Indexing the archive…
Your Universe of Digital Possibilities
The system that cannot lose: bet one, and after every loss double the bet, so the next win repairs everything plus one. Every rule is printed — bankroll 1 000, table limit 500, a 2% house edge. The system wins, and wins, in tiny steps up a long staircase. Then one losing streak meets the table limit.
Stop a fair game by any rule that cannot see the next coin and its expected value does not move; give the house an edge and the expected loss is exactly the edge times the expected stake (Wald’s identity). A betting system chooses only when to stop and how much to stake — so it can change the shape of the outcome, never its mean.
The exact central interval for K, the count of wins in n independent plays at win probability p — computed from the rules alone, before a single play. A record inside it is the rules working; a record outside it happens one time in twenty, and the band says so in advance. When the chance of each play is fixed before it is played but differs from play to play, the same band comes from the Poisson-binomial sum.
A betting system is a stopping rule and a stake schedule. Neither touches the coin, so neither touches the drift: every unit staked loses 2% in expectation, and the total you expect to lose is exactly 2% of the total you expect to stake. The martingale only changes the shape — a tall spike of small wins and a far bar of ruin. The Wager owns the other cliff, staking too much of a favourable bankroll; The Ensemble owns time against the crowd; and The Ticket, next door, carries the lesson about shape and mean into infinity.