Roulette Strategy Math
Every roulette strategy is ultimately a few formulas. If you understand the math once, you can evaluate any new system on first sight - no testing required.
This page collects the key formulas with worked examples. For interactive tests of every system, use the Roulette Strategy Tester.
1. Probability of any bet
For any bet covering k numbers on a wheel with N pockets, the hit probability is k/N.
| Bet | Numbers (k) | European (N=37) | American (N=38) |
|---|---|---|---|
| Straight up | 1 | 2.70% | 2.63% |
| Split | 2 | 5.41% | 5.26% |
| Street | 3 | 8.11% | 7.89% |
| Corner | 4 | 10.81% | 10.53% |
| Six line | 6 | 16.22% | 15.79% |
| Dozen / Column | 12 | 32.43% | 31.58% |
| Red / Black / Odd / Even / Low / High | 18 | 48.65% | 47.37% |
2. Payout-to-true-odds gap
For a bet covering k numbers, the fair payout would be (N−k)/k to 1. The actual payout is fixed by the casino. The gap is the house edge.
- Straight up on European: fair = 36:1, actual = 35:1, gap = 1 unit per 37 spins = 2.70%.
- Red/black on European: fair = 19:18, actual = 1:1, gap = 1 unit on the zero = 2.70%.
- Straight up on American: fair = 37:1, actual = 35:1, gap = 2 units per 38 spins = 5.26%.
3. Expected value per spin
For stake s, hit probability p, payout multiplier m:
EV = p × (s × m) + (1 − p) × (−s)
Example: $10 on red, European. EV = (18/37 × 10) + (19/37 × −10) = −$0.27.
4. Expected loss across a session
Expected loss = total wagered × house edge
European: house edge = 1/37 ≈ 2.7027% for all standard bets.
American: house edge = 2/38 ≈ 5.263% for all standard bets except five-number bet (7.89%).
5. Variance and standard deviation
Variance measures the spread of possible outcomes. For a single $10 bet on red on European:
Var = p(1−p)(payout difference)2 = (18/37)(19/37)(20)2 ≈ 99.93 dollars-squared.
Standard deviation per spin ≈ $10. Across n spins, total SD scales with sqrt(n). For 100 flat $10 red bets, total SD ≈ $100. So results clustering within $200 either side of expected loss is normal.
6. Probability of N losses in a row
For a bet with loss probability q, P(N losses) = qN. Red on European has q = 19/37 = 51.35%.
| Streak length | P(streak) | Expected once per N spins |
|---|---|---|
| 3 | 13.55% | ~7 |
| 5 | 3.58% | ~28 |
| 7 | 0.94% | ~106 |
| 10 | 0.13% | ~771 |
| 13 | 0.018% | ~5,560 |
7. Martingale absorption capacity
Bankroll B and base bet b: maximum absorbed losses n = largest integer where b × (2n+1 − 1) ≤ B.
For B=$1,000, b=$10: 10 × (27 − 1) = $1,270 > $1,000, but 10 × (26 − 1) = $630 ≤ $1,000. So n = 5 (the 6th loss is affordable, the 7th is not - total absorbed = 6 losses before busting on the 7th).
8. Risk of ruin (flat betting)
For flat betting on an even-money bet with house edge e, starting with K units and betting until ruin or double-up:
P(ruin) = 1 − ((1−e)/(1+e))K
Example: K = 100 units, e = 0.027 (European). P(ruin before doubling) ≈ 49.4%. So about half the time flat betting reaches double bankroll before going broke.
9. ROI vs edge
ROI = (Final − Initial) / Total wagered × 100
Over enough sessions, mean ROI converges to −house edge. Single-session ROI is dominated by variance and tells you almost nothing about edge.