Probability of Streaks for Roulette
Long runs of one colour dey common. Dem also no mean anytin for di next spin.
Five reds one afta di oda: 2.7%. Ten: 0.074%. For session of 200 spins, run of six of one colour dey show most of di time.
How often losing streak dey come
Even-money bet for single-zero wheel dey lose 19 spins out of 37. Each row na di chance sey one particular run of spins go lose finish, and how much Martingale player, wey dey double afta every loss, don lose by dat time.
| losses wey follow each oda | chance | about 1 for | wetin you don lose so far | di next bet |
|---|---|---|---|---|
| 1 | 51.4% | 2 | $1 | $2 |
| 2 | 26.4% | 4 | $3 | $4 |
| 3 | 13.5% | 7 | $7 | $8 |
| 4 | 7.0% | 14 | $15 | $16 |
| 5 | 3.6% | 28 | $31 | $32 |
| 6 | 1.8% | 55 | $63 | $64 |
| 7 | 0.94% | 106 | $127 | $128 |
| 8 | 0.48% | 207 | $255 | $256 |
| 9 | 0.25% | 403 | $511 | $512 |
| 10 | 0.13% | 784 | $1,023 | $1,024 |
| 11 | 0.065% | 1527 | $2,047 | $2,048 |
| 12 | 0.034% | 2974 | $4,095 | $4,096 |
| 13 | 0.017% | 5792 | $8,191 | $8,192 |
| 14 | 0.0089% | 11279 | $16,383 | $16,384 |
| 15 | 0.0046% | 21965 | $32,767 | $32,768 |
Di chance na 19/37 wey you multiply by imself once for every loss, counting from one particular spin. Long session get plenty spins where dat kind run fit start, so for one evening e dey happen more often well well pass wetin one row show.
One Run
Di probabilitymathsProbabilityDi chance of one outcome, from 0 to 1 or from 0% to 100%. One number: 1/37 = 2.70%.Read di whole tin → sey di next k spins go all be redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → na (18/37)ᵏ. Two: 23.7%. Five: 2.7%. Ten: 0.074%, about 1 out of 1,350. Twenty: 1 out of 1.8 million. Twenty-six, di Monte Carlo record: 1 out of 137 million. Dis ones na di probabilities of one particular run wey start for one particular spin, and na di question wey almost nobody dey really ask.
Inside One Session
Di question wey pipo mean na “run of k go show somewhere for my session?”, and di answer big well well pass (18/37)ᵏ becos session of 200 spins dey give almost 200 places where run fit start. Run of six or more of di same colour, red or blackbetBlackEven-money bet for di 18 black numbers. For French: noir.Read di whole tin →, dey show for session of 200 spins about 94% of di time; eight or more about 47%; ten or more about 14%. If you name di colour before, di chances go fall, to about 75%, 27% and 7%. Player wey sit down for one evening go see long streaks every evening.
Why Martingale Dey Worry
Every negative 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 → get run length wey go break am: six losses from $10 base with $500 limit, ten for FibonaccistrategyFibonacciProgression wey dey follow di sequence 1, 1, 2, 3, 5, 8, 13…: one step forward afta you lose, two steps back afta you win.Read di whole tin →, five for tripling on dozensbetDozen1–12, 13–24 or 25–36, e dey pay 2:1.Read di whole tin →. Di probability of losing run of six for even money na (19/37)⁶ ≈ 1.8% for every attempt, and session of 200 spins get maybe 100 attempts, so di break dey come for about four sessions out of five of dat length. Di probability of streaks for roulette na di reason why di cliff no rare.
Hot and Cold Numbers
Di same arithmetic dey apply to straight upsbetStraight upOne single number, e dey pay 35:1.Read di whole tin →. For 300 spins, di count wey we expect for every number na 8.1; di standard deviationmathsStandard deviationSquare root of variance; di normal distance between one result and di expected value.Read di whole tin → na about 2.8. Some number go show 15 hits and some go show 2 by chance alone, every time. Results board wey dey highlight “hot” numbers dey highlight di top of normal distribution. Di Lab dey explain how many spins e go take before hot numbermythHot numberNumber wey don show pass wetin dem expect recently. For fair wheel e no dey predict anytin.Read di whole tin → go mean anytin (thousands, from one wheel wey you identify).
Independence Na Di Whole Point
Run of ten reds no dey make black more likely, and e no dey make red more likely either. Both na 18/37. Streaks na property of random sequences, and if dem no dey, na dat one go be suspicious: sequence of 200 spins wey no get run longer pass four go be much stronger evidence sey pipo don tamper with am pass one wey get run of twelve.
