Mathematics · Main idea · 7 min

Expected Value for Roulette

Roulette expected value na di average result of one bet if you fit repeat am forever. For European roulette e dey di same for every standard bet: you dey lose 1/37 of each stake, for average, for every spin.

For simple language

Imagine sey you put di same $10 bet on red one million times. Some spins go win $10, some go lose $10. Add every result togeda and divide by one million: you go get about −$0.27. Dat number na di expected value of roulette bet. E no be wetin happen for any one spin; na di long-run average for each spin.

Short answerEV = probability to win × amount wey you win − probability to lose × amount wey you lose. For every standard roulette bet, e dey come out as −2.70% of di stake.

Di formula for di EV of roulette bet

EV = p · W − (1 − p) · S
p = n/37 · W = S·(36/n − 1) · n = numbers wey di bet cover · S = stake

Put di roulette payout rule, 36/n − 1, inside am, and n go cancel. Na di whole story of di house edge be dat for one line:

EV = (n/37)·S·(36/n − 1) − ((37−n)/37)·S
= S·(36 − n − 37 + n)/37
= −S/37

Try am: calculator for roulette expected value

Pick how many numbers di bet go cover. Di probability and di payout go change; di expected value no go change.

1 number wey e cover · e dey pay 35:1stake $10

lose 97.3%

p(win)1/37 = 2.70%

win dey pay+$350

loss dey cost−$10

EV = p·W − (1−p)·S0.0270·$350 − 0.9730·$10

expected value−$0.27 for each spin · −2.70%

How e be as time dey go

Average result for each spin for $10 bet on red, as di spins dey add up. For di beginning, luck dey control; di running average dey settle go −$0.27 as di sample dey grow. Na dis long-run convergence di house dey depend on.

Spin afta spin, di running average go stop to swing and e go settle small below zero.

Wrong ideas wey pipo get

“Negative EV mean sey I go lose every session.”

No. E mean sey di average over plenty sessions na loss. Each session dey different well well, and more so for bets wey get high variance like straight up.

“Di casino need long time before e win.”

Di edge of di casino dey work for every spin; wetin di long run dey do na to make di result easy to predict. For one session, luck na di bigger force.

“Some bets get better EV pass oda ones.”

No for European roulette, except one: under la partage or en prison, even-money bets get half of di normal house edge.

Small laugh

$No be wetin happen for one spin. Na di average of all of dem.$

No be wetin happen for one spin. Na di average of all of dem.