The strip that repeats
Play roulette for a few minutes and the history strip starts to look as if it is saying something. Say a 17 lands and, four spins later, lands again. The instinct is to read a pattern into it: a hot number, a wheel in a groove, a sign.
The repeat is the expected case. The strip keeps the last 16 spins, and a full strip contains a repeated number 97.82% of the time; 16 different numbers, the thing that would actually be unusual, turn up about once in 46 strips. Just over one spin in three, 35.49%, lands on a number already on the strip. This is the birthday problem, which asks how many people a room needs before two of them probably share a birthday (23, to most people’s surprise). It needs one input, the number of equally likely results, and for our games we read that number straight from the engine source.
Three questions that sound alike
Three different questions hide inside one feeling. Has any result come up twice, anywhere in the session? Has a result come up twice in a row? Has a number you picked come up at all? Only the first is the birthday problem, and the answers are far apart.
| QUESTION | DICE | ROULETTE |
|---|---|---|
| Any repeat | 119 rolls | 8 spins |
| Back-to-back | 6,932 rolls | 27 spins |
| Your number | 6,931 rolls | 26 spins |
Calculation for this article: the first session length at which each chance passes 50%, with N = 9,999 results for signed-in dice and 37 pockets for roulette.
On dice a repeat of something comes 58 times sooner than a hit on a chosen number, 119 rolls against 6,931; on roulette it is 8 spins against 26. Back-to-back needs one result more than a chosen number, because the first result of a session has nothing to repeat. The chosen-number question has its own roulette article, Roulette hit rates, priced; this piece is about the other two.
The numbers
On a game with N equally likely results, a session of n results has no repeat only if every result is new, which happens with chance (1 − 1 ÷ N) × (1 − 2 ÷ N) × … × (1 − (n − 1) ÷ N). One minus that product is the chance of at least one repeat, and it can be computed exactly.
| CHANCE | DICE ROLLS | SPINS |
|---|---|---|
| 50% | 119 | 8 |
| 75% | 167 | 11 |
| 90% | 215 | 13 |
| 95% | 245 | 15 |
| 99% | 303 | 18 |
| 99.9% | 370 | 21 |
Calculation for this article: the first session length that reaches each level, N = 9,999 for signed-in dice and 37 for roulette. One roll fewer than 119 is 49.997%; one spin fewer than 8 is 45.34%.
| SPINS | REPEAT | ROLLS | REPEAT |
|---|---|---|---|
| 5 | 24.57% | 24 | 2.72% |
| 8 | 55.68% | 50 | 11.55% |
| 10 | 73.71% | 100 | 39.15% |
| 13 | 90.89% | 150 | 67.48% |
| 16 | 97.82% | 200 | 86.51% |
| 20 | 99.83% | 300 | 98.92% |
Calculation for this article: roulette spins on the left, signed-in dice rolls on the right. 16 spins fill the roulette strip and 24 rolls the dice strip.
On average the first repeat arrives on spin 8.31 and on roll 125.99. The dice figure is the surprise: there are 9,999 possible results, yet 119 rolls, 1.2% of them, already make a repeat more likely than not, and 303 rolls, 3%, make it 99% likely.
Why it comes so soon
The usual guess is half the results: about 5,000 rolls, or 18 or 19 spins. It is wrong because a repeat is made by pairs of results, not by results. A session of n results holds n(n − 1) ÷ 2 pairs, each matching with chance 1 ÷ N, so the coincidences on offer grow with the square of the session while the session grows in a straight line.
| SESSION | PAIRS | MATCHES |
|---|---|---|
| 8 spins | 28 | 0.76 |
| 37 spins | 666 | 18 |
| 119 rolls | 7,021 | 0.70 |
| 10,000 rolls | 49,995,000 | 5,000 |
Pairs = n(n − 1) ÷ 2. Expected matching pairs = pairs ÷ N, with N = 37 for roulette and 9,999 for signed-in dice.
At the even-odds points just under one match is expected, and the count then runs away: a full wheel’s worth of 37 spins expects 18. The half-the-results guess, 4,999.5 rolls or 18.5 spins, overshoots the true point about 42 times on dice and 2.3 times on roulette.
