| Bet | Covered pockets | Payout (to 1) | Win chance (European) | Win chance (American) |
|---|---|---|---|---|
| Straight Up | 1 | 35 : 1 | 2.7027% | 2.6316% |
| Red / Black | 18 | 1 : 1 | 48.6486% | 47.3684% |
| Dozen | 12 | 2 : 1 | 32.4324% | 31.5789% |
| Six Line | 6 | 5 : 1 | 16.2162% | 15.7895% |
| Split | 2 | 17 : 1 | 5.4054% | 5.2632% |
Let F be covered (winning) pockets and T be total pockets. The win probability is p = F / T and the loss probability is q = 1 − p.
Assuming independent spins, the probability of no wins is q^N. Therefore, P(≥1 win) = 1 − q^N.
P(k) = C(N,k) · p^k · q^(N−k), where C(N,k) is the combination count.
With stake S and payout R:1, win profit is S·R and loss profit is −S. So, E[profit] = p·(S·R) + q·(−S).
- Select the wheel type you want to analyze.
- Pick a bet type that matches your table rules.
- Enter your stake per spin and the planned spin count.
- Optionally set k to evaluate exact win frequencies.
- Press Calculate to view results above the form.
- Use Download CSV or PDF to save the output.
- Probabilities assume a fair wheel with equal pocket likelihoods.
- Payout schedules are standard; individual tables can differ.
- Expected profit is a long‑run average, not a prediction for one session.
- House edge shown is implied by payout and wheel pocket count.
Wheel type and pocket count
European roulette uses 37 pockets, while the American wheel uses 38. For any bet covering F pockets, the single‑spin win chance is F/T. Even‑money bets cover 18 pockets, so the win rate is 18/37 = 48.6486% in Europe and 18/38 = 47.3684% in America. That small change compounds across many spins.
Coverage versus payout balance
Roulette payouts are designed so higher coverage pays less. A straight bet covers 1 pocket and pays 35:1, giving p = 1/37 = 2.7027% on a European wheel. A dozen covers 12 pockets and pays 2:1, so p = 12/37 = 32.4324%. On the American wheel, the five‑number basket covers 5 pockets, so p = 5/38 = 13.1579%. This calculator keeps the coverage and payout linked, so you can compare bets on equal stake inputs.
Session probability for N spins
Players often care about “Will I hit at least once?” The model uses P(≥1 win) = 1 − (1 − p)^N. With an even‑money bet on a European wheel, p ≈ 0.486486. Over 10 spins, P(≥1 win) is about 99.91%. Over 3 spins, it is about 86.41%. The curve rises quickly, then flattens.
Expected profit and house edge
Expected profit per spin combines outcomes: E = p·(S·R) + (1 − p)·(−S), where S is stake and R is payout. For even‑money bets, E is negative because zeros create extra losing pockets. With S = 10 on red/black in Europe, E ≈ −0.2703 per spin, or −2.703% of stake. The implied house edge is about 2.70% for European and 5.26% for American, regardless of common bet type.
Volatility and realistic swings
Variance matters when bankroll is limited. The calculator reports standard deviation per spin and for N spins, using √N scaling. Larger stakes or longer sessions increase dispersion, even when expectation stays negative. Two sessions with the same settings can end far apart, so use the risk metrics to size stakes conservatively.
Exact‑k results for targets
If you enter k, the binomial formula estimates the chance of exactly k wins: C(N,k)·p^k·(1−p)^(N−k). This is helpful for planning hit‑rate goals, like “15 wins in 30 spins” on an even‑money bet. Compare k scenarios across wheels to see how the extra zero shifts frequencies.
1) European vs American: what changes?
American roulette adds a second zero pocket, increasing total pockets from 37 to 38. That lowers every win probability for the same coverage and increases the implied house edge from about 2.70% to about 5.26%.
2) Does picking “lucky numbers” improve odds?
No. A straight‑up bet always covers one pocket, so its probability is 1/T regardless of which number you choose. The chosen pocket field is included for reporting and record‑keeping only.
3) What does “win at least once” represent?
It is the chance that you record one or more wins across N independent spins. The calculator uses 1 − (1 − p)^N, which is easier to interpret than tracking every possible win/loss sequence.
4) How is expected profit calculated?
For stake S and payout R:1, a win earns S·R and a loss costs S. Expected profit per spin is p·(S·R) + (1−p)·(−S). Multiply by N for session expectation.
5) Why can results be positive if expectation is negative?
Expectation is a long‑run average. Short sessions can land above or below that average due to variance. Even with negative edge, random sequences can produce winning streaks, especially over small N.
6) What should I do with standard deviation?
Use it to gauge typical swings around expected profit. Higher standard deviation means outcomes are more spread out. Over N spins, dispersion grows roughly with √N, so longer sessions widen the range of plausible results.