D’Alembert calculator
Add one unit after a loss; remove one after a win, never below one. Enter your bankroll, unit, table limit and walk-away target. The calculator follows every possible spin sequence to the end of the session and returns the exact chance of each finish, the full stake ladder, and what the session costs on average.
How the session ends
Finishes ahead 72.19% of the time. The average session lasts 106.8 spins; the median finish is $1,200.
Read the result
D’Alembert reaches the $200 target in 72.19% of sessions and breaks in 27.81%, ending where the next required stake no longer fits. Weighted together, the expected finish is −$112: about $4,158 goes through the wheel per session and each $100 of it loses $2.70 on average. A high chance of reaching the target and a negative expected result are the same fact seen from two sides: many small wins, paid for by fewer, larger losses.
Cross-check: with these exact rules, the D’Alembert backtest ran 20,000 seeded sessions and reached the target in 71.59%, with a mean result of −$119. The exact figures above are what that sample is estimating.
The losing streak ladder
| Bet | Stake | Net after it loses | Chance to get here |
|---|---|---|---|
| #1 | $5 | −$5 | 100.00% |
| #2 | $10 | −$15 | 51.35% 1 in 1.9 |
| #3 | $15 | −$30 | 26.37% 1 in 3.8 |
| #4 | $20 | −$50 | 13.54% 1 in 7.4 |
| #5 | $25 | −$75 | 6.95% 1 in 14 |
| #6 | $30 | −$105 | 3.57% 1 in 28 |
| #7 | $35 | −$140 | 1.83% 1 in 55 |
| #8 | $40 | −$180 | 0.94% 1 in 106 |
| #9 | $45 | −$225 | 0.48% 1 in 207 |
| #10 | $50 | −$275 | 0.25% 1 in 403 |
| #11 | $55 | −$330 | 0.13% 1 in 784 |
| #12 | $60 | −$390 | 0.07% 1 in 1,527 |
| #13 | $65 | −$455 | 0.03% 1 in 2,974 |
| #14 | $70 | −$525 | 0.02% 1 in 5,792 |
| #15 | $75 | −$600 | <0.01% 1 in 11,279 |
| #16 | $80 | −$680 | <0.01% 1 in 21,965 |
| #17 | $85 | −$765 | <0.01% 1 in 42,773 |
| #18 | $90 | −$855 | <0.01% 1 in 83,295 |
| #19 | $95 | −$950 | <0.01% 1 in 162,206 |
| #20 | $100 | over bankroll | <0.01% 1 in 315,874 |
19 losses in a row from one unit leave a $100 stake that the remaining bankroll can’t cover. “Chance to get here” is the probability that every earlier bet in the streak went the losing way, from a fresh start. The stake climbs one unit per net loss, so the ladder is slow. The real exposure is drift: losses outnumber wins 19 to 18 on a European wheel, so the level wanders upward over a long session.
How the calculator works
A D’Alembert session has no memory beyond two numbers: the current bankroll and where the stake sits in the progression. The calculator starts with all of the probability on your opening bankroll and the first stake, then applies one spin at a time. Each spin sends 18/37 of every state’s probability to the winning branch and the rest to the losing branch. Probability that reaches a stop rule is set aside as a finished session. After the last allowed spin, whatever is left is counted as spin-capped. Nothing is sampled, so the same inputs always give the same figures.
The stop rules are the ones every system page uses: walk away once the bankroll reaches the target, stop when the next stake is larger than the bankroll or the table maximum, and stop at the spin cap. The test suite solves every calculator on the published rules and requires agreement with the 20,000-session seeded backtests within sampling error. It also checks exact agreement with brute-force enumeration of every spin sequence on a small table.
What no input can change
Every red bet returns −1/37 of its stake on average, so the expected result is always that fraction of the expected amount wagered: −$2.70 per $100. A bigger bankroll, a higher table limit, or a smaller target changes how often the session finishes ahead and how large the bad finishes are. It does not change that figure. The house edge page derives it from the payout table.
Limitations
Fair, independent spins, bets on red only, whole-dollar inputs, and standard European or American settlement. French la partage and en prison are not modelled; the wheel rules guide covers what they change. To test a variation the calculator can’t express, run the D’Alembert Lua in Engineer mode and edit it.
Other progression calculators
- Martingale calculator · Double the stake after a loss; back to one unit after a win.
- Grand Martingale calculator · Double the stake and add one unit after a loss; back to one unit after a win.
- Fibonacci calculator · One step up the 1, 1, 2, 3, 5… sequence after a loss; two steps back after a win.
- Reverse Martingale calculator · Double after a win; back to one unit after a loss or when the next stake would pass the table maximum.
- Paroli calculator · Double after a win, bank after three wins in a row; back to one unit after a loss.
- Contra D’Alembert calculator · Add one unit after a win; remove one after a loss, never below one.
- 1-3-2-6 calculator · Stake 1, 3, 2, then 6 units on consecutive wins; back to 1 after a loss or a completed cycle.
For a single flat bet rather than a progression, use the roulette odds calculator.