Try the random walk on a path of length 4.
Here the probability distributions:
Let a(n), b(n), c(n), and d(n) denote the probabilities that the the fields are visited in the nth step. If the right field is absorbing, the recurrence formulas are
with initial conditions a(0)=1, b(0)=0, c(0)=0, and d(0)=0.
When solving this system, we get. The characteristic equation for the resulting recurrence relation, -4x2+3=0 has two zeros, ,and , and when solving the system of linear equations we obtain, we get
Obviously for even n we have d(n)=0, and for odd n we get d(n)=.
For the expected value for the transition time from the leftmost to the rightmost cell, (using a formula for a series) we get
Erich Prisner 2004