Question

A mouse moves along a tiled corridor with 2m tiles, where m > 1. From each tile i ? 1, 2m, it moves to either tile i - 1 or i + 1 with equal probability. From
tile 1 or tile 2m, it moves to tile 2 or 2m - 1, respectively, with probability 1. Each time the mouse moves to a tile i ? m or i > m, an electronic device outputs a signal
L or R, respectively. Can the generated sequence of signals L and R be described as a
Markov chain with states L and R?

Answer 1

Let, be the status of L or R at nth time. Now, given , means, the mouse in any (m+1) to 2m tiles at nth state. Then, the next time, i.e at (n+1) th state, the probbility that will be purely dependent on the actual time position of the mouse. Because, if the mouse is at any of (m+2) to 2m th tiles, then at the next state, there is no chance that the mouse can move to any of 1 to m tiles. Where as, if the mouse is at (m+1) th tile, then the probability that at the next state the mouse can move to mth tile is 1/2 and then will be L. So, knowing only, we can't precisely calculate the probability that  , which means the signals of L and R can't be descrbed as a markov chain with states L and R