Problem 5 Consider the 3 x 3 board of Snakes and Ladders we studied in class. (S

Question

Problem 5 Consider the 3 x 3 board of Snakes and Ladders we studied in class. (See Figure) The rules are: every turn you toss a fair coin. Heads move you one step forward while tails move you two steps forward. If you are at the bottom of a ladder you move to its top right away. If you are at the mouth of a snake you slide down to its tail right away. You start at 1. The game ends when you reach 9. (a) Calculate the average number of steps it takes one player to go from start to finish. (b) Calculate the average number of steps it takes two players to go from start to finish. (Hint: the two players are playing independently, taking turns moving. Hence, we can consider a game where the two players make their moves simultaneously, but using two independent fair coins. The game ends when one of them reaches the finish. Thus, consider a Markov chain with state space being pairs of numbers. The chain starts at (1, 1). States (a, b) with a or b being 9 are absorbing. Now, your task is to calculate the number of steps before absorption. The dimension of the matrix is rather large, and so once you set up the problem correctly you will need to tise a computer to do the computation.) (c) John and Mary are playing the game. John starts and then they take turns moving. Wha.t is the probability John wins? (Hint: If we instead make the two players move simultaneously, as in part (b), and John?s position is represented by the first coordinate, then we are asking for the probability the Markov chain is absorbed at a state of type (9, b). Collapse all these states into one and collapse the remaining ones [of type (a, 9) with a not = to 9] into one and answer the question.)

Explanation / Answer

You can be 95% confident that the mean salary for all production managers with at least 15 years of experience is between what two numbers (the t-statistic with 8 degrees of freedom is 2.306)? What assumption are you making about the distribution of salaries?

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