I have devised a system for playing roulette which gives a pretty good chance of leaving from the table richer than you arrived!

Background
The game of roulette relies on the spin of a wheel bearing 37 numbers (in the French version) or 38 (American). My system does not care which version you play, so please substitute the correct details for your location. I am in France, so I use the French notation.

When the wheel is spun, a ball is introduced and eventually comes to rest on one of the numbers. In this way can the winners be determined.

There are many ways to bet on the result, but my systems assumes you will bet on a single number continuously until you win. After winning, the system is reset and you may choose another number.

Mathematical basis
In this scenario, you have a 1/37 chance of matching the winning number. If you do so, your stake is returned to you, along with winnings of 35 times your stake. Thus in total you receive back 36 times your stake. Over time, you have a return of 36/37, that is YOU LOSE ! The trick is to win in the meantime…

The approach I have taken is to calculate at each spin of the wheel, the sum of cash flows in 2 cases

– case 1, you fail to match the winning number and your money is taken;

– case 2, you win.

In this way can we see the necessary level of stake for the next bet to ensure that when your number is eventually correct, you make an overall profit.

If these calculations are correctly performed, you will be certain that once you have eventually won on a certain spin, then the total you have lost will at worst be equivalent in value to the pile of chips the croupier pushes towards you.

Limitations
Of course, the chances of winning on average are 1/37 – that is, if you stay at the table for more than 37 spins, you should see your number come up. However, that’s life: sometimes your number won’t come up for a very long time.

I have assumed that you are able to play in chips of 1 currency unit and that you have a stack of 1000 currency units in front of you.

With this method, you can survive at the table for 139 unsuccesful spins in succession and still make an overall profit of 19 units on the 140th spin.

A return on capital of <2%, but tax-free and you had a very good time.

If you only have 100 currency units, you can survive 62 unsuccesful spins in succession and still make an overall profit of 9 units on the 63rd spin.

If your number doesn’t come up by the 140th spin, the gods hate you and you should go home !

How it works
Let us work through the example of the first 10 spins:

As you can see, the profit diminishes by 1 each spin

Let us now move ahead to the 36th spin:

The profit on this spin in ZERO, or break even. From now on the stake must be increased like so:

Moving ahead to the 54th spin, we are at a break even point again and need to increase the stake:

Clearly, the effect of increasing the stake at periodic intervals allows the player to always have a profitable result when his number eventually comes up.

The frequency table is as follows: