ROU 407: Labouchere Roulette System

The Labouchere system is a cancellation betting progression. A player writes a line of numbers, bets the sum of the first and last numbers, removes those numbers after a win, and appends the lost bet after a loss. It is usually applied to roulette’s red/black, odd/even, or high/low wagers.

Completing the original line produces the planned profit on an even-money payoff. That feature makes the system feel mathematically organized. The weakness is what happens before completion: losing results lengthen the line, increase later stakes, create more total action, and can run into bankroll or table limits. The system changes bet size, not roulette probability or payout.

Building and reading the line

Suppose the starting line is:

1 - 2 - 3 - 4

The numbers represent betting units. Their sum is 10, so the intended profit is 10 units if the line is eventually cleared without abandoning or modifying it.

The next bet is:

$$Next\ Bet = First\ Number + Last\ Number$$

For the starting line:

$$1+4=5\ units$$

After a win, cross out the first and last numbers. After a loss, leave the line in place and append the amount lost.

If only one number remains, that number becomes the next bet.

Why clearing the line produces the target

The starting numbers represent pieces of the intended profit. A winning bet equal to the first plus last entries earns enough to cancel both entries. If a loss occurs, the lost amount is appended so it must be recovered by a later winning cancellation.

Under ideal even-money settlement, clearing every original and appended entry leaves profit equal to the sum of the original line. That accounting identity explains why completed lines are satisfying. It does not prove that the line will be completed before the player reaches a stop, table maximum, or bankroll limit.

The result also depends on following the rules exactly. Removing entries after a partial win, changing the unit, restarting the line, or accepting a special roulette rule can change the target and exposure.

One sequence that completes successfully

The result sequence below is Loss, Loss, Win, Loss, Win, Win, Win:

The system achieved its 10-unit target. Total action, however, was:

$$5+6+7+7+9+8+4=46\ units$$

On a European single-zero wheel, even-money wagers have a 2.70% house edge. The expected loss on 46 units of action is:

$$46\times\frac{1}{37}=1.243\ units$$

The actual sequence won 10 units, while its expected result was negative. Both statements can be true. The winning line is a short-term outcome; expected value describes the price of repeating the wagers.

Consecutive losses expose the hidden requirement

With the same starting line, consecutive losses produce bets of 5, 6, 7, 8, and so on. After k consecutive losses, total loss is:

$$Cumulative\ Loss = \frac{k(k+9)}{2}\ units$$

After 10 straight losses:

$$\frac{10(10+9)}{2}=95\ units$$

The next required bet is 15 units. With a $10 unit, the player has lost $950 and must place $150 next merely to continue the written plan.

This growth is slower than the doubling pattern of Martingale, but “slower” does not mean safe. The line can expand for many mixed win-loss sequences, and a win often removes only two numbers while earlier losses may have added several large entries.

Table limits can break the line before bankroll does

Assume a $10 unit and a table maximum of $100 on the chosen even-money wager. The consecutive Labouchere bets are:

$50, $60, $70, $80, $90, $100, $110...

After six consecutive losses, the player is down $450. The next required bet is $110, which exceeds the table maximum. The system cannot follow its own rule even if the player still has cash.

Minimums create a second problem. A player may choose a very small written unit, but if the first-plus-last sum falls below the posted minimum, the line must be rescaled. That increases every later exposure.

Roulette probabilities remain unchanged

On a European wheel, an even-money wager wins on 18 pockets and loses on 19 because zero is neither side. On an American double-zero wheel, it wins on 18 and loses on 20 because both 0 and 00 lose.

The written line does not enter these probabilities. It cannot remove zero, improve the payout, or make a win more likely after a loss.

Resetting the line does not erase the loss

Players sometimes abandon a difficult line and start a fresh one because the new numbers look manageable. The bookkeeping resets, but the session result does not. If the first line is abandoned at -40 units and the second line clears for +10, the combined result is still -30 units.

