Walkthrough: A Labouchère Line

The plan

Write 1 2 3 4 on a card. Each spin, stake the sum of the two end numbers, on redbetRedEven-money bet on the 18 red numbers. French: rouge.Read full →. A win crosses both numbers off; a loss adds the stake to the right-hand end. The line clears when nothing is left, which nets the original total, 10 unitsmoneyUnitThe base stake a system or a player works in. Progressions are described in units so they apply at any table.Read full →, $50. A line needs only one win for every two losses to clear, because each win removes two numbers and each loss adds one. Red wins 48.65% of spins, which is more than the one-in-three required, so most lines clear.

Line 1, spins 1–8

1 2 3 4 → stake 5 (1+4): L, line 1 2 3 4 5. Stake 6: W, line 2 3 4. Stake 6: L, line 2 3 4 6. Stake 8: W, line 3 4. Stake 7: L, line 3 4 7. Stake 10: W, line 4. Stake 4: L, line 4 4. Stake 8: W, cleared. Eight spins, four wins and four losses. Net +$50, exactly the target, with the stake never above $50. It works like a small machine.

Line 2, spins 9–39

1 2 3 4 again. Eleven losses in the first twelve spins. Each loss appends a number that is the sum of two others, so the line grows at both length and size: after twelve spins it read 2 3 4 5 6 7 8 9 10 12 14 16 18 and the stake was $100. By spin 23 it had 16 numbers and the cycle stood at −$850. Wins cross off the big end numbers, which helps: nine wins in the next fifteen spins cut the line to 12 14 16 18 and the cycle to −$250 by spin 38, with the stake touching the $200 limit on the way. Then zerowheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full →: the $150 stake lost, the line read 12 14 16 18 30, and the next stake, $210, was over the limit. The cycle stood at −$400. We stopped. The line would have cleared for +$50 eventually, if “eventually” arrived before the limit and the bankrollmoneyBankrollThe total amount set aside for play. Bankroll management is deciding how much of it to risk per bet and per session.Read full → did not.

The accounts

Total wagered $3,770 over 39 spins against $195 flat. Expected lossmathExpected lossExpected value times total amount wagered; the average amount lost over a session or a period.Read full → on $3,770 is $102; actual −$350, about a third of a standard deviationmathStandard deviationSquare root of variance; the typical distance of a result from the expected value.Read full → below, because at these stakes one standard deviation is about $700. A runaway LabouchèrestrategyLabouchèreCancellation system: write a line of numbers, bet the sum of the two ends, cross them off after a win, add the lost stake after a loss.Read full → line is the heaviest turnovermoneyTurnoverThe total amount wagered over a session or period, counting re-bet winnings. Also: handle, action.Read full → of any common system, because each loss raises the stake by the amount of the loss and the line has no ceiling but the table’s.

What the session shows

Labouchère has the rare property that it usually works, which is why it was a conversation piece at Victorian dinners and why it still sells. The cleared lines are small and frequent; the runaway line is large and rare, and its size is open-ended. The expected loss per dollar is −2.70% in both halves, and the system’s only effect is to decide which half you are in on a given night.

Questions