How many spins until every roulette number comes up?
On average 155.5 spins on a European wheel (160.7 on American) before every number has come up at least once. In any 37 spins, about 13.4 numbers don't appear at all.
How long until every number comes up
This is the coupon collector’s problem. With s numbers already seen, the next new one takes 37/(37 − s) spins on average. Add those up from 0 to 36 and you get 37 × (1 + 1/2 + … + 1/37) = 155.46 spins. On an American wheel it is 160.66. The last few numbers do most of the damage: the final one alone takes 37 spins on average.
| Spins | Distinct numbers (avg) | Unseen (avg) | All 37 seen |
|---|---|---|---|
| 37 | 23.6 | 13.4 | <0.1% |
| 74 | 32.1 | 4.9 | 0.2% |
| 100 | 34.6 | 2.4 | 6.4% |
| 150 | 36.4 | 0.6 | 53.1% |
| 200 | 36.8 | 0.2 | 85.5% |
| 300 | 37.0 | 0.0 | 99.0% |
Half of all sessions see every number by spin 147, and nine in ten by spin 215. Probabilities in the last column come from an exact Markov chain over the count of numbers seen.
Sleeping numbers
A “sleeper” is a number that hasn’t come up for a while. The chance a particular number misses n spins in a row is (36/37)n. In any window of 37 spins, more than a third of the wheel sleeps.
| Asleep for | European | 1 in | American | 1 in |
|---|---|---|---|---|
| 37 spins | 36.29% | 2.8 | 37.28% | 2.7 |
| 50 spins | 25.41% | 3.9 | 26.36% | 3.8 |
| 74 spins | 13.17% | 7.6 | 13.90% | 7.2 |
| 100 spins | 6.46% | 15 | 6.95% | 14 |
| 150 spins | 1.64% | 61 | 1.83% | 55 |
| 200 spins | 0.42% | 240 | 0.48% | 207 |
| 300 spins | 0.03% | 3,713 | 0.03% | 2,982 |
An overdue number is not due
A number that has slept 200 spins comes up next with probability 1/37, the same as one that hit on the last spin. The wheel doesn’t keep score. The same holds in reverse for “hot” numbers: on a fair wheel a hot run is variance, not signal. Both ideas are tested as betting systems, on the hot-number method and betting against a streak. Telling real bias from noise takes thousands of logged spins; the wheel-bias sample-size study works out how many.
Repeats come early
The flip side, and the birthday problem in disguise: some number repeats far sooner than intuition says. The chance that the first k spins are all different is 37/37 × 36/37 × … × (38 − k)/37.
| Spins | European | American |
|---|---|---|
| 3 | 8.0% | 7.8% |
| 4 | 15.4% | 15.0% |
| 5 | 24.6% | 24.0% |
| 6 | 34.8% | 34.0% |
| 7 | 45.3% | 44.4% |
| 8 | 55.7% | 54.6% |
| 9 | 65.3% | 64.2% |
| 10 | 73.7% | 72.7% |
| 12 | 86.5% | 85.7% |
Related
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- Study: European versus American roulette, the risk distribution