Roulette Strategy Overview

The one theorem that matters

Expected valuesmathExpected valueThe average result per bet over the long run. −1/37 of the stake for every standard European bet.Read full → add. If every bet you place has an expected value of −1/37 of its stake, then any sequence of bets, however cleverly chosen from past results, has an expected value of −1/37 of the total staked. A system can decide how much to bet and when; it cannot make a single bet positive, so it cannot make the sum positive. This is not an empirical finding that a better system might overturn. It is arithmetic.

Two families

Negative progressionsstrategyProgressionAny rule that changes the stake based on previous results. Negative progressions raise it after losses, positive after wins.Read full → raise the stake after a loss: MartingalestrategyMartingaleProgression that doubles the bet after each loss. See the Strategy section for what it does and does not do.Read full → (double), FibonaccistrategyFibonacciProgression that follows the sequence 1, 1, 2, 3, 5, 8, 13…: one step forward after a loss, two steps back after a win.Read full → (sum of the last two), D’Alembert (add one), 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 → (sum of the ends of a line). They produce frequent small wins and rare large losses. Positive progressions raise the stake after a win: ParolistrategyParoliPositive progression: double after a win, reset after a loss or after three wins in a row. Also: reverse Martingale.Read full → (double up to three times) and its relatives. They produce frequent small losses and rare large wins. Flat bettingstrategyFlat bettingWagering the same amount every spin. The baseline every progression is compared against.Read full → is the control group.

What a system actually changes

Three things: how often you finish a session ahead, how big the typical win is, and how big the rare disaster is. Martingale finishes about 80% of short sessions with a small profit and the other 20% with a loss that wipes out ten of those wins. Paroli finishes most sessions slightly down and a few with a large gain. Flat betting sits in between. In every case the mean, over many sessions, is −2.70% of everything wagered, and the progressions wager more.

Reading the strategy pages

Each page shows the progression, a seeded bankrollmoneyBankrollThe total amount set aside for play. Bankroll management is deciding how much of it to risk per bet and per session.Read full → chart with a dozen sessions, a histogram of 2,000 simulated sessions you can adjust, and the probabilitymathProbabilityChance of an outcome, from 0 to 1 or 0% to 100%. One number: 1/37 = 2.70%.Read full → of ruin at a given table limittableTable limitMinimum and maximum bet allowed. The maximum is what stops doubling progressions.Read full →. Judge systems by those shapes and by whether you can live with the worst case, not by the single lucky run in an advertisement.

Roulette Table Limits

What actually helps

Four things change your expected lossmathExpected lossExpected value times total amount wagered; the average amount lost over a session or a period.Read full → and none of them is a betting pattern: a lower stake, fewer spins, a la partageruleLa partageFrench rule: when zero hits, half of an even-money stake is returned. Halves the house edge on those bets to 1.35%.Read full → table (halving the edge on even-money betsbetEven-money betRed/black, odd/even, high/low. 18 numbers, pays 1:1.Read full → to 1.35%), and a stop rule decided before you sit down. Together they can cut the expected cost of an evening by three quarters. No progression can cut it by a cent.

• Stake: expected loss is proportional to it.
• Spins: expected loss is proportional to them. Slow tables are cheaper per hour.
• Table rules: la partage or en prisonruleEn prisonFrench rule: when zero hits, an even-money bet is held for the next spin instead of lost. Halves the house edge on those bets.Read full → halve the even-money edge.
• Stop rule: the only part of any “system” that works, because it reduces total turnovermoneyTurnoverThe total amount wagered over a session or period, counting re-bet winnings. Also: handle, action.Read full →.

En PrisonLa Partage

Why systems persist

They work most of the time. A Martingale player who plays twenty evenings will win on sixteen of them and remember those. The four losing evenings cost more than the sixteen winning ones made, but memory does not keep a ledger. Add the fact that systems give a player something to do between spins, and the appeal is understandable. The math is not in dispute; the psychology is doing the selling.

Questions