MHB Need help with probability question. probability of dependent events.

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To calculate the probability of landing on each level in a scenario involving dependent events, a finite state Markov chain is the appropriate method. The levels range from -n to n, with specific transition probabilities influencing movement between levels. When reaching the boundaries, the process resets to level 0, complicating the calculations slightly. The steady state probability distribution can be determined by solving for the eigenvector of the transition matrix. This approach will yield the desired percentages for each level in an infinite run of the process.
Mewlove
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Would like to know what method, or distribution to use when solving a problem like this:I start from level 0. There is a probability p chance to drop to level -1 and a (1-p) chance to increase to level 1.

The levels range from level -n to level n. When it reaches level -n or level n, it resets back to 0 on the same cycle.
(Also, if you are at level -1, there is (1-p) chance to go back to level 0)

How do I calculate the percentage of landing on each level (not counting the reset), assuming I continue running this infinitely? Seems like dependent events. Is there any theorem or formula I can use?

Thanks!
 
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what you're looking for here is a finite state (recurrent, time homogenous) markov chain. Depending on whether your n is even or odd, the chain may actually be periodic with period 2 but that is a minor complication-- you want to solve for the eigenvector that gives a steady state probability distribution.

the key thing to search for is 'finite state markov chain'
 

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