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 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.** \[ \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} \\ \end{bmatrix} \]2. **Researchers estimate that the particle is currently 3 times as likely to be in state 1 as state 2. Find the probability vector representing this estimation.** \[ \begin{bmatrix} \frac{3}{4} \\ \frac{1}{4} \\ \end{bmatrix} \]3. **Based on this estimation, what is the probability that the particle will be in state 2 two weeks from now?** \[ \begin{bmatrix} ? \\ ? \\ \end{bmatrix} \]4. **What is the probability that the particle will be in the state 1 three weeks from now?** \[ \begin{bmatrix} ? \\ ? \\ \end{bmatrix} \]

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Description: 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 obs

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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. 1/3 ...

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. 1/3 ...

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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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Question

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 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.**

\[
\begin{bmatrix}
\frac{2}{3} & \frac{1}{3} \\
\frac{1}{3} & \frac{2}{3} \\
\end{bmatrix}
\]

2. **Researchers estimate that the particle is currently 3 times as likely to be in state 1 as state 2. Find the probability vector representing this estimation.**

\[
\begin{bmatrix}
\frac{3}{4} \\
\frac{1}{4} \\
\end{bmatrix}
\]

3. **Based on this estimation, what is the probability that the particle will be in state 2 two weeks from now?**

\[
\begin{bmatrix}
? \\
? \\
\end{bmatrix}
\]

4. **What is the probability that the particle will be in the state 1 three weeks from now?**

\[
\begin{bmatrix}
? \\
? \\
\end{bmatrix}
\]

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