1. As a New Year's resolution, your neighbor Amanda decides to start working out. Each day she can either run, do push-ups, or do sit-ups. If Amanda chooses to run today, she wouldn't run again tomorrow, and she would be 60% likely to do push-ups and 40% likely to do sit-ups. If she chooses to do push-ups today, then she will do push-ups tomorrow with a probability of 0.4, and she is equally likely to run or do sit-ups otherwise. If she chooses to do sit-ups today, then she will run tomorrow with probability 0.7 and will do push-ups otherwise.

(a) Construct a Markov Chain describing the type of exercise performed each day. Define the states and the one-step transition probability matrix.

(b) Indicate the period of each state.

(c) Assume that Amanda decides to go for a run on Wednesday. What is the probability that Amanda will not go for a run on Sunday of the same week?

(d) (Now assume that on Wednesday Amanda is three times more likely to run than to do push-ups (she avoids doing sit-ups on Wednesdays). What is the probability that she will run on Sunday of the same week?