My class has been going over Elementary laws of Probability and probability of certain locations on a monopoly board. We also talked about vectors, matrices, and row sectors to find determinants. Can you please help me solve and understand the questions/concepts on this packet. Thanks so much! and have a great rest of your day!!! :)

Monopoly This worksheet we will compiete the Monopoly analysis we started on Wednesday. THE GAME We model the game by simply numbering the squares 1 through 40. where 1 Is the "Go" square and 40 is \"Boardwalk". We will only work with a simplified version oi the game: up There are no \"Community Chest" or 'Chance" cardsyou simply land on that square 0 You never get sent to 'Jail"if you land on "Go To Jail\" you lust stay on that square; neither do you go to jail if you roll 3 doubles in a row. THE PROBLEM We want to explore how often each square is landed on it we keep rolling dice for a long time. There is folklore wisdom that. when you play with the full rules. some squares get landed on significantly more often than others: is that true in the simplied game, or is the frequency distribution uniform? In any case, what is the frequency distribution to which we converge after many. many rolls? THE MODEL Now that we have talked about probability we can model this problem as follows: . Let's simplify our lives by thinking about one die roll at a time rather than two. Thus. in an actuai game of Monopoly we would only consider the board after rolls 2. 4. 6, and so on. - Considering each roll to be a unit of time, use the variable t to record the number of rolls. 0 Let s(t) be the square we are on after it rolls. Sc 3(t) is between 1 and 4t}, and 3(0) = 1. o For each 7: between 1 and 40, let pn(t) be the probability that 3(t) = 11. So, for example. 191(0) = 1.195(0) = 0. P5(1) = %. 1110(2) = - - Collect all of these individual probabilities together in a list: P(t)=[P1(3) P20): . P39\") . mom] Let's call this list the probability vector for the Monopoly game: Notice, the probabilities must add up to i, so at each time t we will aiways have 1 = 1310:) +1323) + - -- +2493) +P40(t)- Going back to our question, we want to understand what is happening to P(t) for very large it. 1We discussed these last time. but make sure you agree nowI See also Task #1. 1 Task #1 Let's begin, though, with very small t. We know, for example, that P(0) = [ 1, 0, 0, ... , 0, 0]. Write down P(1) and P(2): P(1) = P(2) = VECTORS & MATRICES Our approach will be to think of the various probability vectors as states and successive die rolls as a process that moves us from one state to the next as follows: P(0) ~ P(1) ~ P(2) ... . P(t) ~ ... Our first goal is to encode each transition P(t) ~. P(t + 1) using a single matrix M. To go further we need a little bit of matrix algebra. A matrix is just a 2-dimensional array of numbers. An a x b matrix has a rows and b columns. For example, 7 2' is a 2 x 3 matrix. If a matrix has just one row it is called a (row) vector. If it has just one column it is called a (column) vector. Our probability vector is a 1 x 40 matrix and is therefore a row vector. Matrices of the same size can be added simply by adding corresponding terms: 18 7 21+(2 1]=[239] Matrices can also be multiplied, but this operation is a bit more complicated: (1) You can only multiply a a x b matrix with a b x c matrix. That is, the number of columns of the first matrix must be the same as the number of rows of the second matrix. (2) The result is a a x c matrix. The entry in position (i, j) of the product is computed by taking row i of the first matrix (a 1 x b vector) and column j of the second matrix (a b x 1 vector), multiplying matching entries, and adding these b products together, e.g.THE TRANSITION MATRIX As noted above, the plan is to find a matrix M that represents the transitions w P(t) ~ P(t+ 1) ~ More precisely, we want a matrix M such that P(t) M = P(t + 1) Remember P(t) and P(t + 1) are 1 x 40 matrices, so M must have 40 rows and 40 columns. Task #2 Since P(0) M = P(1), and we know P(0) and P(1), what can you figure out about the entries of M? Task #3 Suppose we start at square 2 instead of square 1. What would P(0) be now? What would P(1) be? We still want M to transform P(0) to P(1). What new entries of M can we write down? Task #4 What if we start in position 3? In position 4? What does this tell us about the entries of M? Can you now write down M completely?STABLE VECTOR At the start of all of this we were asking whether or not P(t) converges to some stable proba- bility state Pauble when t gets really large. If so, the entries in Petable would tell us the long-term probabilities of being on each square. We were further asking whether all of these probabilities are equal (i.e. whether, in the long term, it is equally likely that we are on any of the squares). The answers to these questions lie in properties of the matrix M. We're interested in vectors v (l.e. 1 x 40 matrices) that remain unchanged when we multiply by M, i.e. UM = v. Slightly more generally, we can search for vectors that M scales by a factor X: UM = Av Any vector v that satisfies this equation is called an elgenvector of M, and the scalar > is the corresponding eigenvalue. Task #5 What are the eigenvalues and eigenvectors of the matrix M = 6 ;] You don't really need to know any of this right now-you'll learn about these things eventually in Math 245. We just need to know that we can find them: computation to the rescue!! MARKOV MAGIC The process we have described, whereby state P(t) is transformed to state P(t + 1) via a transition matrix M, is known as a Markov process. Task #6 Apply the matrix M in Task #5 to the vector w = [1 1] a bunch of times. What happens? Does it seem to converge on a particular vector? Again, you don't need to know about these things right now-you can find out more in Math 358 if you take the course. What matters to us is the following lovely fact: . Suppose ) is the eigenvalue of M with the largest absolute value. Then, in the long run, P(t) converges to the corresponding eigenvector. Our computation showed us that the largest eigenvalue of M is 1, and its eigenvector is Pstable = 40 ' 70 40 . .. . 40 The interpretation is that: every square is equally likely in the long run