Markov Chain Weather Model

A Markov chain models daily weather with three states: Sunny, Cloudy, and Rainy. If a given day is Sunny, the probabilities for the next day are 0.7 for Sunny, 0.2 for Cloudy, and 0.1 for Rainy. If a day is Cloudy, the probabilities are 0.4 for Sunny, 0.3 for Cloudy, and 0.3 for Rainy. If a day is Rainy, the probabilities are 0.2 for Sunny, 0.3 for Cloudy, and 0.5 for Rainy. The initial state vector, representing the respective probabilities of Sunny, Cloudy, and Rainy weather, is [0.5, 0.3, 0.2]. What is the probability that the weather will be Rainy after 3 days?