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In a simplified version of a classic murder mystery board game, players move between locations to deduce details about a murder (the murderer, the weapon, and the location). We know the following about the locations and rules for movement: • There are six locations: Kitchen, Ballroom, Dining Room, Cellar, Lounge, and Hall. Dining Room is adjacent to Kitchen, Cellar, and Lounge . Cellar is adjacent to Ballroom and Hall (in addition to the Dining Room) Kitchen and Ballroom are adjacent (in addition to the above) Lounge and Hall are diacent (in addition to the above)

In a simplified version of a classic murder mystery board game, players move between locations to deduce details about a murder (the murderer, the weapon, and the location). We know the following about the locations and rules for movement: • There are six locations: Kitchen, Ballroom, Dining Room, Cellar, Lounge, and Hall. Dining Room is adjacent to Kitchen, Cellar, and Lounge . Cellar is adjacent to Ballroom and Hall (in addition to the Dining Room) Kitchen and Ballroom are adjacent (in addition to the above) Lounge and Hall are diacent (in addition to the above)

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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In a simplified version of a classic murder mystery board game, players move between locations to deduce details about a murder (the murderer, the weapon, and the location). We know the following about the locations and rules for movement: There are six locations: Kitchen, Ballroom, Dining Room, Cellar, Lounge, and Hall. • Dining Room is adjacent to Kitchen, Cellar, and Lounge . Cellar is adjacent to Ballroom and Hall (in addition to the Dining Room) • Kitchen and Ballroom are adjacent (in addition to the above) Lounge and Hall are adjacent (in addition to the above) . All connections are bi-directional (players can move in both directions) Players must move to an adjacent location on each turn and cannot stay in the same location • Players move to adjacent locations with equal probabilities at each time step If you wish, you can simplify the location using their respective starting letters. I.e. K,B,D,C,L,H
Transcribed Image Text:In a simplified version of a classic murder mystery board game, players move between locations to deduce details about a murder (the murderer, the weapon, and the location). We know the following about the locations and rules for movement: There are six locations: Kitchen, Ballroom, Dining Room, Cellar, Lounge, and Hall. • Dining Room is adjacent to Kitchen, Cellar, and Lounge . Cellar is adjacent to Ballroom and Hall (in addition to the Dining Room) • Kitchen and Ballroom are adjacent (in addition to the above) Lounge and Hall are adjacent (in addition to the above) . All connections are bi-directional (players can move in both directions) Players must move to an adjacent location on each turn and cannot stay in the same location • Players move to adjacent locations with equal probabilities at each time step If you wish, you can simplify the location using their respective starting letters. I.e. K,B,D,C,L,H
(a) Construct the Markov diagram for this the locations of this game and show that the associate transition matrix T can be written as the following. T= = 0 0 0 00 000 0 0 0 0 00 2100 NITO O 0001INTIN Hint: Be sure to state your ordering of the states and justify the entries in T (b) Prove (without computing powers of T) that the matrix T is not a regular matrix. (c) A secret tunnel now added between the Kitchen and Lounge to connect the two areas. All existing connections remain and player move with the same rules. I.e. Players cannot stay in the same location and move to adjacent locations with equal probability. i. State (and justify) the new transition matrix R with the new transition probabilities. ii. Show that this new transition matrix R is regular and determine the stable distribution of this new transition matrix. For this part, you may use a computer program/software to assist your calculations. Be sure to state which parts were done using a computer and provide details of the computations you performed. (d) The choice of connecting the Kitchen and Lounge to create a regular transition matrix is not a unique choice. Starting with the original transition matrix T, provide (and justify) a general criteria required to create a regular transition matrix via the introduction of a single secret tunnel.
Transcribed Image Text:(a) Construct the Markov diagram for this the locations of this game and show that the associate transition matrix T can be written as the following. T= = 0 0 0 00 000 0 0 0 0 00 2100 NITO O 0001INTIN Hint: Be sure to state your ordering of the states and justify the entries in T (b) Prove (without computing powers of T) that the matrix T is not a regular matrix. (c) A secret tunnel now added between the Kitchen and Lounge to connect the two areas. All existing connections remain and player move with the same rules. I.e. Players cannot stay in the same location and move to adjacent locations with equal probability. i. State (and justify) the new transition matrix R with the new transition probabilities. ii. Show that this new transition matrix R is regular and determine the stable distribution of this new transition matrix. For this part, you may use a computer program/software to assist your calculations. Be sure to state which parts were done using a computer and provide details of the computations you performed. (d) The choice of connecting the Kitchen and Lounge to create a regular transition matrix is not a unique choice. Starting with the original transition matrix T, provide (and justify) a general criteria required to create a regular transition matrix via the introduction of a single secret tunnel.
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