A math professor has 4 umbrellas. He keeps some of them at home and some in the office. Every morning, when he leaves home, he checks the weather and takes an umbrella with him if it rains. In case all the umbrellas are in the office, he gets wet. The same procedure is repeated in the afternoon when he leaves the office to go home. The professor lives in a tropical region, so the chance of rain in the afternoon is higher than in the morning; it is 1/5 in the afternoon and 1/20 in the morning. Whether it rains of not is independent of whether it rained the last time he checked.
On day 0, there are 2 umbrellas at home, and 2 in the office. What is the expected number of days that will pass before the professor gets wet (remember, there are two trips each day)? What is the probability that the first time he gets wet it is on his way home from the office?