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1. Max and Patty decide to repeatedly gamble by throwing a fair die. Max is declared winner in a throw if the outcome of the die is odd in that throw; otherwise, Patty is declared winner. After each flip, the winner receives 10 dollars from the loser. Initially Max starts with i dollars and Patty starts with j dollars. In this scenario, derive the closed form expression for Pj, which represents the probability that Patty will ultimately wipe Max out during the gambling. 2. Consider a variant of the Josephus problem where the circle of people is traversed along the counter clockwise direction, but the people are still numbered sequentially along the clockwise direction. For this variant of the Josephus problem, formulate the recurrence relation on J(n) and find its closed form expression. 3. Consider a three-stage variant of the shoeshine shop model, where there are three different chairs and an entering customer must enter the system through chair 1,followed by sequential service from chair 2 and then chair 3.Draw the state transition diagram using the state interpretation. Also, provide the system of equations for the steady-state probability.

1. Max and Patty decide to repeatedly gamble by throwing a fair die. Max is declared winner in a throw if the outcome of the die is odd in that throw; otherwise, Patty is declared winner. After each flip, the winner receives 10 dollars from the loser. Initially Max starts with i dollars and Patty starts with j dollars. In this scenario, derive the closed form expression for Pj, which represents the probability that Patty will ultimately wipe Max out during the gambling. 2. Consider a variant of the Josephus problem where the circle of people is traversed along the counter clockwise direction, but the people are still numbered sequentially along the clockwise direction. For this variant of the Josephus problem, formulate the recurrence relation on J(n) and find its closed form expression. 3. Consider a three-stage variant of the shoeshine shop model, where there are three different chairs and an entering customer must enter the system through chair 1,followed by sequential service from chair 2 and then chair 3.Draw the state transition diagram using the state interpretation. Also, provide the system of equations for the steady-state probability.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1. Max and Patty decide to repeatedly gamble by throwing a fair die. Max is declared winner in a throw if the outcome of the die is odd in that throw; otherwise, Patty is declared winner. After each flip, the winner receives 10 dollars from the loser. Initially Max starts with i dollars and Patty starts with j dollars. In this scenario, derive the closed form expression for Pj, which represents the probability that Patty will ultimately wipe Max out during the gambling. 2. Consider a variant of the Josephus problem where the circle of people is traversed along the counter clockwise direction, but the people are still numbered sequentially along the clockwise direction. For this variant of the Josephus problem, formulate the recurrence relation on J(n) and find its closed form expression. 3. Consider a three-stage variant of the shoeshine shop model, where there are three different chairs and an entering customer must enter the system through chair 1,followed by sequential service from chair 2 and then chair 3.Draw the state transition diagram using the state interpretation. Also, provide the system of equations for the steady-state probability.
Transcribed Image Text:1. Max and Patty decide to repeatedly gamble by throwing a fair die. Max is declared winner in a throw if the outcome of the die is odd in that throw; otherwise, Patty is declared winner. After each flip, the winner receives 10 dollars from the loser. Initially Max starts with i dollars and Patty starts with j dollars. In this scenario, derive the closed form expression for Pj, which represents the probability that Patty will ultimately wipe Max out during the gambling. 2. Consider a variant of the Josephus problem where the circle of people is traversed along the counter clockwise direction, but the people are still numbered sequentially along the clockwise direction. For this variant of the Josephus problem, formulate the recurrence relation on J(n) and find its closed form expression. 3. Consider a three-stage variant of the shoeshine shop model, where there are three different chairs and an entering customer must enter the system through chair 1,followed by sequential service from chair 2 and then chair 3.Draw the state transition diagram using the state interpretation. Also, provide the system of equations for the steady-state probability.
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