Solve problem 4 and correct answers only please.

Problem 3. Batman is chasing the joker. There are four places each could be, arranged in the corners of a square. Batman is moving from his corner to an adjacent corner at rate 1 (rate 1/2 to each). The Joker picks some rate v, and moves from his corner at rate v, in the same way. Find the expected time before they meet are at the same corner). Hint: at each time they are either in opposite corners, nearby corners, or the same corner. Find transition probabilities between these. Problem 4. Batman is again chasing the Joker. This time the chase is in discrete time. In each step, Batman stays in his current corner with probability a, and otherwise moves to a random nearby corner. The joker stays in place with probability and otherwise jumps to a nearby corner. (a) As in the previous problem, find the transition probabilities for the distance between Batman and the Joker. (b) If they start at opposite corners, find the expected time to meet. Note: The case a = B = 0 is different! =