Planet Hopping
Here is the H3X system, which contains 7 planets labeled A to G.
Every month, you can move to a planet which is (a) next to you and (b) connected via the H-link (represented by the grey lines). You are currently at Planet C.
If you were to randomly move to a planet each month (as long as it agrees with the rules above), which planet are you most likely to be on after 4 months?
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1 solution
Assuming that you are equally likely to move to each of a planet's neighbors (and never remain on the same planet), this situation can be described by the Markov matrix P = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ 0 0 . 3 0 0 . 3 0 0 0 0 . 5 0 0 . 2 5 0 0 . 3 0 0 0 0 . 3 0 0 . 3 0 . 3 0 . 3 0 0 . 5 0 0 . 2 5 0 0 0 . 3 0 0 0 . 3 0 . 2 5 0 0 0 0 . 5 0 0 0 . 2 5 0 0 . 3 0 0 . 5 0 0 0 0 . 3 0 0 . 3 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ where the rows and columns are arranged alphabetically. That is, the entry P i j is the probability of moving from planet i to planet j , with i , j = 1 corresponding to A, i , j = 2 corresponding to B, etc.
We can represent our starting state as the row vector π π π = ( 0 , 0 , 1 , 0 , 0 , 0 , 0 ) (indicating that we start on planet C with probability 1 ). Then, the probabilities for each planet after four months are π π π P 4 ≈ ( 0 . 1 3 , 0 . 1 2 , 0 . 2 6 , 0 . 1 2 , 0 . 1 2 , 0 . 1 2 , 0 . 1 3 ) Planet C is the most likely planet, with probability roughly 0 . 2 6 .