Negative progressions: martingale, Fibonacci, D'Alembert

A "negative progression" raises the stake after a loss and lowers or resets it after a win, hoping that the next win will repair what the losses have cost. The family includes the martingale, the "grand martingale", the Fibonacci, the D'Alembert and the Labouchère, with minor cousins (the Tiers et Tout, the pyramid) that the page on betting systems lists one row each.

None of them changes the price, the 2.70 per cent that a European wheel takes from each dollar staked (the house edge page lists it for each game), so what we count here is something else. It is the "wall" that each progression meets (the table maximum, or the end of the "bankroll" put aside for the session) and how often a player reaches it.

The martingale on a real table

The martingale doubles the stake after each losing bet and returns to the first stake once a win arrives, so that every completed "sequence" (the run of bets ending with a win) brings in one "unit" (the first stake). On a table from $10 to $500, a player has room for six stakes ($10, $20, $40, $80, $160 and $320), and the "table maximum" refuses the seventh.

Six losses in a row therefore cost $630, the whole "exposure" of the progression (the total lost when the bad run arrives). On a European wheel a bet on red, an "even-money" bet that pays as much as it risks, loses on nineteen numbers out of thirty-seven (eighteen black and the zero), so the probability that six bets in a row all lose is nineteen thirty-sevenths to the sixth power, about 1.8 per cent, or once in 55 sequences.

Why once in 55 is not rare

Once in 55 sounds rare, and it is not. A sequence lasts about two spins on average, so an evening of two hundred spins contains about a hundred, and by our computation the probability of meeting at least one run of six losses in such an evening is 84 per cent, roughly five evenings in six.

In a single sequence, as we count it, the player wins $10 with a probability of about 98.2 per cent and loses $630 in the 1.8 per cent of cases where the run arrives, so the expected result is a loss of about $1.74. That is 2.70 per cent of the $64 that a sequence puts across the table on average; one bad run, meanwhile, wipes out the profit of sixty-three good sequences!

Raising the "ceiling" changes the rate, not the trade. With a $1,000 maximum the player gains a seventh stake ($640) and the exposure becomes $1,270, the profit of 127 good sequences, while the bad run becomes about twice as rare (once in 106).

Since more money crosses the table in each sequence (about $76 instead of $64), the expected loss per sequence rises to about $2.05: a larger bankroll buys a rarer and larger disaster at a higher average price. If you think of the bankroll as insurance, it is insurance that raises its premium with the cover.

The grand martingale, one stake sooner

The grand martingale doubles and adds one unit after each loss, so that a win at the fifth stake closes the sequence five units up, and it is often presented as the "ambitious" version. On the same table its stakes are $10, $30, $70, $150 and $310, the sixth ($630) is refused, and five losses in a row cost $570. That run comes about once in 28 sequences, twice as often as with the plain martingale, which in our view settles the question of ambition.

The Fibonacci, a slower road to the same ceiling

The Fibonacci climbs along the sequence 1, 1, 2, 3, 5, 8, 13 and so on, one step forward after a loss and two places back after each win. At $10 a unit it needs nine losses in a row to reach the ceiling of a $500 table (its stakes run from $10 to $340, and the tenth would be $550), and nine straight losses happen about once in 400 sequences, the figure that makes the system look safe.

The "streak" is not the only road to the ceiling, though, since a Fibonacci also climbs through mixed runs in which losses come more than twice as often as wins, making the real frequency almost four times higher. Counting every road, by our computation a sequence reaches the refused stake about once in a hundred (0.94 per cent), always with the same $880 lost! A completed sequence, on the other hand, ends one unit up at most, and sometimes "level".

The D'Alembert and the square of the gap

The D'Alembert adds one unit to the stake after a loss and takes one away after a win, and its promise of "balance" rests on a piece of arithmetic that is real. Ten losses followed by ten wins, for example, leave the player ten units up, one for each win; the wheel, however, does not deliver equal numbers (nineteen losing numbers against eighteen winning ones on red), and when losses run ahead the cost grows roughly with the square of the gap.

For a player at $10 a unit over a hundred spins, the exact distribution we have computed shows the trade plainly. The session ends ahead about 59 times in 100 (the "median" result, the one in the middle, is a gain of about $130), and yet the average is a loss of about $176, because the sessions that end behind lose about $940 on average.

That average is once again 2.70 per cent of the "turnover" (everything wagered), about $6,500 here, because the stake climbs during every bad stretch. We would read the 59 per cent as a measure of how pleasant the system feels, and the $176 as its price.

Labouchère, the line that grows

The Labouchère, also called the "cancellation" system, starts from a written line of numbers whose sum is the target (1-2-3-4, say, for ten units), stakes the sum of the two ends, crosses both out after a win and adds the lost stake to the end of the line after a loss. Each win removes two numbers and each loss adds one, so the line shrinks on average if the player wins more than one bet in three, which is why the system feels "unbeatable".

The stakes, however, grow with the line. With $10 units, the line 1-2-3-4 and a $500 table, by our computation about one line in twelve meets a stake above the maximum while still unfinished, and the player who stops there has lost about $1,450 on average, against the $100 won by each of the other eleven lines.

What the family really sells

Every system on this page sells one exchange: many small wins against a rare and large loss, at the same price per dollar as any other way of betting. The exchange feels favourable because the wins are frequent; you can go through many evenings remembering only those, and the large loss, when it arrives, looks like "bad luck" rather than the bill for earlier evenings.

There is a further cost, less visible, that we would not leave out. A progression pushes more money over the table than a "flat" stake in the same number of spins, so its expected loss is larger in dollars even when the share is identical: over two hundred spins the martingale described above wagers about $6,310 on average, where a flat $10 player wagers $2,000.

If you have just lost and the thought of doubling to "win it back" has crossed your mind, that thought is a martingale without the name, and it meets the same wall. We would suggest comparing it with flat betting, which fixes the cost of an evening in advance, and with the positive progressions, which make the opposite exchange; a pause, in our view, is the only move on this page that costs nothing.

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By the Rand Casinos editorial team · Last updated October 2026 · Online since 2007