Umbrella Markov Chain: Expected Rainy Commutes Until Getting Wet

A professor commutes between home and office once per day. Each commute, rain occurs independently with probability $p$. The rule: if it rains and you have at least one umbrella at your current location, you must take exactly one umbrella with you; otherwise you get wet. You start with 2 umbrellas at home and 0 at the office. Model the umbrella count at your current location as a Markov chain on states $\{0, 1, 2\}$. 1. Write down the transition matrix and explain the state dynamics. 2. Compute the expected number of rainy commutes until the first time you get wet, as a function of $p$, starting from the initial configuration.