Betting Systems — Why Martingale and D'Alembert Don't Work

Betting systems change the size or timing of bets, but they do not change a game’s house edge. Martingale, Fibonacci and D’Alembert produce different patterns of short-term results while leaving the average result per credit wagered unchanged. This guide explains the distinction between expected value and variance.

What a betting system is

A betting system is any method that dictates how much to wager on each round based on the history of previous rounds. Increase after a loss, decrease after a win, follow a mathematical sequence — the variations are endless. What they all share is the idea that past results should inform future bet sizing.

The Martingale

The Martingale is the best-known system. The rule is simple: after every loss, double your bet. After a win, return to the starting amount. The logic sounds airtight — eventually you must win a round, and when you do, the win pays back all previous losses plus one unit profit.

The problem is the word “eventually.” Here is what happens in practice:

Starting bet 10 credits. Losing sequence: 10, 20, 40, 80, 160, 320, 640, 1,280 credits — eight consecutive losses. Total wagered: 2,550 credits. Ninth bet required: 2,560 credits.

For European roulette red/black, eight consecutive losses have a probability of about 0.74%, or roughly one in 135 eight-spin sequences. This calculation and the progression mechanics follow William Feller’s An Introduction to Probability Theory and Its Applications (Wiley, 1968). A finite balance or table limit can stop the progression before a win occurs.

The Martingale converts many small wins into the occasional catastrophic loss. It does not change the amount the house is expected to extract from you over time.

The Fibonacci

The Fibonacci system uses the Fibonacci sequence to determine bet sizes: 1, 1, 2, 3, 5, 8, 13, 21, 34… After each loss you move one step forward in the sequence; after a win you move two steps back. The system is more conservative than the Martingale but operates on the same flawed premise.

The Fibonacci sequence grows more slowly than exponential doubling, so catastrophic losses build more gradually. But the fundamental mathematics are unchanged. Every bet you place at roulette or Sic Bo or craps has the same house edge regardless of which number in the sequence you are at. Previous results do not alter the probability of the next roll.

The D’Alembert

The D’Alembert increases bets by one unit after a loss and decreases by one unit after a win. It is based on the intuition that wins and losses should balance out over time — a concept from eighteenth-century physics applied, incorrectly, to gambling.

Games like roulette and Sic Bo are independent trials: the outcome of any roll has no relationship whatsoever to previous rolls. The dice or ball has no memory. The D’Alembert’s assumption that a run of losses makes a win more likely is the gambler’s fallacy in disguise. An extended losing run under this system still results in losses that exceed the expected-value prediction based solely on the house edge.

Why no system can work: the math

Consider a game with a 2.78% house edge. For every credit wagered, the expected return is 0.9722 credits. It does not matter whether that credit is wagered on its own, after a previous loss or as part of a sequence. Each bet has the same expected value.

Because expected value is linear, the expected value of a sequence is the sum of the expected values of its individual bets. No arrangement of bet sizes changes this total. Across 1,000 credits wagered at a 2.78% house edge, the mathematical expected loss is 27.80 credits, regardless of the progression.

The relevant result is the optional stopping theorem. Under its stated conditions, stopping a martingale does not create a positive expectation. J. L. Doob sets out the theorem in Stochastic Processes (Wiley, 1953); casino games with a negative expectation do not become positive through a stopping rule or progression.

What betting systems actually do

This is where the analysis becomes useful. While no system changes expected value, they do change variance — the spread of possible outcomes.

• Martingale reduces the probability of a losing session (most sessions end in a small win) but dramatically increases the size and probability of catastrophic losses when they occur.
• Fibonacci and D’Alembert create a similar but less extreme version of the same trade-off.
• Flat betting keeps the wager size constant, so it does not add the escalation risk created by a progression.

No betting system generates an edge where none exists. They are tools for shaping risk, not eliminating it.

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For the underlying mathematics of probability and expected value, see Probability Basics. For a full comparison of house edges across casino games, see Casino House Edge Explained.