A stochastic law, wave seven — chance
RUIN — the walk that always ends
A single stake bets even money, step after step, drifting between broke and target. One bold walk plays out to its barrier inside a fan of every path it could have taken — and the house's edge quietly fixes how it ends.
The mathematics of a certain end
01Even money, no memory
Each step is one bet. Win and the stake rises by a unit; lose and it falls by one. The walk has no memory of where it has been — only where it stands. That single rule, repeated, is enough to decide everything.
02Two absorbing walls
The walk is trapped between zero and the target. Touch either and the game stops — there is no bouncing back from broke, and no reason to keep playing once you've won. Zero is ruin. Every path ends at a wall.
03The drift decides
With a fair coin the odds of ruin are just your distance to each wall. Tilt the coin by even two percent — the house edge — and ruin becomes near-certain. The fan leans down; the exact probability is printed, and the ensemble converges to it.
A hundred lives of one gambler, drawn at once — and almost all of them end at zero.