# Random Walk on a number line and further cases

i) A monkey is sitting on 0 on the real line in period 0. In every period t ∈ {0, 1, 2, . . .} it moves 1 to the right with probability p and 1 to the left with probability 1−p, where p ∈ 1 2 , 1 . Let πk denote the probability that the monkey will reach positive integer k in some period t > 0. The value of πk for any positive integer k is

A)p

B)1

C)0

D)p/1-p

ii) Refer to the previous question. Suppose p = 1/2 and πk denote the probability that the monkey will reach positive integer k in some period t > 0. The value of π0 is

(a) 0

(b) 1/2^k

(c) 1/2

(d) 1

The answer key says the answer to both of them is 1. What is the possible mathematical explanation and also the intuition?