Quick or long, game length with a strong 3-in-4 edge

Question

Each bet wins 1 point with probability p = 3/4 and loses 1 point with probability q = 1/4. You start at 1 point and stop at 3 points (win) or 0 points (broke). On average, how many bets until the game ends?

Accepted Answer

The drift per bet is \mu = p - q = 	frac12, and E[T] = \frac{E[	ext{final}] - 	ext{start}}{p - q}, \qquad E[	ext{final}] = N \cdot hi. With r = q/p = 1/3, the win probability from 1 toward N = 3 is h1 = \frac{1 - 1/3}{1 - (1/3)^3} = \frac{2/3}{26/27} = \frac{9}{13}. So E[	ext{final}] = 3 	imes 	frac{9}{13} = 	frac{27}{13}, and E[T] = \frac{27/13 - 1}{1/2} = \frac{14/13}{1/2} = \frac{28}{13} \approx 2.154. A strong edge and a large drift end the game fast, in just over two bets on average. Takeaway. Large drift shortens games: \frac{N hi - i}{p - q} = 28/13 bets, the strong edge pullin…

Quick or long, game length with a strong 3-in-4 edge

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