Solution
The Uphill Drunk
Show the problem again
A walker on the integer line starts at 0. Each step he moves +1 with probability 2/3 or −1 with probability 1/3, independently. What is the expected number of steps until he first reaches +10? Give an exact number.
Worked solution
The answer is 30. Each step has mean drift 2/3 − 1/3 = 1/3. The hitting time T of +10 has finite mean (positive drift), and S_T = 10 exactly because the walk moves one unit at a time. Wald's identity gives E[S_T] = E[T]·(1/3), so E[T] = 30. Alternatively, reaching +10 means ten successive 'climb one level' times, each with mean m satisfying m = 1 + (1/3)(2m), so m = 3.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.