This page collects the formulas behind every Martingale calculation in one place. They are the same equations used inside the Martingale roulette calculator, written out so you can apply them by hand. For the conceptual background behind them, see the mathematical model page.
Next Bet Formula
Next bet = previous bet × 2, or equivalently base bet × 2^(step - 1). After three losses at a $10 base bet the next bet is 10 × 2^3 = $80. The rule is pure doubling, which is why the numbers grow so fast.
Total Exposure Formula
Total exposure after n losses = base bet × (2^n - 1). This sums every bet from step one to step n. At a $10 base bet, five losses cost 10 × (2^5 - 1) = 10 × 31 = $310, matching the tables on the progression example page.
Bankroll Required Formula
Bankroll required = base bet × (2^n - 1) to survive n consecutive losses. It is identical to total exposure, because the bankroll must cover the full run of losing bets. To also place the (n+1)th bet you need an extra base bet × 2^n on top. The bankroll requirements page works through this in detail.
Losing Streak Probability Formula
P(n losses in a row) = p^n, where p is the single-spin loss probability: 19/37 on European and 20/38 on American roulette. Five European losses in a row is (19/37)^5, about 3.6%. Full figures are on the losing streak probability page.
Maximum Steps From Bankroll
Largest n where base bet × (2^n - 1) ≤ bankroll. Rearranged, n is roughly log2(bankroll / base bet + 1). A $630 bankroll at a $10 base bet gives n = 6 exactly, because 10 × 63 = $630.
Maximum Steps From Table Limit
Largest step where base bet × 2^(step - 1) ≤ table maximum. With a $10 base and $1,000 maximum, step seven ($640) fits but step eight ($1,280) does not, so the table allows seven bets. Your survivable depth is the smaller of this and the bankroll result, as shown on the initial bet size page.
Expected Value Reminder
EV per dollar wagered = -house edge, which is -2.70% on European and -5.26% on American roulette, no matter which staking pattern you use. Martingale rearranges bets but cannot make this number positive. That is the mathematical reason the system cannot beat roulette, covered on the expected value page.
Formula Examples
Combine the formulas for a full picture. At a $25 base bet: the fourth bet is 25 × 2^3 = $200; total exposure after four losses is 25 × 15 = $375; surviving seven losses needs 25 × 127 = $3,175; and the chance of seven European losses in a row is (19/37)^7, under 1%. Every one of these can be reproduced in seconds with the calculator.