KEY TAKEAWAYS
The Martingale system doubles the stake after every loss and returns to the base stake after a win, and the math shows it cannot change the expected value of a roulette bet. Suppose a $10 base stake on red at an American wheel, with room for six bets. A round then ends with a $10 gain 97.87% of the time and a $630 loss 2.13% of the time. The expected result of a round is −$3.60, which is the 5.26% house edge applied to the $68.47 expected to be wagered per round.
The system changes the shape of the results into many small wins and rare, very large losses. It does not change the average. This guide gives the formulas with every symbol defined, a worked example, a table of losing-streak probabilities and required stakes, and the mistakes that make the plan look better than it is. All figures are hypothetical and calculated on the page. Gambling is best treated as paid entertainment, and the math here explains why no staking plan turns it into income.
What the Martingale System Is
The Martingale is a staking plan for even-money bets such as red or black. The bettor picks a base stake, doubles the stake after every loss, and returns to the base stake after a win. The appeal is arithmetic: one win after a run of losses recovers every earlier loss and adds one base stake of profit. Starting at $10, losses of $10, $20 and $40 total $70, and a win on the $80 bet returns $160, a net gain of $10.
That is correct as far as it goes, which is why the plan feels convincing. The hidden assumptions are an unlimited bankroll and no table limit. Real play has both, so the plan hides a rare but very large loss inside a long run of small wins. Each round ends in one of two ways: a win that pays one base stake, or a losing streak that exhausts the bankroll or reaches the table limit.
Two things stay fixed however the stakes move. Each spin is independent, so a losing streak does not make a win more likely on the next spin. And every bet carries the same house edge as before. A staking plan only decides how much to bet on each spin. It cannot alter the probability or the payout of the spin itself, and it reshapes the swings described in the variance guide without moving the average.
The Martingale Formulas
With a base stake s, the stake on the bet after n straight losses, and the total lost so far, follow two short formulas.
total lost after n losses = s × (2ⁿ − 1)
where:
s is the base stake in dollars.
n is the number of consecutive losses so far.
2ⁿ means 2 multiplied by itself n times.
A win on the next bet pays s × 2ⁿ, so the net profit is s × 2ⁿ − s × (2ⁿ − 1) = s. A winning round always nets exactly one base stake, however long the streak was.
The chance of a streak comes from the multiplication rule for independent events.
where:
q is the probability of losing a single bet: 20 ÷ 38 ≈ 0.5263 for red on an American wheel and 19 ÷ 37 ≈ 0.5135 on a European wheel.
If the bankroll or table limit allows N bets, a round ends with a gain of s with probability 1 − qᴺ, or with a loss of s × (2ᴺ − 1) with probability qᴺ.
round EV = −house edge × expected amount wagered in the round
where:
N is the largest number of bets the bankroll or table limit allows.
The second line is the general rule: the round EV always equals the house edge times the amount wagered, so no choice of s or N can lift it above zero.
Worked Example: Six Bets on an American Wheel
Suppose a $10 base stake on red at an American wheel with a $500 table limit. The stakes can be $10, $20, $40, $80, $160 and $320, so N = 6. A seventh bet would be $640, above the limit.
Step 1: q = 20 ÷ 38 ≈ 0.5263.
Step 2: P(6 straight losses) = 0.5263⁶ ≈ 0.021256, or 2.13%.
Step 3: total lost = 10 × (2⁶ − 1) = $630.
Step 4: P(gain $10) = 1 − 0.021256 = 0.978744, or 97.87%.
Step 5: round EV = 10 × 0.978744 − 630 × 0.021256 ≈ 9.79 − 13.39 ≈ −$3.60.
Check: the expected amount wagered is 10 × (1 + 1.0526 + 1.0526² + 1.0526³ + 1.0526⁴ + 1.0526⁵) ≈ $68.47, and 68.47 × 5.26% ≈ $3.60. The two methods agree.
Wins are frequent but the average is negative: about 98 rounds in 100 gain $10, yet the rare $630 loss pulls the average of a round down to −$3.60.
Losing Streaks, Stakes and Table Limits
The table uses a $10 base stake and shows how fast the required stake grows and how likely each streak is. The streak probabilities are qⁿ from the formula above.
| Losses in a row (n) | Total lost | Next stake needed | P(n losses), American | P(n losses), European |
|---|---|---|---|---|
| 1 | $10 | $20 | 52.63% | 51.35% |
| 2 | $30 | $40 | 27.70% | 26.37% |
| 3 | $70 | $80 | 14.58% | 13.54% |
| 4 | $150 | $160 | 7.67% | 6.95% |
| 5 | $310 | $320 | 4.04% | 3.57% |
| 6 | $630 | $640 | 2.13% | 1.83% |
| 7 | $1,270 | $1,280 | 1.12% | 0.94% |
| 8 | $2,550 | $2,560 | 0.59% | 0.48% |
| 10 | $10,230 | $10,240 | 0.16% | 0.13% |
Read down the table: the stake doubles with every loss while the probability shrinks by only about half, so the amount at risk rises as fast as the chance of the streak falls. With a $500 limit the seventh bet cannot be placed, and the six-loss streak is final.
A rare streak still arrives with enough rounds. Over 100 independent rounds on the American wheel, the chance of at least one six-loss streak is 1 − 0.978744¹⁰⁰ ≈ 88.33%, and the expected result is 100 × (−$3.60) ≈ −$360.37. On a European wheel the same plan has a 1.83% bust chance and an expected loss of about $1.74 per round, which is 2.70% of $64.20 wagered.
Common Mistakes and Misconceptions
The most common error is believing that a win is due after a losing streak. Spins are independent, so after five losses the chance of winning is still 18 ÷ 38 ≈ 47.37% on an American wheel.
A second error is reading frequent small wins as proof that the plan works. A plan that wins $10 in 97.87% of rounds feels profitable until the rare loss arrives, and the expected value already includes that loss.
Third, many readers ignore the ratio of risk to reward. The plan risks $630 to win $10, so it needs a win probability of 630 ÷ 640 ≈ 98.44% per round to break even, and the real figure is 97.87%.
Finally, a bigger bankroll does not fix the math. It makes the busts rarer and larger, and the round EV stays equal to the house edge times the amount wagered.
Quick Reference and How to Check Your Answer
| Quantity | Formula |
|---|---|
| Stake after n losses | s × 2ⁿ |
| Total lost after n losses | s × (2ⁿ − 1) |
| Profit on a winning round | s |
| P(n straight losses) | qⁿ |
| Round EV (N-bet cap) | s × (1 − qᴺ) − s × (2ᴺ − 1) × qᴺ |
Check a calculation in three steps. Confirm that the win and bust probabilities add up to 100%. Confirm that a winning round always nets exactly s. Then compute the expected amount wagered and check that round EV equals −house edge × that amount. If the two disagree, a stake or a probability was entered wrongly.
Related Concepts and Next Steps
The Martingale is an application of two ideas covered earlier: the expected value formula, which gives the round EV, and probability in gambling, which explains why streaks do not change the next spin. The edge used above comes from the roulette house edge.
Other doubling and progression plans, such as the Fibonacci and D’Alembert systems, face the same result: they redistribute the wins and losses but leave the expected value untouched.
Frequently Asked Questions
