1. stochastic particle movement probabilities

consider a particle that moves along the set of integers. if it is presently at i then it next moves to i+1 with probability p and i-1 with probability 1-p. starting at 0, let a denote the probability that the particle ever reaches 1.

a. show a=p+(1-p)a^2
b. show: a= 1 if p>= .5
p/(1-p) if p<.5

c. calculate the probability that the particle reaches n, for n>0
d. Suppose p<.5 and the particle does reach n, n>0. if the particle is at i, i<n, and n has not yet been reached show particle will move to i+1 with probability 1-p. OR
P(next at i+1 | at i and will reach n)= 1-p

2. Hi,

what part of the problem have you done?, what have you tried?

for a., if you haven't done it yet, you have to decompose the event according to the first step: either the first step goes to 1, and we are done, or it goes down to -1, hence the particle needs to first go back to 0 and then from 0 to 1. But going from -1 to 0 is "the same" as going from 0 to 1... I let you try to make some sense out of this and conclude.