Probability for Roulette
Every roulette probability na di number of pockets wey you count, divided by 37.
p = numbers wey you cover / 37. One number: 2.70%. Eighteen: 48.65%. Di chance of one particular sequence na di single-spin probabilities wey you multiply togeda.
Single Spins
Each of di 37 pocketsequipmentPocketOne of di 37 small boxes wey get number for di rotor. Dem also dey call am: slot, cup.Read di whole tin → get di same chance, so di chance to win for roulette for one spin na just counting. Bet on four numbers dey win 4/37 of di time, about 10.8%. Notin dey for single-spin probabilitymathsProbabilityDi chance of one outcome, from 0 to 1 or from 0% to 100%. One number: 1/37 = 2.70%.Read di whole tin → pass dat one, and na why you fit analyse di whole game for one napkin.
Many Spins
Spins no dey depend on each oda, so di probabilities of one sequence dey multiply. Two redsbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → one afta di oda: (18/37)² = 23.7%. Red then 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 →: (18/37)(1/37) = 1.3%. Seventeen two times one afta di oda: (1/37)² = 0.073%, about one out of 1,369. Di order matter for one particular sequence, but e no matter for count: “red then blackbetBlackEven-money bet for di 18 black numbers. For French: noir.Read di whole tin →” na 25.0%, but “one red and one black for two spins, any order” na 50.0%.
At Least Once
Use di complement. Di chance sey straight upbetStraight upOne single number, e dey pay 35:1.Read di whole tin → no go hit for 37 spins na (36/37)³⁷ = 36.3%, so di chance of at least one hit na 63.7%, no be di “certainty” wey many players dey expect from 37 chances of 1 out of 37. Di same calculation tok sey one number get about 1 out of 3 chance sey e no go show at all for 37 spins, and na why di results board dey always show some “cold” numbers.
Di Waiting Time Wey We Expect
For average, straight-up number dey show once every 37 spins, but di waiting dey change well well: di median wait na about 25 spins (half of di time e dey come sooner) and one wait out of twenty dey pass 110 spins. Waiting times for geometric process get long tail, and player wey “sabi” sey one number don due afta 60 spins dey describe tin wey dey happen one time out of five.
Common Mistakes
To add probabilities for events wey fit happen togeda (red or odd no be 36/37; na 26/37, becos ten numbers na both). To multiply for events for di same spin (red and odd for one spin na 10/37, no be 18/37 × 18/37). To treat past spins as part of di sequence wey you dey bet on now: afta ten reds, di chance of di eleventh na 18/37, no be (18/37)¹¹.
Wetin Be Di Odds for Roulette, and How Dem Take Different from Probability?
Casinos dey quote payoutsmoniPayoutWetin bet wey win dey return, wey dem dey quote as “X to 1” on top di stake wey dem return.Read di whole tin → as odds. One single number dey win with probability of 1/37, so di odds to win for roulette on one single number na 36 to 1 against. Payout of 35 to 1 dey short by 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 →, and na dat one be di house edgemathsHouse edgeDi part of each stake wey di casino dey keep for average: 2.70% for European roulette.Read di whole tin → for im original form. To convert: odds against = (1 − p)/p; probability = 1/(odds + 1).
Formula
Questions wey pipo dey ask
(1/37)² = 1/1369 ≈ 0.073%.
36/37 = 97.3%. Zero na di rest.
All 37 na 1/37. Seventeen na di number wey pipo dey bet pass, and dat one na fact about pipo.
Play with Maths
Try dem by yourself first. Each solution dey folded; open am when you ready.
You dey bet on redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin →, spin afta spin. Wetin be di chance sey your fifth win go come exactly for di eighth spin? And sey you go get five wins inside di first eight spins?
Show di solution
Answer 12.9%, about 1 for 8, for di fifth win for spin eight. 33.4% for five wins by dat time.
Di fifth win for spin eight mean two tins: spin eight win, and di first seven spins get exactly four wins and three losses, for any order. 35 ways dey to put four wins among seven spins.
So di chance na 35 × (18/37)⁵ × (19/37)³ = 35 × 0.02725 × 0.1354 = 12.9%.
For five wins by spin eight, add di same kind of term for di fifth win for spins five, six and seven: 2.7% + 7.0% + 10.8% + 12.9% = 33.4%. Statisticians dey call dis di negative binomial distribution: how many tries one fixed number of successes dey take. Taylor (2021) work di same question for American wheel, 12.2%.
Small laugh
Count pockets. Divide by 37. Multiply for sequences. Finish.
Sources
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.