s 3 and 4 require some thought. You will need to find an elegant solution (not a bruce force solution) to Problem 3, and then it will easily generalize to Problem 4. 1. In a certain state a voter is allowed to change her/his party affiliation (for primary elections) only by abstaining from the primary for one year. Let s1 denote the state that indicates that the person votes Blue, s2 that she/he votes Green, and s3 that she/he abstains, in the given year. Experience shows that a Blue will abstain 1/2 the time in the following primary, a Green will abstain 1/4 of the time, while a voter who abstained for a year is equally likely to vote for either party in the next election. a) Find the Transition Probability Matrix. b) Find the probability that a person who votes Blue this year will abstain 3 years from now. c) Classify the states. (I.e. the communicating classes, their periods, the hierarchy, the ergodic classes, the transient classes, the cyclic subclasses, etc.) d) In a given year of the population votes Blue, Green and the rest abstain. What proportions do you expect in the next primary election? 2. Seven boys are playing with a ball. The first boy always throws it to the second boy. The second boy is equally likely to throw it to the third or seventh. The third boy keeps the ball if he gets it. The fourth boy always throws it to the sixth. The fifth boy is equally likely to throw it to the fourth, sixth or seventh boy. The sixth boy always throws it to the fourth. The seventh boy is equally likely to throw it to the first or fourth boy? a) Set up the transition matrix P b) Classify the states. c) Give an interpretation for the chain ending up in one of the ergodic sets. 3. Suppose the process below in Figure 1 starts in state 2. What is the mean number of times it visits state 2? Figure 1: Markov chain for Problem 3. 4. In Problem 2 suppose the ball is given to the fifth boy. What is the mean number of times that the seventh boy has the ball? What is the mean timespent before reaching an ergodic set? Hint: Generalize the analysis of Problem 4