(a) (10 points) A grasshopper is moving over the integer lattice Z (the set of integers). Every minute it flips a coin and moves either one step to the right (from its current state k to \(k+1\)) if the coin shows H or one step to the left (from its current state k to \(k-1\)) if the coin shows T. Thus the grasshopper always moves to a neighboring site. Assume that the coin flippings are independent and the coin is biased, \(P(H) = \frac{3}{4}\) while \(P(T) = \frac{1}{4}\). What is the probability that the grasshopper never visits negative integers? \((Hint:\) Let \(a_n\) be the probability that the grasshopper never visits negative integers when it starts its random walk from the site n. Use the first step analysis, namely condition on the first move of the grasshopper. Express first \(a_0\) in terms of \(a_1\), and then \(a_1\) as a function of \(a_0\). For the latter calculation, use that when it starts at 1, in order to ever get to -1, the grasshopper has to return to 0 first.)
