Paroli System Calculator

Defaults: Base Bet $10, Win Streak 3, Payout Odds 1 (even money). Paroli is a positive progression: you let a winning even-money stake ride, usually for a fixed number of wins, then pocket. The engine’s closed form: profit = base × ((1 + payout)^wins − 1), finalStake = base × (1 + payout)^wins. With 10, 3, 1 that is profit = 10 × (2^3 − 1) = 10 × 7 = $70.00 and finalStake = 10 × 8 = $80.00. Path: bet $10, win, let $20 ride; win, let $40 ride; win, the position is worth $80. You risked only the original $10 (if you were going to stop after three anyway). The $70 is the net if you take the $80 down after the third win versus walking away before the first spin.

If any of the three spins loses, a strict Paroli resets to $10 and the streak profit is 0 for that attempt (you may have lost only the base, or a later multiple if you count a second-spin loss as −$20 depending on how you mark the books). This calculator does not price the losing paths. It only names the pot at the end of a successful streak of `wins` at the stated payout. A 3-win even-money run on European roulette has probability (18/37)^3 ≈ 11.54 percent from a standing start. Most Paroli cycles end earlier. The $70 headline is the size of the rare completed parlay, not the expected value of trying.

Common mistakes: treating $70 as average profit per cycle; letting the streak run past the table’s “paroli” cap (many houses cap lets-it-ride at three); and using payout 1 on a column bet, which is not even money. Next, set wins to 1: profit $10, final $20 (a single even-money hit). Set wins to 4: profit 10×(16−1)=$150, final $160, probability (18/37)^4 ≈ 5.61 percent. Set payout to 2 (for a 2-to-1 bet) with three wins: profit 10×(3^3 − 1)= $260, final $270 — and a much smaller chance of three 2-to-1 hits in a row.

Do not read $70 as money you are owed. It is the completed-streak profit if three even-money wins occur at a $10 start. House edge still clips every spin, including the $40 third stake. Try wins 2 versus 3 to see why historical Paroli stopped at three: the extra double looks pretty and is the spin the wheel most enjoys taking back.

Paroli is the reverse Martingale: press winners, not losers. The word comes from Italian parlari / paroli, related to parlay — let the stake ride. Sixteenth- to eighteenth-century European gambling houses already knew the three-coup paroli on even chances; later roulette rooms standardised “Paroli of three” as a house-friendly way to let customers feel they were using a system while the zero kept working. This calculator prices a completed win streak: what the pile is worth and what the net is versus the starting unit.

Who uses it: players who dislike chasing losses and prefer to speculate with money “the house is holding”; writers contrasting positive and negative progressions; teachers showing that pressing winners still has negative expectation because the large bet is placed after the account is already lucky, and the next spin still carries 19/37 of a miss. Who should not: anyone who thinks a 16th-century Italian name changes 18/37. There is no wheel field here. Probability is not computed by the engine. You bring (18/37)^n yourself.

What it does not do: subtract the cycles that die on spin 1 or 2, compute expected value of a “stop at three” rule over a session, or handle mixed payouts mid-streak. Final Stake is the size sitting on the layout after `wins` successful parlays, which is also what you would lose on a further spin if you failed to take down. That is the standard criticism of Paroli: the advertised $70 is realised only if you walk, and the next impulse is to go for four.

Limitations. Payout Odds default 1 is even money. Using 1 on a dozen bet is the wrong model. Win Streak is capped in the UI because longer parlays are mostly theoretical. Table maximums clip Final Stake. French even-money rules (la partage) change losing-zero outcomes but are not in this formula. The calculator will not tell you to press. It will tell you that three even-money doubles turn $10 into an $80 pile and $70 of paper profit. Educational, not a forecast that three reds are coming, not financial advice.

Paroli is the reverse Martingale: a parlay of three on even money, pressing winners rather than chasing losers. Write b for the base, n for the number of consecutive wins required, and r for net payout odds, with r = 1 for even money. After n successful parlays the position has been multiplied by (1 + r) each time. After that sentence the closed form is final stake = b(1 + r)^n and profit versus never having bet is b((1 + r)^n − 1). Defaults base 10, wins 3, payout 1 give (1 + 1)^3 = 8, so final stake 10 × 8 = 80 dollars and profit 10 × 7 = 70 dollars. Path: 10 wins and rides as 20; 20 wins and rides as 40; 40 wins and the position is worth 80.

Geometric-series bookkeeping recovers the same 70 without multiplying by eight first. Successive winning returns are 10 × 1, then 20 × 1, then 40 × 1, summing to 70. Equivalently the pile 80 minus the original 10 that was at risk equals 70 if one takes down after the third coup. Early modern Italian and French houses already knew a three-coup paroli; eighteenth-century French sources use the word for a doubled stake left to ride. No inventor solved roulette with it. Each spin remains a negative-expectation bet. Session expected value under a stop-at-three rule equals house edge times expected turnover. The stop truncates the right tail of huge parlays and some of the give-back; it cannot flip the sign of 1 / 37.

Probability of the advertised path on European even money is (18 / 37)^3 ≈ 11.54 percent from a standing start. The complementary 88.46 percent of attempts do not produce 70 dollars. Most cycles die on the first or second spin. Strict Paroli then resets to 10. The 70-dollar headline is the size of a completed parlay of three, not the expected value of trying. Using payout 1 on a column or dozen is the wrong model; those bets are not even money. A four-win even-money paroli would print profit 150 and final stake 160 at probability (18 / 37)^4 ≈ 5.61 percent. Houses historically liked a cap at three because the extra double is the spin the wheel most enjoys taking back.

Expected profit of a single attempt — reset to 10 after any loss, take down after three wins — is negative. One wins 70 with probability p^3 and loses some prefix of the geometric stakes with probability 1 − p^3; with p = 18 / 37 < 1 / 2 the weighted sum is below zero. House edge European even money about 2.70 percent, American about 5.26 percent, untouched by the parlay. Table maxima clip the 80. French even-money rules such as la partage change losing-zero outcomes and are not in (1 + r)^n. The standard criticism is behavioural: the advertised 70 is realised only if one walks, and the next impulse is to go for four.

Assumptions: every one of the three spins pays 1-to-1 and the whole position rides; no table maximum; no zero-special rules; take-down after n = 3. Reverse martingale / parlay of three with base 10 and payout 1 produces profit 70 and final stake 80 on the completed path. It is not an expected value, not a forecast that three reds are coming, and not a sixteenth-century Italian name that changes 18 / 37. Educational closed form (1 + r)^n, nothing more.