Formula
Questions wey pipo dey ask
26 blacks for Monte Carlo on 18 August 1913 na di one wey popular. Dem report 32 reds one afta di oda for one American casino for 1943. Both of dem fit happen by chance, if you look how many spins casinos dey run.
Na 18/37 either way. Bet wetin your plan tok, or no bet at all.
No. Bias dey show as some pocketsequipmentPocketOne of di 37 small boxes wey get number for di rotor. Dem also dey call am: slot, cup.Read di whole tin → wey dey hit more often over thousands of spins, no be as runs of one colour.
Play with Maths
Try dem by yourself first. Each solution dey folded; open am when you ready.
How many spins of European wheel e dey take, for average, before every one of di 37 numbers don come out at least once?
Show di solution
Answer About 155 spins for average; di median na 147.
Count di wait one new number at a time. Di first spin dey always show new number. Once you don see 1 number, 36 pocketsequipmentPocketOne of di 37 small boxes wey get number for di rotor. Dem also dey call am: slot, cup.Read di whole tin → out of 37 na new, so di next new number dey take 37/36 spins for average. Once you don see k numbers, e dey take 37/(37 − k).
Add di waits: 37/37 + 37/36 + 37/35 + … + 37/1 = 37 × (1 + 1/2 + 1/3 + … + 1/37) ≈ 37 × 4.20 = 155.5 spins.
Most of am na di end. Di last number alone dey take 37 spins for average, and di last five about 85. For American wheel, di same sum dey give about 161. Dis na di coupon collector problem; di same mean dey follow from di standard calculation.
For 37 spins of European wheel, one spin for each pocket, how many different numbers you suppose expect to see?
Show di solution
Answer About 23.6. Around one third of di numbers, about 13, no go come out at all.
Take one number, like 17. For each spin, e no dey come out with chance 36/37, so e dey miss all di 37 spins with chance (36/37)³⁷ ≈ 0.363.
Every number get di same chance, so for average 37 × 0.363 ≈ 13.4 numbers dey miss and 37 − 13.4 ≈ 23.6 dey come out, plenty of dem two times or more.
Roulette players dey call dis di law of thirds. E no be law, na just di arithmetic of repeats: for run wey long reach di wheel, about one third of di numbers no dey show, for fair wheel like any oda one. For 38 pockets di share na (37/38)³⁸ ≈ 0.363 too; for big wheel e dey settle near 1/e ≈ 0.368.
One friend show you two lists of 50 colours, redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → or blackbetBlackEven-money bet for di 18 black numbers. For French: noir.Read di whole tin →, and e tok sey each one come from wheel wey dem remove di zeroswheelZeroDi 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 →. One dey change colour 25 times and im longest run na six. Di oda one dey change 35 times and e no dey ever run pass three. Which one dem make up?
Show di solution
Answer Di second one. Real list dey change about 24 or 25 times. List wey change 35 times dey show about once for 500 lists, and list wey no get run wey long pass three, about two times for 100.
Each new colour dey either repeat di last one or change, half and half, so 49 chances to change dey give 24.5 changes for average, plus or minus about 3.5. Thirty-five na three of dose steps above di average; fair wheel dey produce dat many changes about once for 500 lists.
Di runs dey tell di same story. Real list of 50 dey get run of four or more about 98% of di time, and im longest run dey usually be five or six.
Pipo wey dem ask to invent random list dey change too often, becos run of four or five dey look like pattern for dem. Taylor (2021) show di same sign for lists of coin tosses wey students make up: real randomness get streak pass wetin pipo expect.
One book of casino stories tok about one player wey im number come out five times one afta di oda, and e call am record. How rare dat one be for European wheel? And how often e suppose happen somewhere for one casino wey get 20 tables, each one dey spin 40 times every hour, day and night?
Show di solution
Answer One number wey you name, five times one afta di oda: 1 for 69 million. Any number wey come out four more times afta e first show: 1 for 1.87 million. Dat casino dey spin about 7 million times every year, so e suppose see run of five three or four times every year.
One number wey you name dey come out with chance 1/37 for each spin, so five one afta di oda get chance (1/37)⁵ = 1 for 69,343,957. But nobody dey name di number before. Anytin wey come out first, di next four spins must repeat am: (1/37)⁴ = 1 for 1,874,161.
20 tables × 40 spins × 24 hours × 365 days = 7,008,000 spins every year. Each spin fit start run, so di casino dey expect 7,008,000/1,874,161 ≈ 3.7 runs of five every year.
No official record of dis kind dey, and nobody need am: rare event for one table na normal tin across one casino, and pipo dey tell di stories about di table where e happen. Di story, for Goldman (2007), happen for double-zero wheel, where di figures na 1 for 79 million and 1 for 2.09 million.
Small laugh
Long runs na how random dey look.
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.