(Absent-Minded Professor). A mathematics professor has two umbrellas, both of which were originally at home. The professor walks back and forth between his home and office, and if it is raining when he starts a journey, he carries an umbrella with him unless both his umbrellas are at the other location. If it is clear when he starts a journey, he does not take an umbrella with him. We assume that at the time he starts a journey, it rains with probability p and the states of weather are mutually independent.
(a) Find the limiting proportion of journeys in which the professor gets wet.
(b) What if the professor had three umbrellas to begin with, all of which were originally at home?
(c) Is the limiting proportion affected by how many umbrellas were originally at home?