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For the following problems use Matlab or the Java program MarkovChain.java on Canvas (or software of your choice that supports matrix computations). (1) A gambler starts with k dollars, and engages in a series of bets. On each bet, the gambler wins one dollar with probability 1/2, and looses one dollar with probability 1/2. If he ever reaches a fortune of 6 dollars, he quits and his fortune stays at that level, and similarly if he ever hits 0 he quits and he stays broke. What is the approximate probability that the gambler ends up broke? Can you guess at how this probability should depend on the starting fortune k ? Identify the communications classes and state for each class whether it is transient or recurrent, and whether it is periodic (if so, give the period). (2) Six positions, numbered 0 to 5, are arranged on a circle. A particle moves around the circle as follows. With probability 1/3 the particle moves counter-clockwise one position, with probability 1/3, the particle moves clockwise one position, and with probability 1/3 the particle remains at its current position. If the particle starts off at position 0, what is the distribution of the position after 5 transitions? What about after 50 transitions? If the particle starts off at a random position, uniformly distributed from 0 to 5 , what is the distribution of the position after 1 transition? How do your answers change if the particle moves counter-clockwise two positions with probability 1/7, counter-clockwise one positions with probability 2/7, and with probability 4/7 the particle moves 1 position clockwise? Identify the communications classes and state for each class whether it is transient or recurrent, and whether it is periodic (if so, give the period)