No edge, no wall, is ruin certain?

Question

Each bet wins 1 unit or loses 1 unit with probability $1/2$ each, a perfectly fair game. You start with 12 units and play forever against a house with unlimited money, so there is no upper target.

What is the probability you eventually go broke?

Accepted Answer

With an edge the ruin probability is (q/p)^i, but here p = q = 1/2, so q/p = 1 and that formula gives 1^{12} = 1. The clean way to see it: a fair one-dimensional walk is recurrent, meaning it is certain to visit every level eventually, including 0. There is no ceiling to save you, so sooner or later the walk touches 0 and you are wiped out. P(	ext{ruin}) = 1. The catch is timing: ruin is certain but can take an enormous number of rounds, and the expected time to reach 0 is actually infinite. Takeaway. Against an unlimited house, a game with no edge means certain ruin; only a genuine per-bet …

No edge, no wall, is ruin certain?

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