Consider a simple stochastic model of the weather in Syracuse in which each day is classified as either Sunny, Rainy or Windy. The weather tomorrow ONLY depends on the weather today, but NOT on the weather of previous days. If today is Sunny, then tomorrow is Sunny with probability 40%; tomorrow is Windy with probability 60%.
If today is Rainy, then tomorrow is Sunny with probability 25%; tomorrow is Rainy with probability 50%.
If today is Windy, then tomorrow is Sunny with probability 33%; tomorrow is Windy with probability 33%.
a. Find the transition matrix P describing the model.
b. If today is Sunny what is the probability that it will be Rainy after 3 days?