How the Martingale system works

1) The essence of the strategy in a minute

Martingale is the doubling of the bet after each loss in betting 1:1 (red/black, even/odd, etc.). The idea: one win would cover all past losses and give + 1 base rate.

The problem: The mathematical expectation (EV) of each bet does not change (the casino is negative), and the size of the required bank and the risk of a "long tail" grow exponentially.

2) How the martingale cycle works

Let base rate = 1 unit. Sequence after (n) losses:

(1,;2,;4,;8,\dots, 2^n).

Total loss before next step: (1 + 2 +\dots + 2 ^ {n-1} = 2 ^ n-1).

If the next step wins, the net total of the cycle is (+ 1) one.

To physically reach the step (n! +! 1), you need a minimum bank

(\textbf{BR}_{\min}=2^{n+1}-1).

Example: if you want to withstand 10 consecutive losses and still have a chance to "block" - you need BR ≥ (2 ^ {11} -1 = 2047) c.e., despite the fact that the bet itself at the 11th step will be (2 ^ {10} = 1024) c.e.

3) Table limits make "infinity" impossible

The casino always has a maximum bet. If min = 1 CU, max = 512 CU, the chain stops at 512:

(1,2,4,\dots,256, \mathbf{512}).
Nine losses in a row - and you can't double further. One such "tail" will reset dozens of "successful" short cycles.

4) Long run probability (not that small)

Denote (p) - probability of winning one bet 1:1, (q = 1-p) - loss. In European roulette to "red" (p = 18/37\approx 0. 4865), (q=19/37\approx 0. 5135).

The probability of losing exactly (k) times in a row is (q ^ k).

For (k = 10): (q ^ {10 }\approx 0. 5135^{10}\approx 0. 001275) (0. 1275%).

For a long session, the chance to see at least one such series is close to

(1- (1-q ^ {k}) ^ {T}), where (T) is the number of "starting positions" (approximately the number of bets).

For 500 spins, the chance of a series of 10 minuses in a row ≈ (1- (1-0. 001275)^{500}\approx 46%).

5) Expectation remains negative

Key formula: Total ≈ − edge × Turnover, where (edge = 1-RTP).

Martingale does not change (edge) - he only inflates the Turnover (the sum of all bets). Therefore, the average "entertainment price" at a distance is higher, not lower.

6) How much one "small plus" actually costs

The cycle that ends with a victory in step (n! +! 1) scrolls the turnover of (2 ^ {n + 1} -1) units for the sake of profit + 1. This is a very "expensive" one unit, especially when you consider the edge. One "fatal" cycle to the limit eats up many such "plus" cycles.

7) Risk of Ruin (RoR)

Even without formulas, it is clear: with a EV≤0 and unlimited horizon, RoR → 1. With time and rate limits, the RoR is determined by how long the series you survive to the "wall."

Practical assessment: if your BR is less than (2 ^ {n + 1} -1) before the limit, a series of (n! +! 1) losses is guaranteed to "break" the strategy.

8) "Soft" progressions (d'Alembert, Fibonacci, Labouchere) - better?

They raise the rate more slowly, but the problem is the same: EV rates do not change, turnover grows, limits and bankroll are finite. In the short term, the trajectory is "smoother," in the long term - the same mathematical minus.

9) Martingale psychology

The illusion of control: "I control the bet - I control the result." In fact, you only manage risk and turnover.

Selective memory: frequent small victories are remembered, a rare huge drain is forgotten.

Escalation of obligations: an increase in the rate after a drawdown pushes to tilt.

10) Where is martingale appropriate?

As a case study of variance and exponent - yes. As a real strategy against casinos - no. In products with a real advantage (rarely sports/exchange) it is more correct to scale the bet with fractional Kelly, and not with progression.

11) Safe alternatives to "doubling"

Rate as% of current bankroll:

High-Vol: 0. 25–0. 75% BR; Mean: ~ 1% BR; Low/1: 1: 1-2% BR.
Play in series: record the time and limit of attempts.
Stop levels: SL/TP (for example, − 20... − 40 %/+ 30... + 150% of the session budget).
Choice of "cheap" bets/games: European roulette instead of American, basic strategy in blackjack, high actual RTP in slots.
Bonus hygiene: consider 'Bonus × Wager × edge' - sometimes conditions partially compensate for edge (but not progressions).

12) Fast cheat sheets

Required bank for (n) losses + next step: (\boxed {BR _ {\min} = 2 ^ {n + 1} -1}).

Cycle turnover to win step (n + 1): (\boxed {2 ^ {n + 1} -1}).

Probability of (k) cons in a row: (\boxed {q ^ k}).

The chance to see such a series in (T) attempts: (\boxed {1- (1-q ^ {k}) ^ {T}}).

Average wait for a series: (\boxed {\text {Total }\approach -edge\times\text {Turnover}}) - regardless of the betting pattern.

13) 'before doubling' checklist

Do I know the table limit and how many doubles does it really allow?

Will the bank be enough to the step before the limit (formula (2 ^ {n + 1} -1))?

Am I psychologically ready to bet (2 ^ n) after a long series?

Do I understand that the expectation of each bet is negative, and doubling only accelerates the implementation of the minus?

Martingale is not a "loophole," but a beautiful paradox of variance: frequent small victories mask rare destructive plums. Exponential rate increases, table limits and negative EVs make "infinite" doubles mathematically and practically untenable. Want control - play percentage from bankroll, series, with stop levels and choose products with a lower edge.