What a progression is
Casino games are games of chance with a built-in house edge. No strategy, system, pattern or timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose. A progression plan is one kind of betting system: a rule for raising or lowering the stake after each result. This article describes how two well-known plans work and what they do and do not change.
Progressions are usually described for even-money bets, such as red or black on a roulette wheel, where a win pays the same amount as the stake. The same principles apply elsewhere.
Positive and negative progressions
The two families are mirror images.
- Positive progressions raise the stake after a win and drop back after a loss. The idea is to profit from streaks while risking only a small amount when the run ends.
- Negative progressions raise the stake after a loss, aiming to recover earlier losses with one win. The doubling plan, called the martingale system, is the best-known example.
Neither idea gives the player an advantage. Each bet is judged separately by the wheel, so the stake for the next bet cannot make the chance of winning higher.
The Paroli plan
The Paroli plan is a positive progression. A common version is:
- Start with a base bet.
- After each win, double the stake.
- After three wins in a row, or any loss, return to the base bet.
Take a hypothetical $10 base bet on red on a single-zero wheel, where the chance of red is 18/37. The possible outcomes of one cycle are:
| Result | Bets placed | Net result |
|---|---|---|
| Lose the first bet | $10 | −$10 |
| Win, then lose | $10, $20 | +$10 − $20 = −$10 |
| Win, win, then lose | $10, $20, $40 | +$10 + $20 − $40 = −$10 |
| Win three times | $10, $20, $40 | +$10 + $20 + $40 = +$70 |
The chance of winning three in a row is (18/37)^3, which is about 0.115, or 11.5 percent. Every other path ends with a loss of $10, which has a total probability of about 88.5 percent.
The expected result of one cycle is 0.115 × $70 − 0.885 × $10, which is about $8.06 − $8.85 = −$0.79. Checking this a second way: the expected total wagered is $10 + $20 × 0.486 + $40 × 0.237 ≈ $29.20, and 1/37 of that is about $0.79. The two methods agree.
So Paroli turns an ordinary stream of small coin-flip results into a pattern of frequent $10 losses and occasional $70 wins. It feels like a bargain because the loss is capped and the win is large. The average cost per cycle is still exactly what the edge dictates for the total amount wagered. The pattern has changed; the expected value has not.
The Fibonacci plan
Fibonacci is a negative progression based on the sequence 1, 1, 2, 3, 5, 8, 13, 21, and so on, where each number is the sum of the previous two. The player bets a number of units according to a position in the sequence:
- After a loss, move one step to the right (a larger bet).
- After a win, move two steps to the left (a smaller bet), or stop if at the start.
Suppose a unit is $1. Three losses in a row mean bets of $1, $1 and $2, a total of $4 lost. A win on the next bet of $3 gets back $3, leaving the player $1 behind, not even. That differs from doubling, where one win always returns the player to a profit of one unit. After a longer losing run the shortfall can be larger, so one win does not reliably recover everything.
The appeal is slower growth. The total wagered after a run of losses is the sum of the first terms:
| Losses in a row | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Latest bet (units) | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 |
| Total lost (units) | 1 | 2 | 4 | 7 | 12 | 20 | 33 | 54 |
After eight straight losses, doubling would require a bet of 128 units on the eighth bet and would have lost 127 units before it. Fibonacci’s eighth bet is 21 units, and the total lost is 54 units. That sounds gentler, but it is a matter of degree. A long enough losing run still produces large losses, and a later win does not necessarily offset them.
Other plans
There are many variations. The D’Alembert plan adds one unit after a loss and subtracts one after a win. The Labouchere plan has the player cross numbers off a list. All of them follow the same pattern: rules that change the stake according to past results. Because the wheel has no memory, the rules cannot improve the chance of winning any individual bet.
What progressions really do
The expected loss over a series of even-money bets equals the edge times the total amount wagered. A progression changes the total wagered and the distribution of results, which is the statistical variance, but not the per-dollar cost.
- Positive progressions tend to give frequent small losses and occasional large wins.
- Negative progressions tend to give frequent small wins and occasional large losses.
Neither shape is a way to profit. Where the shape matters is in how it feels: a plan with frequent small wins can give a false sense of reliability, and a plan with rare large wins can feel unlucky even when it is performing exactly as the math predicts. Measures such as standard deviation describe the spread, and expected value describes the average, which stays negative.
Also remember that a plan cannot be followed forever. Table limits and a finite budget cap how far any escalating sequence can go, and a tempting plan can encourage people to wager more in total than they intended.
If you find that a plan is drawing you into betting more than you can afford, or it is hard to stop, support services in your area can offer help.