Martingale
Double the bet after every loss; return to the base bet after a win. Each win recovers all previous losses plus one base unit. The page below shows what that does to a bankroll, and why it cannot change the expected loss.
How does the Martingale work?
Lose, and double. Win, and go back to the base bet. That is the whole system. With a $10 base bet and a $500 table limit the sequence runs out after six losses. The last column is the probability of reaching that step on an even-money bet, where each loss has probability 19/37.
Base bet $10, table limit $500. After six losses the next bet is $640, above the limit: the progression breaks with $630 lost. The chance of six straight losses on red is (19/37)^6 = 1.83%, roughly once every 55 attempts.
Walk the system
A loss doubles the stake. A win goes back to the start.
Stakes and results are in units of the base bet.
What does a session look like?
Small steps up, then a trapdoor. Every finished run of doubling adds one base bet, so the line climbs steadily. The run that hits the $500 limit takes back $630 at once: 63 of those steps.
- p10 −$1,000
- p25 −$1,000
- median −$1,000
- p75 +$930
- p90 +$1,830
Simulator
Each setting runs 2,000 sessions of even-money bets from a $1,000 bankroll. Results update immediately.
Most sessions finish slightly ahead (40.8%), which is the appeal. The mean is negative (−$66.44) because the losing sessions are large: the progression hit the table limit in 59.3% of sessions. A flat $10 bettor over the same spins expects to lose about $27.
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.
A run of fifteen losses comes about once in 22,000 tries. That sounds safe. It is not, for two reasons. Every spin starts a new try, and an evening has hundreds of spins. And nobody gets as far as fifteen: from a $10 base the $500 limit ends the doubling after six losses, which come about once in 55 tries.
Why can it not change the odds?
Expected value adds. If every individual bet has an expectation of −2.70% of its stake, then any sequence of bets, however the stakes are chosen, has an expectation of −2.70% of the total staked. The choice of stake can depend on past results; it cannot depend on the next one.
bᵢ chosen before spin i; Xᵢ = ±1 independent of bᵢ