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An Uber driver operates in three parts of a city: A, B, C. Suppose that you keep track of their destination after every trip and model it as a Markov chain, i.e., the state corresponds to the location of the driver when the next request is accepted and then the system transitions according to the destination of the next ride. Transition matrix P is given by Note that P4- = p² = P = 0.2 0.8 0 0.3 0.2 0.5 0.1 0.5 12 0.28 0.32 0.4 0.17 0.53 0.3 p³ 0.21 0.38 0.41, 0.2168 0.4112 0.372 0.2007 0.4493 0.35 p5 0.2095 0.4244 0.3661 = = 0.4/ 0.192 0.488 0.32 0.223 0.392 0.385 0.197 0.449 0.354/ /0.20392 0.44168 0.3544 0.20993 0.42542 0.36465 0.20583 0.42553 0.35864/ i. Suppose that the driver begins the day in B. What is the probability that she will be in C after completing 2 rides? ii. Suppose that the first ride ended in A and the second ride was from A to A. What is the proba- bility that the driver will stay in A for the next 2 rides after that? iii. Suppose that half of the time the driver begins the day in A and half the time in B. What is the probability that after first 3 rides of the day she will be in C?

An Uber driver operates in three parts of a city: A, B, C. Suppose that you keep track of their destination after every trip and model it as a Markov chain, i.e., the state corresponds to the location of the driver when the next request is accepted and then the system transitions according to the destination of the next ride. Transition matrix P is given by Note that P4- = p² = P = 0.2 0.8 0 0.3 0.2 0.5 0.1 0.5 12 0.28 0.32 0.4 0.17 0.53 0.3 p³ 0.21 0.38 0.41, 0.2168 0.4112 0.372 0.2007 0.4493 0.35 p5 0.2095 0.4244 0.3661 = = 0.4/ 0.192 0.488 0.32 0.223 0.392 0.385 0.197 0.449 0.354/ /0.20392 0.44168 0.3544 0.20993 0.42542 0.36465 0.20583 0.42553 0.35864/ i. Suppose that the driver begins the day in B. What is the probability that she will be in C after completing 2 rides? ii. Suppose that the first ride ended in A and the second ride was from A to A. What is the proba- bility that the driver will stay in A for the next 2 rides after that? iii. Suppose that half of the time the driver begins the day in A and half the time in B. What is the probability that after first 3 rides of the day she will be in C?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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An Uber driver operates in three parts of a city: A, B, C. Suppose that you keep track of their destination after every trip and model as a Markov chain, i.e., the state corresponds to the location of the driver when the next request is accepted and then the system transitions according to the destination of the next ride. Transition matrix P is given by Note that P4= p² P = /0.2 0.8 0 0.3 0.2 0.5 0.1 0.5 0.4/ 12 0.28 0.32 0.4 0.17 0.53 0.3 , p³ = 0.21 0.38 0.41 0.2168 0.4112 0.372 0.2007 0.4493 0.35 p5 0.2095 0.4244 0.3661 " = 0.192 0.488 0.32 0.223 0.392 0.385 0.197 0.449 0.354/ 0.20392 0.44168 0.3544 0.20993 0.42542 0.36465 0.20583 0.42553 0.35864, i. Suppose that the driver begins the day in B. What is the probability that she will be in C after completing 2 rides? ii. Suppose that the first ride ended in A and the second ride was from A to A. What is the proba- bility that the driver will stay in A for the next 2 rides after that? iii. Suppose that half of the time the driver begins the day in A and half the time in B. What is the probability that after first 3 rides of the day she will be in C? iv. Suppose that you do not know the location that the driver begins the day. Can you estimate the probability that she will finish her day in C? Provide an estimate, or explain which information you are missing in order to give an answer.
Transcribed Image Text:An Uber driver operates in three parts of a city: A, B, C. Suppose that you keep track of their destination after every trip and model as a Markov chain, i.e., the state corresponds to the location of the driver when the next request is accepted and then the system transitions according to the destination of the next ride. Transition matrix P is given by Note that P4= p² P = /0.2 0.8 0 0.3 0.2 0.5 0.1 0.5 0.4/ 12 0.28 0.32 0.4 0.17 0.53 0.3 , p³ = 0.21 0.38 0.41 0.2168 0.4112 0.372 0.2007 0.4493 0.35 p5 0.2095 0.4244 0.3661 " = 0.192 0.488 0.32 0.223 0.392 0.385 0.197 0.449 0.354/ 0.20392 0.44168 0.3544 0.20993 0.42542 0.36465 0.20583 0.42553 0.35864, i. Suppose that the driver begins the day in B. What is the probability that she will be in C after completing 2 rides? ii. Suppose that the first ride ended in A and the second ride was from A to A. What is the proba- bility that the driver will stay in A for the next 2 rides after that? iii. Suppose that half of the time the driver begins the day in A and half the time in B. What is the probability that after first 3 rides of the day she will be in C? iv. Suppose that you do not know the location that the driver begins the day. Can you estimate the probability that she will finish her day in C? Provide an estimate, or explain which information you are missing in order to give an answer.
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