The Media Police have identified six slates associated with television watching:

Question

The Media Police have identified six slates associated with television watching: 0 (never watch TV), 1 (watch only PBS), 2 (watch TV fairly frequently), 3 (addict), 4 (undergoing behavior modification), 5 (brain dead). Transitions from state to state can be modelled as a Markov chain with the following transition matrix: Which states are transient and which are recurrent? Starting from state 1, what is the probability that state 5 is entered before state 0; i.e.. what is the probability that a PBS viewer will wind up brain dead?

Explanation / Answer

OK, in Excel I made a 6x6 matrix multiplier to help with this. I call the first row row 1 and the first column column 1. Where you started is read vertically, where you finish is read horizontally. read like this state zero next turn state one next turn state zero start here is a number here is a number state one start here is a number here is a number so the 0.7 in the matrix is the chance to go from state three (on the side) to state three (from the top) in one turn. The 1/3 under the .7 is the probability to go from state four (on the side) to state 3 (from the top) in one turn. (a) Transient states and recurrent states . A transient state is a state that you can't get back to, or there is a non-zero chance that you can't get back. A recurrent state is one that you can always get back to - even if the odds are not good that you get back. State 0 is what we call absorbing, once you get in, you can never get out. This is the ultimate persistant state State 1 is transient, this is the second column of zeros, if you start in the second row you can never end up in the second column. In fact, you are out in the very next turn (why do taxpayer dollars fund this if the transistion probability is zero?). State 2 is transient, there is a 50:50 chance that you will be there next time if you are there now (0.5 in column 3 row 3), but states 0 and 5 absorb state 2, so in the long run, state 2 empties out since there is a 10% chance of going to state 0 and a 10% chance of going to state 5 (0.1 in column 1 and column 6 row three) State 3 has a 70% of still being in state 3 next turn (0.7 in column 4 row 4), but there is a 20% chance of going to state 5 (0.2 in column 6 row 4) and that absorbs all of the viewers who start in state 3. State 4 has a 1/3 of being in state four again next time, but the 1/3 chance of being absorbed into state 0 empties out this state in the long run. State 5 has a 100% of staying in state five (1 in column 6 row 6) so this is the ultimate in persistent states, and since other states have a finite probability of coming into this "black hole" it's membership grows until it reaches steady state. (b) To find out the probabilities after many turns, we multiply the matrix many times. I multiplied the matrix out a bunch of times and got 1 0 0 0 0 0 0.66 0 8.63617E-78 3.1421E-29 7.10771E-30 0.34 0.32 0 8.63617E-78 4.87279E-29 1.10227E-29 0.68 0.2 0 0 4.47327E-29 1.01189E-29 0.8 0.6 0 0 3.37297E-29 7.62996E-30 0.4 0 0 0 0 0 1 Starting in state one means we start in the second row 0.66 0 8.63617E-78 3.1421E-29 7.10771E-30 0.34 are the long-term probabilites we are looking at. So if I start in state one, a long time later I have a 66% chance of being in state 0, a 0% chance of being in state 1, a 0% chance of being in state 2 (8.63617E-78 is basically computer speak for zero), a 0% chance of being in state 3, a 0% chance of being in state 4 and a 34% chance of being in state 5. Once a viewer enters state 0 or state 5, there is no way out, so a viewer who starts in state 1, a PBS viewer, has a 34% chance of winding up brain dead.

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