Mathematics of Roulette · For practice · 7 min

Expected Loss

Expected value times di total wey you bet: wetin di session dey cost for average.

Short answer

Expected loss = total wey you bet × 1/37. Stake $10, 100 spins: $1,000 wey you bet, $27 expected loss. Progressions dey bet more, so dem dey lose more.

for every $1 wey you bet
$0.027
$10 × 100 spins
$27
$10 × 50/h
$13.50 for one hour
for one year, every week
≈ $700

Turnover, No Be Bankroll

Expected lossmathsExpected lossExpected value times di total moni wey you bet; di average amount wey you go lose for one session or one period.Read di whole tin → dey depend on how much moni pass through di table, no be on how much you bring. $100 bankrollmoniBankrollDi total moni wey you set aside for play. Bankroll management na to decide how much of am you go risk for each bet and for each session.Read di whole tin → wey you bet again many times fit carry $2,000 of turnovermoniTurnoverDi total moni wey you bet for one session or one period, including winnings wey you bet again. Dem also dey call am: handle, action.Read di whole tin → and $54 of expected loss; $10,000 bankroll wey you bet once on redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → dey carry $10,000 of turnover and $270. Di income of di casino na toll on traffic, and di traffic na every chipmoniChipToken wey stand for one fixed amount, wey you put on top di layout to make bet. Dem also dey call am: check.Read di whole tin → for every spin.

Di Formula

E[loss] = Σ stakes / 37. For person wey dey bet flat, dat one na stake × spins / 37. For progressionstrategyProgressionAny rule wey dey change di stake based on di results wey don pass. Negative progressions dey raise am afta losses, positive ones afta wins.Read di whole tin → na di average stake × spins / 37, and di average stake high pass di base, becos stakes dey rise during losing stretches. MartingalestrategyMartingaleProgression wey dey double di bet afta every loss. Check di Strategy section to see wetin e dey do and wetin e no dey do.Read di whole tin → from $10 base with $500 limit dey average about $28 for every spin over long session, so e dey lose almost three times as much as flat $10 betting, without to change di odds of any single bet.

playaverage stakewetin you bet for 100 spinsexpected loss
Flat $10$10$1,000$27
D’Alembert, $10 base≈ $14≈ $1,400≈ $38
Fibonacci, $10 base≈ $17≈ $1,700≈ $45
Martingale, $10 base, $500 limit≈ $28≈ $2,800≈ $75
Flat $10, table wey get la partage$10$1,000$13.50
Averages wey we simulate. Di edge for every dollar na 2.70% for every row except di last one.

Flat Betting Strategy for Roulette

Speed

Live tables dey run 40–60 spins for one hour, casino floors 30–50, speed tables 90 or more, RNG tables as fast as you fit click. With di same stake, speed na di biggest single tin wey dey move expected loss for every hour: if you move from slow floor table go RNG table wey you play for 200 spins for one hour, di cost of di evening go multiply by four without any change for di odds.

Expected loss for every hour with $10 for every spin$ for every hour
Casino floor · 40/h
$10.81
Live table · 55/h
$14.86
Speed live · 100/h
$27.03
RNG · 200/h
$54.05
$ for every hour
027.03$54.05
stake × spins for every hour / 37. Speed na di biggest tin wey dey decide wetin one hour go cost.

Live Roulette and RNG Roulette

For One Year

Regular player wey dey play two hours every week with $10 for live table of 50 spins dey turn over $52,000 for one year and e dey pay about $1,400 for expected loss for European wheel, $700 for French one, $2,700 for American one. Di arithmetic dey boring, but di conclusion no boring: di choice of wheel and table worth more pass any system by factor of infinity, since systems worth exactly zerowheelZeroDi one green pocket wey no even-money, dozen or column bet dey cover; na im be di source of di house edge. Dem also dey call am: single zero, 0.Read di whole tin →.

Expected loss for one year: 2 h every week, $10 for every spin, 50 spins/h$
French (even money)
$702
European
$1,405
American
$2,737
Speed table wey be European, 100/h
$2,811
$
01405.50$2,811

Game Wey Worth More Pass Any Bank

Di St. Petersburg game, wey Nicolaus Bernoulli pose for 1713, dey toss coin until e land heads and e dey pay $2 if dat one take one toss, $4 if two, $8 if three, e dey double every time. Every possible length dey add $1 to di expected valuemathsExpected valueDi average result for each bet for di long run. −1/37 of di stake for every standard European bet.Read di whole tin →, so for paper di game worth infinite amount. No bank fit pay infinite prizes, and once dem cap di prizes for wetin di bank get, di value go collapse: with bank of $1 million di game worth about $21, with $1 billion about $31. Di Martingale na di same game wey you see from di oda side. E dey win one unitmoniUnitDi base stake wey system or player dey work with. Dem dey describe progressions in units so dat dem go fit work for any table.Read di whole tin → almost every cycle and e dey lose everytin for one rare long run; without limit, ever bigger wins for pay for di long runs, and with limit dem no go pay, and na why di table limittableTable limitDi minimum and maximum bet wey dem allow. Na di maximum dey stop progressions wey dey double.Read di whole tin → dey decide how Martingale go fail but no be whether e go fail.

di bank fit pay reachfair price of di game
$100$7.56
$10,000$14.22
$1 million$20.91
$1 billion$30.86
Di fair price na di expected payout when dem pay any prize wey pass di bank as di whole bank.
Di St. Petersburg game: e worth infinite amount for paper, about $21 once di bank get $1 million.

Table Limits for Roulette

Expected Loss No Be Di Result

Any single session go different from im expected loss, often by pass di expected loss imself. For 100 even-money spins with $10, di standard deviationmathsStandard deviationSquare root of variance; di normal distance between one result and di expected value.Read di whole tin → na $100 against di expected −$27; winning session no strange. Di expected loss na di price; di variancemathsVarianceOne measure of how wide di results dey spread around di expected value. E high for straight up, e low for even-money bets.Read di whole tin → na di packaging; di result for one night na sample of one.

Formula

E[loss] = Σ stakes / 37
E[loss for every hour] = stake × spins for every hour / 37

Questions wey pipo dey ask

Di casino dey take 2.70% of my buy-in?

No. Na 2.70% of everytin wey you bet. If you bet your winnings again, you don bet dem again.

Smaller stake dey always cheaper?

For every hour, yes, with di same speed. For every target wey you reach, no: see risk of ruinmathsRisk of ruinDi probability sey you go lose di whole bankroll before you reach one target or one number of spins.Read di whole tin →.

How comps dey work?

Casinos dey estimate your expected loss from your average bet and di time wey you play, and dem dey return small part of am as comps. Dem dey rebate part of di price, dem no dey give you edge.

Small laugh

Turnover × 1/37. Di clock dey inside di formula.

Turnover × 1/37. Di clock dey inside di formula.

Sources

  1. Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Chapter 13, roulette.
  2. Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013.
  3. Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed. CRC Press, 2021.
we check am 2026-09-28model: European wheel, single zerohow we work am ↗where we get am ↗