More Compaction for the Cat and Mouse Problem

  If one looks at the numbers for P in the 2nd compacted state diagram in the last post one sees that there is even more symmetry present since P₁₂=P₂₁ and P₃₂=P₃₁. This allows us to combine states 1 and 2 to get the P'₁₁ and P'₂₁ of a simpler transition matrix. The probabilities of the two original states add to get the probability of state 1 in the second diagram.

The corresponding transition matrix is 2x2 and the calculation again produces the same results as before.

Summing just over the survival states results in the sum of an infinite geometric series with r=1/2 and a sum equal to 2 which already has been noted.