Random Walk and Statistics, last time we hit 0

[frinny913]

This is for my Statistics and Stochastic Processes class, we are learning about random walk

1. Let p<q where p is success (+1) and q is failure (-1). Let T=last time we hit 0 (T$$\geq$$0) Find P(T=t)

Then using the answer from the above question, make up a formula for $$\sum(2n choose n)*x^{n}$$

The Attempt at a Solution

I know the law of large numbers says that the random walk will eventually enter a downward cone and never leave, meaning that at a long enough time the amount of loses will be greater than the wins because of p and q. But I don't know where this gets me.

 

[lippyka]

For the second question, the formula for \sum(2n choose n)*x^{n} is:\sum_{n=0}^{\infty} (2n \choose n) x^n = \frac{1}{\sqrt{1-4x}}