Big edge, risk of ruin with a 75 percent bet and 4 units

Question

Each bet wins 1 unit with probability p = 0.75 or loses 1 unit with probability q = 0.25. You start with 4 units and play forever against a house with unlimited money. What is the probability you eventually go broke?

Accepted Answer

With an edge and no ceiling, the eventual-ruin probability from i units is P(	ext{ruin}) = \left(\frac{q}{p}ight)^i. Here q/p = 0.25/0.75 = 1/3 and i = 4: \left(	frac13ight)^4 = \frac{1}{81} \approx 0.0123. A strong edge makes each unit of bankroll cut the risk to a third of what it was, so even a modest 4 units drives ruin down near 1%. Takeaway. A large edge gives a small ratio q/p = 1/3, and (q/p)^i = 1/81; strong edges make ruin vanish quickly as bankroll grows.

Big edge, risk of ruin with a 75 percent bet and 4 units

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