Solving Markov Chain Question on Two Switches

[Jenny123]

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?

 

[Stephen Tashi]

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.