Variance
Why straight up and red bet wey get di same 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 feel like different games.
Variance dey measure how far results dey spread around di expected value. For every unit wey you stake, e be about 34 for straight up and about 1 for red.
Di Formula for One Bet
For bet wey dey pay W to 1 with probabilitymathsProbabilityDi chance of one outcome, from 0 to 1 or from 0% to 100%. One number: 1/37 = 2.70%.Read di whole tin → p, di result for every 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 → na +W with probability p and −1 with probability 1 − p. 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 → na pW − (1 − p) = −1/37. 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 E[X²] − EV² = pW² + (1 − p) − (1/37)². For straight upbetStraight upOne single number, e dey pay 35:1.Read di whole tin →: (1/37)(1225) + 36/37 − 0.0007 ≈ 34.08. For redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin →: 18/37 + 19/37 − 0.0007 ≈ 0.999.
Wetin E Mean for One Session
Variances dey add up across independent spinsmathsIndependent spinsEach spin get di same probabilities no matter wetin happen before. Na di base for every calculation for dis site.Read di whole tin →, so for N spins with fixed stake, di variance of di total na N × stake² × σ², and di standard deviationmathsStandard deviationSquare root of variance; di normal distance between one result and di expected value.Read di whole tin → na σ × stake × √N. 100 spins with $10: di standard deviation na about $584 for straight ups and $100 for red, around di same expected −$27. One player go see results wey dey swing by hundreds; di oda one go see small slow movement. Both of dem dey pay di same price.
Variance and Systems
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 → no dey change di EV for every unit wey you bet, but e dey change both di amount wey you bet and di shape of di distribution. 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 → dey reduce di variance of most sessions (plenty small wins) and e dey add fat tail (di rare big loss); di total variance across all sessions high pass di one of flat bettingstrategyFlat bettingTo bet di same amount every spin. Na di baseline wey dem dey compare every progression with.Read di whole tin → becos of dat tail. ParolistrategyParoliPositive progression: double afta you win, start again afta you lose or afta three wins one afta di oda. Dem also dey call am: reverse Martingale.Read di whole tin → dey do di opposite. To choose system na to choose variance profile; di mean no dey di menu.
- p10 −$500
- p25 −$500
- median −$500
- p75 +$40
- p90 +$1,120
Flat Betting Strategy for RouletteParoli Strategy for Roulette (Reverse Martingale)
Why E Matter to Player
Variance na wetin dey make short sessions hard to predict and long ones easy to predict. With σ = $100 and EV = −$27 for 100 spins, di edge na one quarter of di noise and winning session na normal tin. For 10,000 spins di EV na −$2,700 and σ na $1,000; now di edge don reach almost three times di noise, and winning run na three-sigma event. Variance na di player friend for short run and di house friend for long run.
Variance of Combination
For many bets for di same spin, di outcomes dey connect (hit on 17 na also hit on blackbetBlackEven-money bet for di 18 black numbers. For French: noir.Read di whole tin → and di second dozenbetDozen1–12, 13–24 or 25–36, e dey pay 2:1.Read di whole tin →), so di variances no dey just add up. Di Bet Builder tool dey calculate di exact distribution for any set of chipsmoniChipToken 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 → by listing all 37 outcomes, and na di only correct way to do am.
Formula
Questions wey pipo dey ask
No. Every bet get di same expected value. Higher variance mean sey di swings around am go wide pass.
Even-money betsbetEven-money betRed/black, odd/even, high/low. 18 numbers, e dey pay 1:1.Read di whole tin →, with σ ≈ 1 for every unit. E go even low pass with la partageruleLa partageFrench rule: when zero come out, dem go return half of even-money stake. E dey cut di house edge for dose bets into half, to 1.35%.Read di whole tin →, wey also dey improve di mean.
Di expected profit na di same for every dollar; variance only dey affect how di casino income go dey come in lumps, and casino get thousands of players to smooth am.
Play with Maths
Try dem by yourself first. Each solution dey folded; open am when you ready.
Two players each dey bet $1 every spin for 100 spins for European wheel, one dey always bet on redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin →, di oda dey always bet on 17. Who likely pass to dey ahead for di end?
Show di solution
Answer Di player on 17: 50.9%, against 35.5% for red. Both dey lose $2.70 for average.
Red: di player dey ahead with 51 wins or more out of 100, each with probabilitymathsProbabilityDi chance of one outcome, from 0 to 1 or from 0% to 100%. One number: 1/37 = 2.70%.Read di whole tin → 18/37. Di binomial distribution dey give 35.5%, and exactly 50 wins, wey be break even, anoda 7.7%.
Number 17: each win dey return $36, so two wins (−$28) no reach and three (+$8) reach. Di chance of three hits or more for 100 spins for 1/37 na 50.9%.
Di straight-up player dey ahead more often, but di results dey come in lumps: one hit more or less dey move di result by $36, and di players wey dey behind dey usually far behind. Bollman (2024) give di American figures, 26.5% and 49.2%.
Small laugh
Di same expected loss. One of dem dey shout am for your ear.
Sources
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.