The classic version runs on 365 birthdays: 23 people share one 50.73% of the time, 22 people 47.57%. In a 1989 paper on coincidences in the Journal of the American Statistical Association, the statisticians Persi Diaconis and Frederick Mosteller give a rule of thumb for c equally likely categories: about 1.2√c draws for an even chance of a match and 2.5√c for 95%. Here 1.2√9,999 is 119.99 against the exact 119, and 1.2√37 is 7.3 against the exact 8. As far as they know, they add, the earliest mention of the problem is by Richard von Mises in 1939.
With a large enough sample, any outrageous thing is likely to happen.Persi Diaconis and Frederick Mosteller, “Methods for Studying Coincidences”, 1989
They call it the law of truly large numbers, and illustrate it with a lottery. The New York Times had put a woman who won the New Jersey lottery twice on its front page as a 1 in 17 trillion long shot. That figure, they note, is the chance that someone holding one ticket in each of exactly two draws wins both, and she had bought many tickets, many times. The question that fits the surprise is whether anyone in the United States wins twice: two Purdue statisticians put that at better than even within seven years, and better than 1 in 30 within the four months between her wins.
Twice in a row
The repeat that feels most pointed is the immediate one, and its chance is plain. The next spin repeats the last with chance 1 in 37, 2.70%, and the next signed-in roll with chance 1 in 9,999, 0.0100%: exactly the chance of any other particular result. A 17 after a 17 is no rarer than a 4 after a 17. It is only the one outcome the eye has a name for.
| SESSION | ROULETTE | DICE |
|---|---|---|
| 10 | 21.85% | 0.09% |
| 37 | 62.71% | 0.36% |
| 100 | 93.36% | 0.99% |
| 1,000 | > 99.99% | 9.51% |
| 10,000 | > 99.99% | 63.21% |
Calculation for this article: 1 − (1 − 1 ÷ N)ⁿ⁻¹ over n spins or rolls. Even odds arrive at spin 27 and roll 6,932, 90% at spin 86 and roll 23,024, and 99% at spin 170 and roll 46,046.
The two kinds of repeat live on different scales. A 119-roll session has a back-to-back repeat only 1.17% of the time, against 50.59% for any repeat; an eight-spin session has one 17.45% of the time, against 55.68%. A long session on a small wheel still collects them: 100 spins expect 2.68, and 93.36% of 100-spin sessions hold at least one. Three in a row is 1 in 1,369 for any given three spins and passes even odds at spin 976; on dice it is 1 in 99,980,001. Losing runs are a different question again, priced in The longest losing streak, priced.
Hot numbers and near misses
Numbers that repeat a lot are just as normal. Thirty-seven spins give each pocket one landing on average, yet some pocket lands three or more times 98.38% of the time and four or more times 51.00%. In 100 spins some pocket must land at least three times, because 37 pockets cannot absorb 100 spins at two apiece.
| TIMES | 37 SPINS | 100 SPINS |
|---|---|---|
| 3+ | 98.38% | 100% |
| 4+ | 51.00% | > 99.99% |
| 5+ | 10.96% | 99.97% |
| 6+ | 1.63% | 91.91% |
| 7+ | 0.20% | 53.96% |
Calculation for this article: exact counts over every way 37 equally likely pockets can fill 37 or 100 spins. Some pocket reaches three landings with even odds by spin 21.
Seen from the other side, 37 spins show 23.57 different pockets on average and leave 13.43 untouched; all 37 different has a chance of about 1.3 × 10⁻¹⁵. A missing number and a hot one are the two halves of the same distribution.
On dice, with 9,999 results, an exact repeat on the 24-roll strip is rare: 2.72% of full strips hold one, and the next roll matches the strip 0.240% of the time. What the eye catches instead is the near miss. Two of 24 rolls share the same whole number, 42.17 and 42.80 say, 95.05% of the time, and that chance passes 50% by the 13th roll.
Diaconis and Mosteller name four principles that, once coincidences with an evident cause are set aside, account for large numbers of the rest: hidden causes; psychology, including memory and perception; the law of truly large numbers; and multiplicity of endpoints, which includes treating nearly alike events as the same. The whole-number match is that last principle in miniature. Widen “the same” from one result in 9,999 to about one in 100, and a 2.72% coincidence becomes a 95.05% one.
