Risk of ruin against an infinitely rich house

Question

Each bet you win 1 unit with probability $p = 0.6$ or lose 1 unit with probability $q = 0.4$ . You start with 5 units and play forever against a house with unlimited money, so there is no upper target, only the floor at 0.

What is the probability you eventually go broke?

Accepted Answer

With no upper wall, the only question is whether the walk ever slides all the way down to 0. When you have an edge (p q), the probability of eventual ruin starting from i units is P(	ext{ruin}) = \left(\frac{q}{p}ight)^i. Each extra unit of cushion multiplies your risk by the same factor q/p < 1. Here q/p = 0.4/0.6 = 2/3 and i = 5: \left(	frac23ight)^5 = \frac{32}{243} \approx 0.132. So about a 13% chance of ruin despite the edge, and the risk shrinks geometrically as you add bankroll. (If instead p \le 1/2, the ratio q/p is at least 1 and ruin becomes certain.) Takeaway. Against an unl…

Risk of ruin against an infinitely rich house

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