Devin has N umbrellas that he wears during his commute to and from his house and work. Some of the umbrellas are at his house and some are at work at any point. Devin only takes an umbrella with him during his walk if it is raining, which occurs with probability p independently. At t=0, Devin is at home. At each time step after, Devin travels to the location, taking an umbrella with him if it is raining and one is available at his location. If all umbrellas are at one location while Devin is at the other and it rains, then Devin gets wet.

a) If N = 1, what is the probability that Ben gets wet at t = 2?

b) Draw a Markov chain and provide the transition probability matrix with N = 4. Identify the state and event that describes Ben getting wet in the Markov chain