A game piece is moving randomly on a 3 x 3 board: in each step it...

 

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A game piece is moving randomly on a 3 x 3 board: in each step it can move up, down, left, or right (if it stays on the board), choosing with equal probability among the possible steps. (E.g. if it is in a corner then it has two possible steps each with probability 1/2, if it is in the middle of the board then it has four possible steps, each with probability 1/4.) (a) Suppose that the game piece starts at the middle of the board at time 0. Let A, be the probability that it is standing in the middle of the board at time n. Find lim,o P(A,). (b) Suppose that the game piece starts at the lower right corner of the board at time 0. Let B, be the event that the game piece is not at the middle of the board at time n and at time n+1. Find lim,- P(Bn). 1-00 A game piece is moving randomly on a 3 x 3 board: in each step it can move up, down, left, or right (if it stays on the board), choosing with equal probability among the possible steps. (E.g. if it is in a corner then it has two possible steps each with probability 1/2, if it is in the middle of the board then it has four possible steps, each with probability 1/4.) (a) Suppose that the game piece starts at the middle of the board at time 0. Let A, be the probability that it is standing in the middle of the board at time n. Find lim,o P(A,). (b) Suppose that the game piece starts at the lower right corner of the board at time 0. Let B, be the event that the game piece is not at the middle of the board at time n and at time n+1. Find lim,- P(Bn). 1-00

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