A biased walk on four steps

Topic: Random walks and first passage · more questions on this topic
‹ A fair walk between 0 and 8 Checking the duration formula ›

Take p=2/3, d=4, and x=1. Compute the probability of hitting 4 before 0, and compare it with the fair-walk value.

Worked answer (click to show)

Here q=1/3 and q/p=1/2, so

h_1=\frac{1-(1/2)}{1-(1/2)^4}=\frac{1/2}{15/16}=\frac{8}{15}.

The fair-walk value at the same point is 1/4. Since 8/15>1/4, the upward drift lifts the hitting probability above the fair line, as the sign check requires when p>q.