Drifting toward the goal from the halfway mark

Question

A player holds 2 chips and plays a favorable game: each round she wins a chip with probability 	frac23 and loses a chip with probability 	frac13. She stops at 4 chips (goal) or 0 chips (broke). What is the probability she reaches 4 chips before going broke?

Accepted Answer

With p0 = 0, p4 = 1, first-step analysis gives pk = 	frac23 p{k+1} + 	frac13 p{k-1}. The ratio is r = q/p = 	frac12, so pk = \frac{1 - (1/2)^k}{1 - (1/2)^N}, \qquad N = 4. At k = 2: p2 = \frac{1 - 1/4}{1 - 1/16} = \frac{3/4}{15/16} = \frac{4}{5} = 0.8. Takeaway. A favorable drift (r = 	frac12) plus a head start (halfway to goal) gives a strong 4/5 chance. The same recursion covers every start, edge, and target.

Drifting toward the goal from the halfway mark

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