How the "reverse Martingale" strategy works

1) The essence of the strategy in a minute

Reverse Martingale (Paroli) - you increase the bet only after winning, and after losing you return to the base bet. The idea is "to give profits to grow, and cut losses": many small cons (− 1 base rate), occasionally - large pluses if you catch a series of victories.

Base (unit) = 'b'.

Scheme "let it ride": 'b → 2b → 4b →...' while you win; for any loss - reset to b.

Usually they put the ceiling of the series (cap) in 2-3 victories in a row and fix the profit.

💡 Important: the strategy does not change RTP/house edge and does not create + EV. It changes the trajectory of the result (distribution of winnings), and not the average over the distance.

2) Two popular Paroli options

A) Full "let it ride"

After winning, leave the win on the table and double the bet.

Risk: if you lose at any step, you lose all the current bet on the table; the loop often ends with a minus' − b'if the loss occurred before reaching the cap.

B) "Free-bet Paroli" (gentle)

After the first victory, you take the original bet 'b' and play only with winnings.

Then the worst thing after the first victory is to return to zero in the cycle, and not go into minus more '− b'.

Almost always limit the series 2-3 victories, then fix the profit.

3) Series probabilities (at equal odds 1: 1)

Let the probability of a single victory'p ', loss'q = 1 − p'.

Probability of exactly N consecutive wins: 'p ^ N'.

In European roulette at rates 1:1: 'p = 18/37 ≈ 0. 4865`.

2 wins in a row: 'p ^ 2 ≈ 0. 237`.

3 in a row: 'p ^ 3 ≈ 0. 115`.

4 in a row: 'p ^ 4 ≈ 0. 056`.

The larger the target for the length of the series, the less often you reach it.

4) Series Cap Payouts

Suppose cap N wins, base 'b', payouts 1:1.

If you get to N victories and fix: cycle profit (full let-it-ride) ≈ 'b· (2 ^ N − 1)' (the increasing rate brought such a delta in total).

If you lose before the cap: the cycle usually ends around '− b' (or ≈0 with a careful "free-bet" after the first victory).

Intuition: you exchange many small minuses for rare large pluses. The distribution becomes positively asymmetric (right "fat" tail), but the average expectation remains the same (negative in casinos).

5) The pros of Paroli versus martingale

Controlling maximum loss: one loss after a series is not a disaster; typical loss per cycle ≈ '− b' rather than "exponent."

Psychologically easier: you increase the rate on the positive, and not after pain.

Better under "hit hunting" (high volatility): a rare skid fits into the framework of the strategy (especially with a 2-3 victory cap).

6) Main cons and traps

Same EV. RTP/edge games do not change: over a long distance, the result ≈ '− edge × Turnover'.

Rarity of long series. A cap of 3-4 wins sounds beautiful, but is achieved infrequently (see 'p ^ N').

False sense of "free play." Even "free-bet" after the first victory does not give a positive expectation - it only limits the depth of the minus per cycle.

Discipline of fixation. Without a hard cap, the series easily turns into "greed" and is reset to the next loss.

7) How to put into practice (presets)

Cap selection:

Universal: N = 2-3 wins in a row. Then the frequency of achieving the goal drops sharply and the chance to give everything back grows.

Rate: From the current bankroll (BR):

High-Vol slots/bets: 0. 25–0. 5% BR to base 'b'.
Average volatility: ~ 1% BR.
Low/rates 1:1: 1-2% BR (conservative).

Mode:

Free-bet Paroli after first win + cap 2-3 + instant fix and reset to 'b'.

Stop rules per session:

SL/TP (e.g. − 20... − 40 %/+ 30... + 150% of session budget) or 30-60/90 min timer.

8) Mini-examples

Roulette 1: 1, 'b = 2 cu.', cap 3, free-bet

Probability of reaching 3 wins: '≈ 11. 5%`.

Cycle success: profit of the order of 'b· (2 ^ 3 − 1) = 14 standard units' (if the winnings were completely "rolled").

The rest ~ 88. 5% of cycles you lose near the base rate (or close to zero if you managed to fix 'b' after the first victory).

Slot (freespin/bonus series unpredictable):

Paroli is good as a style: increase the bet only after winning and immediately roll back after losing. But the frequency of consecutive drifts is lower than in the 1:1 net rates, and the variance is higher → keep less% BR and shorter than the session.

9) Comparison with alternatives

Classic martingale: faster "kills" the bank-roll with an exponent and rests on limits. Paroli limits losses, but "winning" streaks are rare.

Flat rate (% BR): most predictable on risk; there is no "series hunt," but it is easier to manage the budget.

Fractional Kelly (only at EV> 0): scales the preponderance of the model; not for negative EV casino games.

10) Cheat sheet formulas

Probability of reaching cap'N ':' p ^ N '.

Expected distance total (any betting style): '≈ − edge × Turnover'.

Estimate of the "price of the hour": 'Loss _ hour ≈ edge × b × (bets/min) × 60' (the faster the pace, the more expensive the hour of play).

Base for high-vol products: 0. 25–0. 5% BR; for medium - ~ 1% BR.

11) Paroli checklist (before kick-off)

I chose the cap of the series (2-3) and fix the profit without exception?

Use free-bet after the first victory?

Base 'b' is X% of current BR (not a fixed amount)?

I understand that EV does not change, and the strategy is only about the form of distribution?

Do I have an SL/TP and a time limit per session?

The reverse Martingale is about the form of risk, not about the "breaking" of mathematics. It creates a positively asymmetric curve: many small disadvantages and rare large advantages if you keep the cap 2-3 victories and the discipline of fixation. For conscious play, use free-bet, a small base in% of BR, short sessions and remember: with negative EV over a long distance, it is not the "system" that decides, but the management of turnover and risk.