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In addition, squares 2 and 4 are \"for sale\" and can be purchased by the players on a rst-come rstserved basis (each player may only own one square). As in Monopoly, if you land on the square owned by the other player, you must pay that player \"ren \". The rst player to run out of money loses the game. The goal of this question is to see how we can encode aspects of the game using the language of linear algebra. Later we will see how this will help us determine which of the two squares is more likely to be landed on and help us determine a winning strategy for the game. (a) (b) (d) (f) Because the 4-sided die was constructed poorly, it is not a fair die (1'. e. the probability that you roll a 1 may not be the same as the probability that you roll a 2). The probability of rolling each number using the game board's die is shown below: R011 4/ Probability 1 1/8 2 2/8 3 4/8 4 1/8 If you are on square 1, what is the probability that you will end up on square 2 after you roll the die? Square 4'? Square 1? (Note: (i) Remember that rolls which have you stop on square 3 result in you ending up on square 1 and (ii) since you must end up somewhere on the board, the sum of your three answers should add up to 1.) Let A = [(15,] be the 3 X 3 matrix whose ijth entry, 0.55;, is equal to the probability that you will end up on square 1' after one roll of the die provided you started on square 3'. Write down the matrix A. 3n Let xn = yn be the vector such that Zn 0 {En is the probability you will end up on square 1 after 11 rolls. 0 yn is the probability you will end up on square 2 after 11. rolls. 0 zn is the probability you will end up on square 4 after 11 rolls. 1 Since you initially start on square 1, x0 = 0 . Calculate Axe and interpret the meaning 0 of each of its entries in the context of the game. Now calculate A2xo and interpret the meaning of each of its entries in the context of the game. If you know the vector x\2. Imagine a 2player board game consisting of a board with four squares and one 4-sided die. Rules of the game. The game is fairly simple: both players start on square 1 and take turns rolling the die to advance their positions on the board (after square 4, players will continue back to square 1). For example, if a player starts on square 2 and rolls a 3, they will advance back to square 1. However, if a roll results in a player stopping on square 3, they will automatically be transported back to square 1. So, if a player starts on square 2 and rolls a 1, they will end up back on square 1. If they instead roll a 2, they will move past square 3 and end up on square 4. D Clarification to Question 2b: 0 For i = 1 or 2, (113 is the probability of landing at Square 4 after one roll of the die, if you started from Square i; 0 Forj = l or 2, agj is the probability of landing at Square j after one roll of the die, if you started from Square 4. 0 Note: There was a mistake in the original statement of the meaning of at}- as it should be the probability of landing on square 1' provided you started on square j