American roulette keeps 38 pockets on the wheel, and European roulette keeps 37. Paroli goes the opposite way from doubling after a loss. The chip grows only after a win, and any loss closes the series. At the start of a series one’s own money is one unit: everything that stands on the table after that is gathered from chips already won.
The bet goes on an even chance: red or black, odd or even, 1–18 or 19–36. On the RoulCoin wheel there are 18 winning pockets out of 38, and 20 losing ones, with 0 and 00. The payout equals the amount staked. Below is a paroli of three steps: 1, then 2, then 4. Stopping after the third win is the player’s rule. The wheel knows no such limit.
French roulette with half the stake returned on zero does not soften this paroli: on the RoulCoin wheel there are two zeros, and both take the chip in full. A loss of a step because of 0 or 00 closes the series the same way as a loss because of the other color.
Three steps and one unit of one’s own chips
Before the first spin, 1 is taken from the bankroll. A win leaves the series at +1, and the next bet is 2: one unit of one’s own and one unit just won. The second win brings the series to +3, and 4 goes on the table. The third win gives +7, and the series is taken. A loss on any of the three steps returns the bankroll to minus 1 relative to the start.
| Step | Bet | If the win is taken | If the step is lost |
|---|---|---|---|
| 1 | 1 | +1 | −1 |
| 2 | 2 | +3 | −1 |
| 3 | 4 | +7 | −1 |
The large chip on the third step was not taken from the bankroll separately. Four on the table is +1 after the first win and +2 after the second. So a loss of the four does not make the series minus 4. The bankroll ends up one unit below the mark from which the series was started.
Four outcomes on American roulette
On 38 pockets a series of three steps breaks into four outcomes. The shares are rounded to hundredths. The second column is the same rule on 37 pockets, so there is something to compare the frequency with, not the shape of the minus.
| Outcome of the series | 38 pockets | 37 pockets | Result |
|---|---|---|---|
| A loss at once | 52.63% | 51.35% | −1 |
| A win, then a loss | 24.93% | 24.98% | −1 |
| Two wins, then a loss | 11.81% | 12.15% | −1 |
| Three wins | 10.63% | 11.51% | +7 |
Three wins in a row on an American wheel are (18/38)³, 10.63%, about one series out of nine. The expectation of one series is counted like this: 8×(18/38)³ − 1 ≈ −0.15 units. Over 100 series about 15 units fall to the table. On average about 2.84 units are staked per series: one unit always, a two with probability 18/38, a four with probability (18/38)². A share of 5.26 per 100 staked, applied to this sum, gives the same 0.15.
A long paroli up to the ceiling
The trio of steps can be continued. While wins keep coming, the bet doubles again, and a loss of any step that was staked still leaves the series at −1. The plus accumulates as 2ⁿ − 1. The ceiling in the table is conditional: 500. In an account the limit is checked separately, and it can be either lower or higher.
| Step | Bet | If the win is taken | If the step is lost |
|---|---|---|---|
| 4 | 8 | +15 | −1 |
| 5 | 16 | +31 | −1 |
| 6 | 32 | +63 | −1 |
| 7 | 64 | +127 | −1 |
| 8 | 128 | +255 | −1 |
| 9 | 256 | +511 | −1 |
| 10 | 512 | not staked at a maximum of 500 | — |
The ninth step, a bet of 256, still fits under a ceiling of 500. The tenth, 512, no longer does. If the ninth win happened, the series stands at +511, and the table will not accept the next double: the plus stays with the player, because there is nowhere to place the bet. A loss on the ninth step, if it was staked anyway, returns not minus 256 of new money, but minus 1 against the starting bankroll. The 256 on the cloth is the previous wins of this series.
Any lost step of a paroli closes the series at −1. Plus 7 happens only when the bet of 4 has also won.
European roulette leaves the series the same minus of 1
European roulette changes the frequency, not the contract of the series. There are 19 losing pockets out of 37, and three wins in a row are 11.51%, not 10.63%. The expectation 8×(18/37)³ − 1 ≈ −0.08 units, about 8 units over 100 series. On average about 2.92 units are staked per series there, and a share of 2.70 per 100 staked agrees with these 0.08. A broken paroli here too costs minus 1, not the minus of the current chip from the bankroll.
For a player in England three wins in a row are counted on the same 18 pockets out of 38. A time zone does not shorten the series. Playing online for money is possible after an account is created: a live balance does not add a single winning pocket to the wheel. While series are run on virtual chips, this is exactly what shows: the large bet on the third step is made up of the first two wins.
- The series’ own chip is one, in the very first step. After that the win is returned to the table.
- A loss of any step closes the series. The next one starts from 1 again.
- The number of steps is written down before play. The wheel does not stop by itself on the third win.
- A step that does not fit the table ceiling is not staked. The plus already gathered is taken.
Why plus 7 does not cancel the table’s share
Plus 7 is a rare outcome of one series, about 11 out of 100 on a wheel of 38 pockets. The other series of this trio end at −1. 11×7 − 89 = −12, and an exact count by the shares, without rounding to “11 and 89,” gives about −15 over 100 series. Rare sevens do not cover the frequent single-unit minuses.
Does a loss on the third step take 4 from the bankroll?
No. The series closes at −1. The four on the table is gathered from the wins: +1 after the first step and +2 after the second.
Do two wins already give plus 7?
No. If one stops after two, the result is +3. Plus 7 appears when the third bet also wins, that is 4.
Must one always stop on the third win?
That is the player’s choice. The fourth bet equals 8. Its loss still leaves −1 against the starting bankroll, and a win gives +15.
Does paroli in this form suit a Dozen?
No. Doubling here rests on a payout of 1 to 1. A Dozen pays 2 to 1 and covers 12 pockets out of 38, so a loss of a step no longer reduces the series to minus one of one’s own bets.
Do three wins in a row come up more often than once out of ten?
On 38 pockets, no: 10.63%, about one series out of nine. On 37 pockets, 11.51%. Both numbers are about three wins in a row, not about one spin.
Paroli moves the large bet onto chips that are not one’s own, already won, and because of that the minus of one series looks small. Neighboring playing strategies pay for the same attempt differently: somewhere the bet grows after a loss, and somewhere the aim of the cycle is one unit. The number of steps is better written down before the first spin, while there is not yet a four from past wins on the table.