The system and the wheel
The martingale is the oldest betting system in the casino and the easiest to explain. Bet one unit on an even-money chance. If it wins, start again. If it loses, double the stake and bet again, and keep doubling until a win arrives. Because each stake is larger than all the previous losses combined, the first win always leaves the player exactly one unit ahead. It feels less like gambling than like collecting.
That is all the input the calculation needs. A single bet wins with probability 18/37 and loses with probability 19/37. A series of at most n bets fails only if all n lose, which happens with probability (19/37)ⁿ, and when it fails the player has lost 1 + 2 + 4 + … + 2ⁿ⁻¹ = 2ⁿ − 1 units. Every other series ends one unit up. Nothing below is simulated.
One series, priced
| MAX BETS | BANKROLL NEEDED | SERIES BUSTS | EXPECTED STAKED | EXPECTED RESULT |
|---|---|---|---|---|
| 3 | 7 | 1 in 7.4 | 3.08 | −0.08 |
| 5 | 31 | 1 in 28 | 5.28 | −0.14 |
| 7 | 127 | 1 in 106 | 7.59 | −0.21 |
| 10 | 1,023 | 1 in 784 | 11.31 | −0.31 |
| 12 | 4,095 | 1 in 2,974 | 13.95 | −0.38 |
| 14 | 16,383 | 1 in 11,279 | 16.75 | −0.45 |
| 17 | 131,071 | 1 in 83,295 | 21.22 | −0.57 |
Units of the first stake. A series wins +1 unless every bet loses, in which case it loses the whole bankroll. Expected result divided by expected staked is −2.70% on every row.
Read across any row and the shape of the system is plain. A deeper martingale busts far less often — ten doublings fail once in 784 series, seventeen once in 83,295 — and the price of that reliability is the bankroll, which doubles with every row, and the size of the loss when it comes. The expected result of a series gets worse, not better, as the system gets safer: −0.31 units for ten bets, −0.57 for seventeen. The reason is in the last two columns. A deeper series stakes more on average, and the wheel takes its 1/37 of everything staked.
The algebra is short enough to show. A series wins +1 with probability 1 − (19/37)ⁿ and loses 2ⁿ − 1 with probability (19/37)ⁿ, so its expected result is 1 − (38/37)ⁿ. Since 38/37 is greater than one, that number is negative for every n and grows with n. The expected amount staked in the series works out to 37 × ((38/37)ⁿ − 1). Divide one by the other and n cancels: the series loses exactly one thirty-seventh of what it stakes, whatever its depth.
The session that usually wins
| MAX BETS | BANKROLL | TARGET | REACHES TARGET | BUSTS FIRST | EXPECTED RESULT | EXPECTED STAKED |
|---|---|---|---|---|---|---|
| 7 | 127 | +100 | 38.8% | 61.2% | −13.3 | 493 |
| 10 | 1,023 | +100 | 88.0% | 12.0% | −28.7 | 1,062 |
| 10 | 1,023 | +1,000 | 27.9% | 72.1% | −172.8 | 6,393 |
| 17 | 131,071 | +1,000 | 98.8% | 1.2% | −570.2 | 21,096 |
Units of the first stake. The session stops at the target or at the first busted series. Expected result divided by expected staked is −2.70% on every row.
This is why the martingale has admirers. A player with 1,023 units who plays until they are 100 up walks away a winner 88 times in 100. A player with 131,071 units who aims for 1,000 walks away a winner almost 99 times in 100. Those are true statements, and a player who has used the system a few times has probably only ever seen the winning branch.
The expected result column is the other half of the same truth. The rare losing sessions give back so much that the average session loses 28.7 units in the first case and 570 in the second. The system does not lower the house edge; it converts a steady small loss into frequent small wins and an occasional catastrophic loss, and the catastrophe is sized precisely so that the average stays at 2.70% of everything staked.
Streaks are not rare
| RUN OF LOSSES | WITHIN 100 SPINS | WITHIN 1,000 SPINS |
|---|---|---|
| 7 or more | 36.3% | — |
| 10 or more | 5.6% | 46.2% |
| 12 or more | — | 15.0% |
| 17 or more | — | 0.57% |
Exact, by dynamic programming over the current run length. A loss is any of the 19 losing pockets, zero included.
Ten losses in a row sounds like something that would make the news. Over a thousand spins — a long evening at a fast table — it happens close to half the time. Seven in a row inside a hundred spins is better than a one-in-three chance. The streak that ends a martingale is not an outlier the system is unlucky to meet; for any bankroll a person actually has, it is the expected end of a long enough session. The famous black run at Monte Carlo in 1913 is the story people remember, but the arithmetic of streaks says the ordinary ones are the ones that matter.
The limit that ends it
Even an unlimited bankroll would not rescue the system, because tables have limits. The doubling ladder runs into the maximum stake long before it runs into infinity.
At seventeen bets the numbers are the last row of the first table: 131,071 units risked to win one, a bust once in 83,295 series, and an expected loss of 0.57 units per series. A table limit is often described as the casino’s defence against the martingale. It is better described as a floor under how deep the system can go, and the table shows that depth was never the thing standing between the player and a profit.
Single-zero roulette, every spin verifiableEven chances pay ×2 on 18 of 37 pockets — the same 2.70% whatever system you bringRoulette →
Does the martingale strategy work in roulette?
It wins small amounts often and loses large amounts rarely, and the average is always a loss. On a single-zero wheel every martingale, at any depth, loses exactly 2.70% of the total amount staked. With 10 doublings a series busts once in about 784 series and loses 0.31 units on average.
How much bankroll does the martingale need?
2ⁿ − 1 units to survive n bets. Ten bets need 1,023 units, seventeen need 131,071. On this engine’s demo table (minimum 1, maximum 100,000) seventeen bets is the most the limit allows.
How likely is a losing streak of 10 in roulette?
On an even-money bet on a single-zero wheel, a run of at least 10 losses appears somewhere in 100 spins 5.6% of the time and somewhere in 1,000 spins 46.2% of the time. The next spin after such a run still wins 18 times in 37.
If a martingale session usually ends in profit, how can it lose on average?
Because the losing sessions are much larger. With 1,023 units and a +100 target, 88.0% of sessions reach the target, but the other 12.0% lose most of the bankroll, and the expected result of a session is −28.7 units.
Is there a betting system that beats the house edge?
No system that only changes stake sizes can. Each spin is independent and returns 36/37 of its stake on average, so any sequence of stakes returns 36/37 of their total. Stake sizing changes the variance and the risk of ruin, not the edge.
- Betkyo engine source: roulette/derive.ts (floor(u × 37), even chances ×2, single zero, 2.7% edge)
- Betkyo engine source: demoContext() in _shared/coinGame.ts (demo minBet 1, maxBet 100000); real-money limits from coinMinBet and coinMaxBet
- Exact calculations written for this article: series bust probabilities and expected results, session outcomes by first-step analysis, and losing-run probabilities by dynamic programming over run length
