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