Solving Markov Chain Question on Two Switches

In summary, the question is about finding the transition matrix for a process where each of 2 switches can be on or off during the day, with a probability determined by the previous day's number of on switches. The states for this process would be none-on, one-on, and two-on. The goal is to determine the fraction of days where both switches are on and both are off.
  • #1
Jenny123
2
0
Hi, I need help with answering this question. Firstly, I'm not sure what the transition matrix should like. Should there be 2 states? One where both switches are off and one where both switches are on?

The question is:
Suppose that each of 2 switches is either on or off during the day. On day n, each switch will independently be on with probability (1+ number of on switches during day n-1)/4
For instance, if both switches are on during day n-1, then each will independently be on during day n with probability 3/4. Let Xn be the process that counts the number of switches that are on during day n. Find P, the transition matrix and hence find what fraction of days are both switches on? What fraction are both off?
 
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  • #2
Jenny123 said:
Let Xn be the process that counts the number of switches that are on during day n. ?
This suggests that you try a process where the states are possible number of switches that are "on". Those would be: none-on, one-on, two-on.
 

1. How do Markov Chains work?

Markov Chains are mathematical models that are used to describe the probability of transitioning from one state to another over a series of discrete time steps. They work by representing a system as a set of states and the probabilities of transitioning between those states.

2. How do I set up a Markov Chain for a problem?

To set up a Markov Chain, you will need to identify the states of the system, the transition probabilities between those states, and the initial state. These can be represented in a transition matrix, which shows the likelihood of transitioning from one state to another at each time step.

3. How do I solve a Markov Chain problem with two switches?

To solve a Markov Chain problem with two switches, you will need to set up a system of equations using the transition matrix. These equations will represent the probabilities of being in each state at each time step. Once the system of equations is set up, you can use matrix operations to solve for the probabilities of being in each state at any given time.

4. What is the significance of two switches in a Markov Chain problem?

Two switches in a Markov Chain problem indicate that there are two independent systems that can affect the transition probabilities between states. This adds an additional layer of complexity to the problem and requires the use of multiple transition matrices to accurately represent the system.

5. Are there any limitations to using Markov Chains?

While Markov Chains can be powerful tools for modeling and solving problems, they do have limitations. One limitation is that they assume the probabilities of transitioning between states remain constant over time, which may not always be the case in real-world systems. Additionally, Markov Chains can become computationally complex for large systems with many states.

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