Labouchere Roulette Calculator

What is the Labouchere system in roulette?

Labouchère, also called the cancellation system or split Martingale, turns a target into a line of positive numbers. The next even-money wager is the first value plus the last value; after a win those outside values are removed, and after a full loss the wager value is appended to the end of the line.

The sequence can organize stake sizes and target accounting, but it does not change the probability of the next roulette result. The roulette strategy and bankroll-risk guide compares Labouchère with other progression systems and flat betting without treating any staking rule as a way to remove the house edge.

Does the Labouchere system work?

It works as a defined cancellation method: if the line is eventually emptied under the stated 1:1 settlement rules, the cycle finishes at the original line-sum target. Completion is not guaranteed. Losses can lengthen the line and enlarge later bets until bankroll or table limits prevent the next required wager.

Why line order and outcome order both matter

The starting line’s outside values determine the first bet, so two sequences with the same sum can begin with different stakes. The lines 1, 2, 3, 4 and 1, 3, 4, 2 both target ten units, but their opening bets are five and three units respectively.

The W/L order then changes which values are removed and which bets are appended. A win can shorten the line by two numbers, while a loss lengthens it by one. The same counts of wins and losses can therefore produce different maximum wagers and different unfinished lines.

The same target can require different peak stakes. Both paths above complete the line with three wins and two losses. With a table maximum of 70 and bankroll of 500, however, LWLWW stops before its 80-unit wager while LLWWW can finish. A completed cycle always earns its original target; the route and ability to complete it differ.

The calculator keeps the chronological state rather than estimating from counts alone. Extra path symbols are ignored once the line is empty because the completed cycle has ended.

Labouchere system example: 1-2-3-4 sequence

This example uses a ten-unit base and a four-number line targeting 100 units. The path LWLWW develops as follows:

The cycle requires 320 units of action to obtain its 100-unit target on this particular path. A different path may finish sooner, remain incomplete or reach a stake that cannot be placed.

French la partage and the H convention

Under French la partage, zero returns half of an eligible even-money stake. The calculator uses H for that half-loss and assigns it probability 1/37 under the French model.

The H update is a bookkeeping choice for this calculator: it appends half of the current bet value because the monetary loss is half the stake. The accounting identity remains exact: cumulative net plus the remaining line value still equals the initial target.

This is a transparent calculator convention, not a claim that every player or casino applies the same progression rule. En prison is excluded because it preserves an unresolved stake across a later spin and therefore needs a separate multi-spin state model.

Does Labouchere roulette change expected value?

No. European even-money wagers retain a 1/37 house edge, American even-money wagers retain 2/38, and eligible French la partage wagers reduce the expected cost to 0.5/37. The line changes future stake amounts but not the return percentage on each unit wagered.

The edge-cost card applies the selected house edge to the realised action. It is a baseline for the path that was actually processed, not a fixed forecast for every cycle, because losses and wins alter later wagers.

Mathematical work on the system emphasizes the tail risk behind frequent completed targets. In an idealized unlimited game, Han and Wang prove that the expected largest bet is infinite when the win probability is at or below one half (paper and abstract). That result concerns ordinary full-win/full-loss cancellation, not the H extension. A finite bankroll or table limit can stop a real cycle before completion; the remaining line records the shortfall from its target. The roulette RTP and expected-loss calculator separates return percentage from stake volume.

Labouchere vs Martingale and Fibonacci

These systems all react to earlier results, but they store state differently. Labouchère keeps a changing cancellation line, Martingale keeps a doubling level, and Fibonacci keeps a position in a predefined sequence. The Martingale calculator shows exponential recovery-bet growth, while the Fibonacci roulette calculator models recursive sequence movement through the same kind of exact W/L path.

Reverse Labouchere: follow the same path with opposite updates

The reverse version keeps the outside-number wager but swaps the line updates: a win appends the bet value; a full loss removes the first and last values. With one number left, bet it once and remove it after a loss. The starting line describes a single-cycle loss boundary, not a profit target.

This panel shares the starting line, base unit, bankroll, table maximum and chronological path above. It accepts standard European or American W/L outcomes only: zero, and American 00, count as L. La partage, surrender and en prison require different rules and are excluded here.

Enable JavaScript to compare the entered path. The worked example below remains available.

An empty reverse line is a loss, not a completed profit target

Write T for the starting line sum multiplied by the base unit, N for net profit, and S for the current line sum multiplied by the base unit. Ordinary cancellation keeps N + S = T. Reverse keeps S − N = T: wins raise both S and N; losses lower both. Emptying a reverse line therefore ends at N = −T. Stopping early can leave a smaller loss or a profit.

For line 1, 2, 3, 4 at base 1, with enough bankroll and a table maximum of at least 7, the path WWLLL produces reverse stakes 5, 6, 7, 7, 7. Net profit moves through +5, +11, +4, −3, −10, then the line empties. With a preselected profit stop of 10, the same reverse path ends after its second win at +11. Ordinary cancellation ends after WW at +10, leaving the last three symbols unused.

The −T boundary belongs to one unmodified cycle with positive line values, full 1:1 settlement and no restart or fees. Restarting an empty line creates a new cycle and can add further losses. Neither the starting line nor a selected winning path establishes a profitable system.

Fixed-stake comparison

The flat result beside the calculator uses one base unit on each processed outcome. It does not run a separate bankroll or stopping rule, and it excludes symbols after completion or a blocked bet.

Actual minimum stakes, chip denominations and zero rules affect whether a sequence is playable. Our table-selection comparisons are at https://roulettecalc.com/casinos/.

For a baseline without Labouchère's changing line state, use the roulette session simulator. It keeps the stake fixed, so it is useful for comparing ordinary session variance with the state-dependent progression modeled here; it is not a Labouchère simulator.