seed 20260821, edge 3%
  target       rounds  win rate   theory   EV/bet  longest lose streak
    1.01    2,000,000  96.0511% 96.0396% -2.9884%                    4
     1.1    2,000,000  88.1717% 88.1818% -3.0111%                    7
     1.5    2,000,000  64.6587% 64.6667% -3.0119%                   12
       2    2,000,000  48.5099% 48.5000% -2.9802%                   23
       3    2,000,000  32.4084% 32.3333% -2.7750%                   37
       5    2,000,000  19.3967% 19.4000% -3.0165%                   54
      10    2,000,000   9.6850%  9.7000% -3.1500%                  114
      20    2,000,000   4.8674%  4.8500% -2.6510%                  237
      50    2,000,000   1.9336%  1.9400% -3.3200%                  807
     100   20,000,000   0.9674%  0.9700% -3.2555%                 1331
    1000   20,000,000   0.0959%  0.0970% -4.0950%                 9961

$100 bankroll, $1 flat bets, cap 2,000 rounds — 50,000 sessions per target
  target busted<cap  median rounds (busted only)  alive@500  alive@2000  ever 2x bank
     1.5      14.6%                         1715     100.0%       85.4%         0.00%
       2      25.6%                         1554     100.0%       74.4%         0.06%
       5      48.7%                         1080      95.7%       51.3%        11.14%
      10      61.0%                          780      83.3%       39.0%        28.39%
     100      84.6%                        200.0      35.6%       15.4%        40.25%

Martingale at 2x target: $1 base, double after loss, $100 bankroll, 500-round sessions x 100,000
  wiped out (cannot cover next doubled bet): 88.88%
  finishers: 11.12%, median end balance $348.00

rounds until a multiplier first appears (10,000 trials each)
  >=    10x: median 7.0 rounds, mean 10
  >=   100x: median 71.0 rounds, mean 103
  >=  1000x: median 726.0 rounds, mean 1,034

total simulated rounds: 451,614,756