Finding the value of ##P(X_3 = 1|X_1 = 2) = ?## in a Markov Chain

Thread starter user366312
Start date May 6, 2019
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Chain Markov chain Value

In summary, a Markov chain is a mathematical model that represents a sequence of events where the probability of transitioning from one state to another depends only on the current state and not on any previous states. The value of ##P(X_3 = 1|X_1 = 2)## in a Markov Chain represents the probability of being in state 1 at time 3, given that the chain started in state 2 at time 1. It is calculated by multiplying the probability of transitioning from state 2 to state 1 at time 2, with the probability of transitioning from state 1 to state 1 at time 3. This value can change over time in a Markov Chain as the transition probabilities between

May 6, 2019

user366312

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Homework Statement: If ##(X_n)_{n≥0}## is a Markov chain on ##S = \{1, 2, 3\}## with initial distribution ##α = (1/2, 1/2, 0)## and transition matrix

## \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix},##

then ##P(X_3 = 1|X_1 = 2) = ?##.

Relevant Equations: Markov Chain

##P^2=\begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix} \begin{bmatrix} 1/2&0&1/2\\ 0&1/2&1/2\\ 1/2&1/2&0 \end{bmatrix}=
\begin{bmatrix}
1/2 & 1/4 & 1/4\\
1/4 & 1/2 & 1/4\\
1/4 & 1/4 & 1/2
\end{bmatrix}##

So, ##P(X_3 = 1|X_1 = 2) = 1/4##.

Is this solution correct?

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May 6, 2019

Orodruin

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Yes.

May 6, 2019

user366312

Gold Member

Orodruin said:

Yes.

i have another thread. kindly see that also.

Related to Finding the value of ##P(X_3 = 1|X_1 = 2) = ?## in a Markov Chain

1. What is a Markov Chain?

A Markov Chain is a mathematical model that describes a sequence of events where the probability of each event depends only on the previous event. It is often used to model random processes in various fields such as physics, biology, economics, and computer science.

2. How do you find the value of ##P(X_3 = 1|X_1 = 2)## in a Markov Chain?

To find the value of ##P(X_3 = 1|X_1 = 2)## in a Markov Chain, you need to first determine the transition probabilities between states. Then, you can use the Markov Chain property to calculate the probability of the third event (X_3) being in state 1, given that the first event (X_1) was in state 2.

3. What does ##P(X_3 = 1|X_1 = 2)## represent in a Markov Chain?

##P(X_3 = 1|X_1 = 2)## represents the conditional probability of the third event (X_3) being in state 1, given that the first event (X_1) was in state 2. It can also be interpreted as the probability of moving from state 2 to state 1 in two steps.

4. Can the value of ##P(X_3 = 1|X_1 = 2)## change over time in a Markov Chain?

Yes, the value of ##P(X_3 = 1|X_1 = 2)## can change over time in a Markov Chain. This is because the transition probabilities between states can change over time, which can affect the probability of moving from one state to another.

5. How is the value of ##P(X_3 = 1|X_1 = 2)## useful in a Markov Chain?

The value of ##P(X_3 = 1|X_1 = 2)## can be useful in predicting future events in a Markov Chain. By knowing the probability of moving from one state to another, we can make informed decisions about the likelihood of certain outcomes and plan accordingly.