No memory, same price
None of this means the game is doing anything. A roulette spin is a function of the server seed, your client seed and the round’s nonce, and of nothing else; the nonce moves on by one with every bet, so each round reads a fresh hash. The server seed’s SHA-256 hash is shown before you bet, and your client seed enters every result.
So a repeat carries no information. After 17 lands, even twice, the next spin lands on 17 with chance 1 in 37, like any other number. A bet on “that number again” is an ordinary straight-up bet: ×36 total return on a 1-in-37 chance, a 97.30% return and a 2.70% cost, the same as every bet on the wheel. On dice the multiplier is (1 − house edge) ÷ win chance whatever the last roll was; Dice, priced exactly lists it for every target.
What the arithmetic offers is a way to stop being surprised. A 16-spin strip shows a repeat 97.82% of the time, and a 100-spin session holds a back-to-back pair 93.36% of the time. The gambler’s fallacy reads a missing number as due, and the hot-number habit reads a repeated one as live. The hot hand debate is about people, whose form can change; a hash has no form. The repeat was always coming.
Thirty-seven pockets, no memorySingle zero, every bet returns 97.30%, every spin drawn from a committed server seed and your client seedRoulette →
How many roulette spins until a number repeats?
On a single-zero wheel of 37 equally likely pockets, some number has repeated with 55.68% chance by spin 8, 90.89% by spin 13 and 99.33% by spin 18; the first repeat comes on spin 8.31 on average. Calculations for this article.
How many dice rolls until a result repeats?
On signed-in Betkyo dice, one of 9,999 equally likely results per roll, some result has repeated with 50.59% chance by roll 119, 90.15% by roll 215 and 99.02% by roll 303. The first repeat comes on roll 125.99 on average.
What are the odds of the same number twice in a row in roulette?
1 in 37, 2.70%, on each spin: the chance of any other number. At least one back-to-back repeat has happened with even odds by spin 27, and 93.36% of 100-spin sessions hold one. Three in a row is 1 in 1,369 for any given three spins.
Does a repeated number mean the wheel is hot or rigged?
No. Repeats are what independent draws produce: 97.82% of 16-spin strips hold one, and in 37 spins some pocket lands four or more times 51.00% of the time. Each spin comes from the server seed, your client seed and the round’s nonce, with no earlier result as an input.
What is the birthday problem?
It asks how many draws from N equally likely outcomes it takes before two probably match. For 365 birthdays the answer is 23 people (50.73%). The rule of thumb is about 1.2√N; the exact even-odds answers for 9,999 and 37 outcomes are 119 and 8.
Does betting the number that just came up change the odds?
No. It is a straight-up bet: ×36 total return on a 1-in-37 chance, a 97.30% return and a 2.70% house edge, the same as every other bet on the wheel.
- Betkyo engine source: deriveRouletteSpin(), ROULETTE_ORDER, ROULETTE_RED and rouletteMultOf() in roulette/derive.ts; the 16-spin history strip in rouletteGame.tsx; RouletteEngine.derivePocket() and U64SeedStream.intBelow(37) in the house service
- Betkyo engine source: OriginalBaseDiceService.kt getResult() (9,999 results, 0.01–99.99) and acquireTicketSeed; SeedPairRandom.nextIntBelow(); PfHashUtils.normalize() (N_BITS = 52); the 24-roll history strip in diceGame.tsx; consumeNonce() in _shared/demoLocal.ts
- Diaconis, P. and Mosteller, F. (1989), “Methods for Studying Coincidences”, Journal of the American Statistical Association 84(408): 853–861 — abstract and sections 7.1–7.2
- Calculation for this article: exact products ∏(1 − k ÷ N) for N = 9,999, 10,001, 37 and 365; pair counts n(n − 1) ÷ 2 and matches per pair 1 ÷ N; back-to-back and chosen-number chances 1 − (1 − 1 ÷ N)ⁿ⁻¹ and 1 − (1 − 1 ÷ N)ⁿ; exact counts for the most landings by one pocket, three in a row, coverage and the dice whole-number match; cross-checked by simulation