Starting a new line can therefore hide cumulative damage. A separate record of bankroll, total action, and maximum drawdown is necessary. The written sequence should never be allowed to replace the actual cash ledger.

The same problem appears when a player reduces the unit after losses or increases it after a completed line. Those changes create a new strategy path. Results from altered lines cannot be used as evidence that the original Labouchere rules succeeded.

Why completed lines are persuasive

Many result patterns clear the line and deliver the target. This creates a high proportion of sessions that appear controlled, especially when the target is modest and play stops immediately after completion.

The risk is concentrated in the unfinished lines:

• the line becomes long;
• the largest numbers remain at the end;
• required stakes rise;
• stopping locks in a loss larger than the planned profit;
• restarting creates another layer of action;
• the player treats completion as an obligation rather than a choice.

A system can therefore produce frequent small completed targets while leaving occasional large failures. Win frequency alone does not establish positive expectation.

Total action is the cost driver

For any fixed wheel and wager type:

$$Expected\ Loss = Total\ Action\times House\ Edge$$

A player using Labouchere with $2,500 of total even-money action has approximate expected loss of:

• European wheel: $2,500 × 2.70% = $67.57;
• American wheel: $2,500 × 5.26% = $131.58.

The line can change how that action is distributed across spins and how volatile the session feels. It cannot change the expected loss per dollar wagered.

What a simulation should report

A useful simulation should not report only the percentage of lines completed. It should also define the wheel, zero rule, starting line, unit, bankroll, table minimum and maximum, stop conditions, and treatment of incomplete lines. At minimum, it should show:

• average final result;
• median final result;
• total action;
• maximum wager;
• maximum drawdown;
• probability of bankroll failure;
• probability of table-limit failure;
• distribution of abandoned-line losses.

Without those outputs, a high completion rate can conceal rare but severe failures. The system’s expected value remains negative because every underlying roulette wager remains negative. Simulation can reveal the shape of the risk; it cannot turn the progression into an advantage.

Nevada’s official Roulette Rules of Play describe the wheel, permitted wagers, and settlement structure. Labouchere is not a special roulette wager under those rules; it is a private method of choosing how much to place on ordinary wagers.

Common execution errors

Labouchere also creates practical errors that flat betting avoids:

• crossing out the wrong entries;
• appending the intended bet rather than the amount actually lost;
• forgetting how a zero result affects the chosen roulette wager;
• changing unit size mid-line;
• treating a partial payout or special rule as ordinary even money;
• placing or changing bets after the dealer has closed betting;
• losing track of total action while focusing on the remaining line.

At a live table, the dealer is responsible for legal wagers and correct settlement, not for maintaining a player’s sequence. A calculation dispute about the line does not alter the posted wager or payout.

Labouchere versus reverse Labouchere

Reverse Labouchere uses the line in the opposite emotional direction: wins add exposure and losses remove entries, depending on the chosen rules. It is sometimes marketed as a way to press a winning streak rather than recover losses. That changes the distribution of bet sizes, but it still leaves the roulette edge unchanged.

Because several versions circulate under similar names, any analysis must state exactly what happens after a win, after a loss, and when one number remains. Mixing rules from different versions makes both the target and bankroll calculation meaningless.

What a disciplined comparison looks like

To evaluate Labouchere honestly, compare it with flat betting over the same wheel, number of spins, starting bankroll, stopping rules, and unit size. Record:

• final result;
• total action;
• largest wager;
• maximum drawdown;
• number of table-limit failures;
• percentage of lines completed;
• size of losses when a line is abandoned.

Looking only at completed lines selects the successes and hides the unresolved exposures.

The clean conclusion is that Labouchere is a bet-sizing and bookkeeping system, not an advantage strategy. It can shape the path of wins and losses, but the roulette house edge continues to apply to every dollar wagered. Compare it with Martingale, D’Alembert, and Betting Progressions Compared, then test session paths with the variance simulator.