Solving Markov Chain Question on Two Switches

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SUMMARY

The discussion focuses on solving a Markov Chain problem involving two switches, where each switch's state (on or off) depends on the previous day's states. The transition matrix is constructed based on the probabilities derived from the number of switches that were on the previous day. Specifically, the states are defined as none-on, one-on, and two-on, leading to a detailed analysis of the probabilities for both switches being on or off over time.

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Jenny123
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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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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.
 

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