unsymetric random walk

start at 0 on a number line, fli a coin, move +2 on heads and -1 on tails, what is the probability that it will hit -1?

Solution 1:

p: the prob from x= k to x=k-1 (markov chain is homogeneous)
Look at x=0
p=0.5* 1 + 0.5*p^3
Solve p.

Solution 2:

x_n = x_{n-1} + z_n
x_0 = 0, z_n = +2, -1 with probability 1/2.
Let y_n = t^x_n, select t so that y is a martingale.

The condition for y_n to be martingale is 1/2(t^2 + 1/t) = 1.
Solve this equation to get t = (sqrt(5) -1)/2.
y_0 = 1, y_n goes to 0 or stops at 1/t.
Due to optional stopping time theorem, y_n stops at 1/t with probability p, and
p/t + (1-p) 0 = 1.

So p = t.