Walkthrough: A Paroli Run
Ten dollars on black, doubled after every win, banked after three in a row. The reverse Martingale, spin by spin.
A simulated 80-spin Paroli session. Thirteen first-spin losses, twenty-four broken runs and three completed triples that paid $70 each. Net −$160 after being up $50. The system that loses often and wins rarely, in the shape the arithmetic predicts.
Walk the system
A win moves one step on. A loss, or a third win, goes back to the start.
Stakes and results are in units of the base bet.
The plan
Bet $10 on blackbetBlackEven-money bet on the 18 black numbers. French: noir.Read full →. After a win, let the $20 ride. After a second win, let the $40 ride. After a third win, take the $80 down and start again with $10. A loss anywhere restarts at $10. Each attempt risks $10 of our own money and wins $70 if black comes three times running, which happens (18/37)³ = 11.5% of the time. We expect about one completed run in nine attempts.
Spins 1–25
R B B R R B R B B R B B R R B R 0 B B B B R B B R. Twelve attempts: four lost at the first spin (−$10 each, one of them to zerowheelZeroThe single green pocket that no even-money, dozen or column bet covers; the source of the house edge. Also: single zero, 0.Read full →), three lost at the second spin (−$10 each, because the $20 was the house’s money plus ours), four lost at the third spin (−$10 each, the same way), one completed triple at spins 18–20 (+$70). Net −$40. The completed run felt like proof. The eleven failures felt like nothing, which is the point of the design.
Spins 26–55
Two more triples, at spins 28–30 and 37–39, each part of a run of four blacks that we cashed after three: the fourth black won the fresh $10 and the next redbetRedEven-money bet on the 18 red numbers. French: rouge.Read full → took the $20. Fifteen attempts, net for the stretch +$10; running total −$30; peak +$50 at spin 39. Three triples in 27 attempts is the expected 3.1; at the peak the session is up $50 and feels like a method.
- p10 −$780
- p25 −$510
- median −$210
- p75 +$90
- p90 +$370
Spins 56–80
Thirteen losing attempts in a row, nine of them at the second or third spin where the ride ended one or two blacks short. Net −$130 for the stretch; final −$160. The streak of failures has probabilitymathProbabilityChance of an outcome, from 0 to 1 or 0% to 100%. One number: 1/37 = 2.70%.Read full → about 20%, which is to say it will happen to a ParolistrategyParoliPositive progression: double after a win, reset after a loss or after three wins in a row. Also: reverse Martingale.Read full → player roughly every fifth session of this length.
The accounts
Total wagered $1,460 over 80 spins against $800 for a flat $10 bettor: the rides add turnovermoneyTurnoverThe total amount wagered over a session or period, counting re-bet winnings. Also: handle, action.Read full →. Expected lossmathExpected lossExpected value times total amount wagered; the average amount lost over a session or a period.Read full → on $1,460 is $39; actual −$160 is less than one standard deviationmathStandard deviationSquare root of variance; the typical distance of a result from the expected value.Read full → below, unremarkable. The flat bettor beside us had the same expectation per dollar and a result with about half the swing.
What the session shows
Paroli is MartingalestrategyMartingaleProgression that doubles the bet after each loss. See the Strategy section for what it does and does not do.Read full → upside down: it loses one unitmoneyUnitThe base stake a system or a player works in. Progressions are described in units so they apply at any table.Read full → most of the time and wins seven units occasionally, instead of winning one unit most of the time and losing sixty-three occasionally. The distribution is less dangerous because the big number is a win, not a loss, and the stake never exceeds $40 of which only $10 is ours. The expected loss per dollar is unchanged. It is the better system for a player who wants variancemathVarianceA measure of how widely results spread around the expected value. High for straight up, low for even-money bets.Read full → without a cliff, and it still costs 2.70% of turnover.
Questions
Four in a row is 5.6% and pays $150. The target is a taste; the edge is the same at every target.
No. After one win the $20 on the table is yours. Paroli feels cheap because the loss of a ride does not look like a loss of your own money. It is.
Same expectation, smaller maximum stake, no table-limit cliff. Better in every way except the one that matters, which is unchanged.
Comic relief
Lose small often, win big rarely. The reverse of Martingale, and the same price.
Sources
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.