The five-number bet (top line) in roulette: the worst odds on the table
The top line covers 0, 00, 1, 2 and 3, exists only on American tables, and pays 6 to 1. It wins 13.16% of spins (1 in 7.6), which leaves a 7.89% house edge: the worst bet on either layout. The single-zero first four (0, 1, 2, 3) pays 8 to 1 at the usual 2.70%.
Where the chip goes
Distinct placements: One, and only on a double-zero layout. The single-zero equivalent is the first four (0, 1, 2, 3), which pays 8 to 1.
Odds and payout on both wheels
The win chance is pockets covered over pockets on the wheel. True odds are what the bet would have to pay for the house to make nothing; the gap between true odds and the actual payout is the edge.
| Top line, American | First four, European | |
|---|---|---|
| Numbers covered | 5 of 38 | 4 of 37 |
| Win chance | 13.16% | 10.81% |
| As odds | 1 in 7.6 | 1 in 9.3 |
| Pays | 6 to 1 | 8 to 1 |
| True odds | 6.60 to 1 | 8.25 to 1 |
| House edge | 7.89% | 2.70% |
| Expected result, $10 bet | −$0.79 | −$0.27 |
| Std. deviation per $1 | $2.366 | $2.795 |
| Edge outweighs luck after | 898 spins | 10,692 spins |
The last row is where the expected loss grows past one standard deviation of luck; the house edge page derives it and compares every bet.
How often it hits
The chance of at least one hit in n spins is 1 − (1 − p)n. On average the bet comes in once every 7.6 spins on an American wheel. Half of all players see a hit within 5 spins, nine in ten within 17, and 99 in 100 within 33.
| Within | Top line, American | First four, European |
|---|---|---|
| 1 spin | 13.16% | 10.81% |
| 5 spins | 50.61% | 43.56% |
| 10 spins | 75.60% | 68.15% |
| 25 spins | 97.06% | 94.27% |
| 37 spins | 99.46% | 98.55% |
| 50 spins | 99.91% | 99.67% |
| 100 spins | >99.99% | >99.99% |
| 200 spins | >99.99% | >99.99% |
Droughts
A drought is a run of spins without a hit. The chance that the next L spins all miss is (1 − p)L. Seeing a drought that long somewhere in a long session is far more likely, and the last column is computed exactly by dynamic programming over the current run of misses, not by simulation.
| Misses in a row | Next L spins | Somewhere in 1,000 spins |
|---|---|---|
| 15 | 12.049% | >99.99% |
| 23 | 3.898% | 99.69% |
| 38 | 0.470% | 45.86% |
| 61 | 0.018% | 2.26% |
| 91 | <0.001% | 0.03% |
Chance of being ahead after n spins
Flat stakes, the same bet every spin. After n spins with w wins the net is w × 7 − n stakes, and w is binomial, so these are exact sums. Breaking even counts as not ahead. The chances of winning page compares bets side by side and has a calculator for any session length.
| Spins | Top line, American | First four, European | Expected, $10 bets |
|---|---|---|---|
| 1 | 13.16% | 10.81% | -$0.79 |
| 10 | 38.64% | 29.54% | -$7.89 |
| 37 | 35.90% | 37.09% | -$29.21 |
| 100 | 33.46% | 39.67% | -$78.95 |
| 500 | 22.29% | 41.05% | -$394.74 |
| 1,000 | 15.35% | 36.05% | -$789.47 |
| 10,000 | 0.05% | 16.36% | -$7,894.74 |
Why 7.89%
Every other payout on the layout follows one formula: a bet on k numbers pays 36/k − 1 to 1. For five numbers that is 6.2 to 1, which the house rounds down to 6. A $1 top line therefore returns $7 on 5 of 38 spins: $35 back per $38 staked, a loss of $3, or 3/38 = 7.89%. The usual American edge is 2/38. The extra point and a half is the rounding.
Cover the zeros another way
On a double-zero table, a split on 0–00 plus a street on 1–2–3 covers the same five numbers at the normal 5.26%. On a single-zero table the first four (0, 1, 2, 3) pays 8 to 1 at 2.70%. Both are the same bet in substance and cheaper in fact.
Questions
What does the top line pay in roulette?
6 to 1. A winning $10 top line bet pays $60 and the $10 stake comes back too. For the bet to be fair on a American wheel it would have to pay 6.60 to 1; the difference is the house edge.
How often does the top line win?
13.16% of spins on an American wheel, or 1 in 7.6. On average it hits once every 7.6 spins, and there is a better-than-even chance of at least one hit within 5 spins.
Is the top line a good bet?
No. At 7.89% it carries the highest house edge of any bet on either layout, half again the 5.26% of every other American bet.
Related
- Roulette house edge: where 2.70% and 5.26% come from
- What are the odds of red 10 times in a row in roulette?
- What are your chances of winning at roulette after n spins?
- How many spins until every roulette number comes up?
- Every bet’s odds and payout, both wheels
- Does any betting system work? 20 systems on the same seeds
- Study: European versus American roulette, the risk distribution
Every bet on the layout
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