Why you can't double the rate indefinitely

1) What the strategy promises - and what really happens

The idea of martingale is simple: put 1 ×, lose - double to 2 ×, lose again - up to 4 ×, etc., "so that one win returns everything and gives + 1 unit."

Sounds logical on paper. In practice, there are three insurmountable barriers:

1. The negative expectation (EV) of each bet goes nowhere.

2. Exponential growth of the required bankroll and rate size.

3. Table/casino limits and ultimate bankroll.

2) Expectation remains negative

For equal payouts of 1:1 (for example, red in European roulette), the probability of winning is less than 50% due to zero: (p = 18/37), (q = 19/37). Single bid EV: (-1/37\approach -2. 70%).

Martingale increases turnover, but does not change EV: the expected total of the series is still (\text {Total }\approach -\text {edge }\times\text {Turnover}). And the turnover at doubling grows many times faster.

3) Exponential growth in rates and bankroll

Betting sequence after (n) consecutive losses: (1,2,4 ,\dots, 2 ^ n).

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

The required bankroll to survive (n) consecutive cons and put the next step: at least (2 ^ {n + 1} -1) units.

Example: if you want to withstand 10 consecutive defeats - you need BR (\ge 2 ^ {11} -1 = 2047) units; the next bet is (2 ^ {10} = 1024). This is without taking into account the table limit.

4) Table limits make "infinity" impossible

If the minimum is 1 CU and the maximum is 512 CU, the chain looks like this: 1, 2, 4,..., 256, 512. We lost 9 times in a row - you cannot double further. One such "tail" will reset dozens of "successful" short cycles.

5) Losing streaks aren't as rare as they seem

The probability of losing exactly 10 times in a row on red is (q ^ {10 }\approx 0. 001275) (0. 1275%).

But in a long session, the chance of at least one such series rises sharply. Over the ~ 500 spins, the probability of meeting 10 cons in a row is about 46%.

It is these rare, but inevitable series that "eat up" all the previous small winnings.

6) "Small frequent pluses" vs. "rare big minus"

Martingale creates a false sense of success: many short episodes end with + 1 unit. But mathematically:

Each series increases turnover, therefore, and the expected loss amount = edge × turnover.
A rare "fatal" drain (a series before the limit) is equivalent to the sum of tens/hundreds of small winnings, zeroing the past "victory statistics."

7) Why "almost fair" games don't save

Even in games with very high RTP/low edge (for example, neat blackjack according to basic strategy) martingale:

does not improve anticipation;
accelerates approach to limit or bankroll wall due to exponential growth;
intensifies the dispersion: occasionally - a big victory, but sooner or later - a catastrophic drain.

8) Formulas and quick scores

The turnover of a series of (n) steps is (1 + 2 +\dots + 2 ^ {n-1} = 2 ^ n-1).

Required BR for (n) consecutive losses + next step: (2 ^ {n + 1} -1).

Probability of (k) consecutive losses: (q ^ {k}) (for independent rounds).

The chance to see such a series in a long session is approximate: (1- (1-q ^ {k}) ^ {T}), where (T) is the number of starting positions (of the order of the number of spins).

Expected session total for any bid control (no edge> 0): (\text {Total }\approx -\text {edge }\times\text {Turnover}).

9) Psychological martingale traps

The illusion of control: "I control the bet, which means I control the result." No: you control turnover, not expectation.

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

Escalation of obligations: the desire to "finish off" with a limit rate and "zero" losses is a direct path to bankroll collapse.

10) What to work instead of doubling

Fixed small% of the current bankroll (rule ~ 1%, high-vol - 0. 25–0. 75%, low-vol — 1–2%).

Limits: stop loss and take profit per session (for example, − 20 %/+ 30% for Medium, − 30-40 %/+ 60-150% for High-Vol).

Turnover/speed control: less auto-spins → less "price of the hour."

Choice of "cheap" bets/games: higher RTP/lower margin.

Kelly only with real EV> 0 (sports/exchange), and then - fractional (¼ - ½), and not full.

11) Checklist before "playing doubles"

Do I know the table limit and my real bankroll? How many doubles can I physically endure?

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

Do I realize that the expectation of each bet is negative, and doubling only accelerates the turnover?

Am I ready to lose the entire sequence with one rare tail?

Martingale is not a way to "beat the system," but a casino math accelerator. It masks negative anticipation with frequent small pluses, but leads to rare destructive plums due to exponential growth in rates, final bankroll and limits. If the goal is risk control and predictability of spending, leave doubles in the textbook and use a reasonable percentage of bankroll, limits and "cheap" games.