Hi, I need an analytical solution to a problem which I believe was solved decades ago.
The problem is as following:
A particle x begins at time t=0, with a value of 0. At each time interval, t=1,2,... it is incremented by 1 with probability p, and decremented by 1 with probability q=1-p. There are two boundaries a>0 and b<0, such that when the particle hits either one it stops.
I would like to know the probability that the particle hits a (or b) before b (or a). I'd also like to have the solution with maximum intermediate steps and explanation.
In a classic text by William Feller "An Introduction to Probability Theory and its Application", this problem is discussed in full detail.