(a) Consider the following problem. You are driving down a long line of parking spaces, looking for a place to park your car. Think of potential parking spaces at locations -100, -99, -98, ... , -1, 0, 1, 2, .... You are currently at location -100, which is unoccupied, and you are considering whether to park there. If you decide not to park there, you will drive a bit further and see whether location -99 is occupied or not; if it is, you must proceed to -98; and if not, you must decide whether to park there or try your luck with -98, and so on. You cannot circle back to empty parking spots; U-turns are strictly forbidden. Each spot is occupied with another car with probability a:
(b) Suppose that when you are at spot n, you can see the occupancy status of spot n and the next five spots ahead. What should you do? What is the expected cost to you?
(c) Suppose that you dont know the value of a. Being very specific, suppose a = .9 or a = .7, with each of these two values being equally likely. (If you know the terminology, this makes the occupancy status of the spots exchangeable instead of being independent.) What should you do? What is the expected cost to you? (Use the formulation where you cannot see ahead. A numerically derived solution is quite adequate. An important first step is to decide on what links the "past" to the "future" in this problem.