4. (30pts) Smurfs. Three smurfs share a dorm. Each has his own bed. Every night, they retire to bed one at a time, always in the same order from the youngest to the oldest. On a particular evening, the youngest smurf, who always retires first, has had too much to drink. He randomly chooses one of the three beds to sleep on. As each of the other smurfs retires, he chooses his own bed if it is not occupied, and otherwise randomly chooses another unoccupied bed. (a) What is the probability that the first smurf sleeps in his own bed? (b) What is the probability that the second smurf sleeps in his own bed? (c) What is the probability that the last smurf sleeps in his own bed? (d) (Bonus) Suppose, instead of three, there are 100 smurfs and 100 beds. Repeat part (c)