the first two strategies and compute and compare the variances.

As is evident, the variances don’t show the whole story. What they do show, is that the relatively “steady” options in the Martingale and Reverse Martingale are the most volatile of them all. I hope this article has given you some ideas for your next casino visit and has talked sense into the people who are adamant they can beat the casino. Remember: no betting strategy is foolproof, so gamble in moderation. Know when to quit!
 

This article is written by Balder Stalmeier.