At any given time, a subatomic particle can be in one of two states, and it moves randomly from one state to another when it is excited. If it is in state 1 on one observation, then it is 2 times as likely to be in state 1 as state 2 on the next observation. Likewise, if it is in the state 2 on one observation, then it is 2 as likely to be in the state 2 as state 1 on the next observation. 1. Find the transition matrix for this Markov chain. 2. Researchers estimate that the particle is currently times as like to be in state 1 as state 2. Find the probability vector representing this estimation. 3. Based on this estimation, what is the probability that the particle will be in state 2 two weeks from now? 4. What is the probability that the particle will be in the state 1 three weeks from now?

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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At any given time, a subatomic particle can be in one of two states, and it moves randomly from one state to another when it is excited. If it is in state 1 on one observation, then it is 2 times as likely to be in state 1 as state 2 on the next observation.
Likewise, if it is in the state 2 on one observation, then it is 2 as likely to be in the state 2 as state 1 on the next observation.
1. Find the transition matrix for this Markov chain.
2. Researchers estimate that the particle is currently 5 times as like to be in state 1 as state 2. Find the probability vector representing this estimation.
3. Based on this estimation, what is the probability that the particle will be in state 2 two weeks from now?
4. What is the probability that the particle will be in the state 1 three weeks from now?
Transcribed Image Text:At any given time, a subatomic particle can be in one of two states, and it moves randomly from one state to another when it is excited. If it is in state 1 on one observation, then it is 2 times as likely to be in state 1 as state 2 on the next observation. Likewise, if it is in the state 2 on one observation, then it is 2 as likely to be in the state 2 as state 1 on the next observation. 1. Find the transition matrix for this Markov chain. 2. Researchers estimate that the particle is currently 5 times as like to be in state 1 as state 2. Find the probability vector representing this estimation. 3. Based on this estimation, what is the probability that the particle will be in state 2 two weeks from now? 4. What is the probability that the particle will be in the state 1 three weeks from now?
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