Roulette Betting Systems Tested by Simulation

Betting systems promise to beat roulette by changing your bet size after wins and losses. We ran three of them through 100,000 simulated sessions each. They all lose the same 2.7% of every euro bet. What they change is the pattern of the losses.

The systems

• Flat betting: bet 1 unit on red every spin.
• Martingale: start with 1 unit and double after every loss. After a win, go back to 1 unit. Every completed sequence wins exactly 1 unit, unless you run out of money or hit the table limit first.
• D'Alembert: add 1 unit after a loss and subtract 1 unit after a win, never going below 1.

The setup

• European single-zero wheel, even-money bets on red
• Starting bankroll of 200 units, table limit of 500 units
• A session ends when the player is 50 units ahead, is broke, or has played 1,000 spins
• 100,000 sessions per system

Results

(The Martingale's loss per unit bet is slightly above 2.70% because of how bets are capped when the bankroll runs short. That is noise from the setup, not a real effect.)

What the numbers show

The Martingale feels great, and that is the problem. Three sessions in four end with a win. The fourth wipes out the whole bankroll. It needs only eight losses in a row, which happens about once every 200 or so sequences. Frequent small wins paid for by rare total losses is exactly what makes the system so persuasive to people who have tried it a few times.

The average session result is misleading. The Martingale's −12 units looks better than flat betting's −27, but Martingale sessions are ten times shorter. Per unit bet, every system lands on the same number: 1/37 = 2.70%, the house edge.

Flat betting rarely wins big and rarely busts. It spreads the same expected loss evenly over many spins.

Why no system can work

Each spin has a negative expected value of −1/37 per unit bet. A betting system only decides how much goes on each spin. A sum of bets that each have negative expectation always has negative expectation, whatever sizes you choose. In probability theory this is the optional stopping theorem: you cannot turn a fair or unfavourable game into a favourable one by choosing when to bet or when to stop.

What a system can do is reshape the distribution of outcomes: many small wins with rare disasters (Martingale), or a steady drift downward (flat betting). Choosing between those shapes is a question of preference, not of profit.

For the reason streaks don't help either, see the gambler's fallacy.