A fair walker starts at position 2 on positions 0, 1, 2, 3, 4, 5, 6, 7. Each step goes up 1 or down 1 with probability 1/2, independently. Positions 0 and 7 are absorbing. What is the expected number of steps until absorption?

Learn / Library / Probability

Average length of ruin game

ProbabilityMedium

A fair walker starts at position $2$ on positions $0$ , $1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ . Each step goes up $1$ or down $1$ with probability $\frac{1}{2}$ , independently. Positions $0$ and $7$ are absorbing. What is the expected number of steps until absorption?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Let D(k) be the expected duration from k; it satisfies a second-difference equation with zero values at both ends.

Solution

Let $D (k)$ be the expected number of steps from position $k$ , with $D (0) = 0$ and $D (7) = 0$ .
For interior positions $D (k) = 1 + \frac{D ( k - 1 ) + D ( k + 1 )}{2}$ .
The solution that vanishes at both ends is $D (k) = k \times (7 - k)$ .
At $k = 2$ this gives $2 \times 5$ .
So the expected number of steps is $10$ .

Answer: $10$

Next question Back to the library