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Part II. Problems 1. A sequence of Os and 1s is generated such that a 0 is followed by a 1 with probability 0.5 and a 1 is followed by a 0 with probability 0.8. a) You see a 0 entry. What is the chance the next two entries are 1, then 0? b) Given 0 , what is the chance the blanks will be filled with 0 0 1 0 1 (in that order)? If a 1 is twice as likely as a 0 now, write a matrix formula that forecasts the fifth entry from now. If you have a matrix calculator, compute it. d) If many terms of this sequence are generated, what proportion of them are a 0? A 1? 2. Where in the world is Carmen Sandiego? Suppose for simplicity that she travels between New York City and Los Angeles. On any given day, if she is in NYC, there is a 40% chance she travels to LA. If she is in LA, there is a 20% chance she travels to NYC. a) It is Monday, and Carmen Sandiego is in NYC. What is the chance she will be in LA on Wednesday? (Do not use matrices.) b) It is Monday, and Carmen Sandiego is in NYC. What is the chance she will be in NYC again on Thursday? (Do not use matrices.) c) Assume Carmen Sandiego is equally likely to be in NYC or LA on Monday. Write a matrix formula for the forecast of where she will be next Monday. If you have a matrix calculator, compute it. d) If this game continues, what proportion of time does Carmen Sandiego spend in each city in the long run? 3. A simple artificial intelligence in a turn-based battle game is in one of three states on any given turn: idle, attack, defend. From one turn to the next, here are the probabilities of the status changes: idle-attack 0.5 idle-defend 0.4 attack-idle 0.2 attack-defend 0.4 defend-idle 0.1 defend-attack 0.9 a) The a.i. charges a "super effective" attack during any turn that it is idle, and it requires three turns before it is ready. What is the chance it will have the super effective attack ready on the third turn from now if it defended on this turn? (No matrices needed.) b) On the current turn, the a.i. is in "attack" state. What is the chance it is in the "attack" state on: (i) next turn? (ii) The second turn from now? (iii) The third turn from now? (Do not use matrices.) c) What proportion of time does the a.i. spend in each state after many turns