Walkthrough: A Martingale Session
Ten pounds on red, doubled after every loss, at a £500 table. Spin by spin, including the spin that ends it.
A simulated Martingale session from a £1,000 bankroll. Forty-one small wins, then six losses in a row and a refused seventh bet. The session ends down £220 after being up £410, which is what the system does.
The plan
Bet £10 on redbetRedEven-money bet on the 18 red numbers. French: rouge.Read full →. After a loss, double. After a win, back to £10. Table limittableTable limitMinimum and maximum bet allowed. The maximum is what stops doubling progressions.Read full → £500, so the stakes are 10, 20, 40, 80, 160, 320 and then a £640 bet that the table will refuse. Six consecutive losses break the system; each attempt has (19/37)⁶ ≈ 1.8% of doing so, and a 100-spin session contains about 50 attempts. Expected: roughly a 60% chance of breaking within 100 spins, and a loss of £630 when we do.
Spins 1–30
Ordinary MartingalestrategyMartingaleProgression that doubles the bet after each loss. See the Strategy section for what it does and does not do.Read full →: strings of one to three losses, each ended by a win that nets +£10. By spin 30 we have had 17 wins, each closing a cycle, and are up £170. The largest stake so far was £80 after three losses. It feels like a machine for printing £10 notes.
Spins 31–60
More of the same, with one four-loss cycle (stakes to £160, then a win) that briefly put £310 on the table and felt like courage. After 60 spins: 41 cycles completed, up £410, and the bankrollmoneyBankrollThe total amount set aside for play. Bankroll management is deciding how much of it to risk per bet and per session.Read full →’s low point since spin 30 was £1,040. The histogram of every Martingale session looks like this for most of its length.
Spins 61–66
L L L L L L. Stakes 10, 20, 40, 80, 160, 320: £630 lost. The next bet would be £640; the limit is £500. The system is broken. We can bet £500 (and if it wins, still be £130 down on the cycle) or reset to £10 and accept the loss. Six blacksbetBlackEven-money bet on the 18 black numbers. French: noir.Read full → or zeroswheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full → in a row has probabilitymathsProbabilityChance of an outcome, from 0 to 1 or 0% to 100%. One number: 1/37 = 2.70%.Read full → 1.8% per cycle. We had run about 45 cycles. The chance of reaching this point somewhere in 45 cycles was about 55%. It was not bad luck; it was the median.
- p10 −£1,000
- p25 −£1,000
- median −£1,000
- p75 +£790
- p90 +£1,690
The accounts
Up £410 after 60 spins, down £220 after 66. Total wagered: about £2,100 over 66 spins, against £660 for a flat £10 bettor. Expected lossmathsExpected lossExpected value times total amount wagered; the average amount lost over a session or a period.Read full → on £2,100 of turnovermoneyTurnoverThe total amount wagered over a session or period, counting re-bet winnings. Also: handle, action.Read full → is £57; our actual −£220 is about a third of a standard deviationmathsStandard deviationSquare root of variance; the typical distance of a result from the expected value.Read full → below that, which is unremarkable for a system whose distribution has a long left tail. Had we stopped at spin 60 with a stop-winstrategyStop-winA pre-set profit target at which you end the session.Read full → of £400, we would have left up £410, and the next session would have carried the same 1.8% per cycle.
What the session shows
Martingale converts a 48.65% bet into a sequence that wins 98% of its cycles and loses 63 unitsmoneyUnitThe base stake a system or a player works in. Progressions are described in units so they apply at any table.Read full → on the other 2%. It does not change the expected loss per pound; it multiplies the pounds. The flat bettor beside us wagered a third as much and lost, on average, a third as much. The player who runs this system for a year will win most evenings and lose the year.
How often a losing streak comes
An even-money bet on a single-zero wheel loses 19 spins in 37. Each row is the chance that a given run of spins all lose, and what a Martingale player, who doubles after every loss, has lost by then.
| losses in a row | chance | about 1 in | lost so far | next bet |
|---|---|---|---|---|
| 1 | 51.4% | 2 | £1 | £2 |
| 2 | 26.4% | 4 | £3 | £4 |
| 3 | 13.5% | 7 | £7 | £8 |
| 4 | 7.0% | 14 | £15 | £16 |
| 5 | 3.6% | 28 | £31 | £32 |
| 6 | 1.8% | 55 | £63 | £64 |
| 7 | 0.94% | 106 | £127 | £128 |
| 8 | 0.48% | 207 | £255 | £256 |
| 9 | 0.25% | 403 | £511 | £512 |
| 10 | 0.13% | 784 | £1,023 | £1,024 |
| 11 | 0.065% | 1527 | £2,047 | £2,048 |
| 12 | 0.034% | 2974 | £4,095 | £4,096 |
| 13 | 0.017% | 5792 | £8,191 | £8,192 |
| 14 | 0.0089% | 11279 | £16,383 | £16,384 |
| 15 | 0.0046% | 21965 | £32,767 | £32,768 |
The chance is 19/37 multiplied by itself once per loss, counted from one given spin. A long session has many spins where such a run can start, so over an evening it is far more likely than one row suggests.
Questions
Nine losses break it instead of six: 0.25% per cycle, and £5,110 lost when it happens. Rarer cliff, higher cliff, same −2.70%.
It would have won 48.65% of the time and left the cycle at −£130 anyway. There is no good move after a broken Martingale; there is only the stake you choose next.
Yes, for that session. Not for the mean over all sessions.
Fun with Maths
Try these yourself first. Each solution is folded; open it when you are ready.
At a £1 to £500 table, a MartingalestrategyMartingaleProgression that doubles the bet after each loss. See the Strategy section for what it does and does not do.Read full → player starts at £1 on redbetRedEven-money bet on the 18 red numbers. French: rouge.Read full → on a single-zerowheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full → wheel. How many losses in a row stop the doubling, how much has been lost by then, and how often does it happen?
Show the solution
Answer Nine losses: £511 lost, and the next bet would be £512. It happens once in about 403 sequences.
The stakes are 1, 2, 4, 8, 16, 32, 64, 128 and 256: together £511. The tenth would be £512, over the limit. A £500 bet that wins still leaves the player £11 down; one that loses leaves them £1,011 down.
Red loses with chance 19/37, so nine losses in a row have a chance of (19/37)⁹ ≈ 0.25%, once in about 403 sequences. Each completed sequence wins £1, so the player expects about £402 of small wins before the one £511 loss, and the 2.70% of everything wagered is still lost on average.
On a double-zero wheel the chance is (20/38)⁹, once in about 323. Bollman (2021) prints “about 1 in 833”, which is (18/38)⁹, the chance of nine wins in a row.
Comic relief
Forty-one small wins and one large loss. That is the whole system.
Sources
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Thorp, E. O. “Optimal gambling systems for favorable games.”