I continue to find the physical reasoning behind the given probability transition impossible to understand despite trying for more than two hours.
To describe the move from state 1 to state 0, we imagine that there is one working machine at the start of the day; a repair man works on another machine. If he fails to repair the machine he is working on and the one working machine breaks down, then the number of machine at the end of this day is 0. Recalling that the probability for a machine to break down is 0.1 and the probability for a broken machine to not be fixed is 0.5, we note that the probability turnout is 0.05. In moving from state 1 to state 2, there exists one working machine at the start of this day and in repairing a broken machine, the broken machine is fixed and no machine breaks down during this period. The probability that the one working machine does not break down is 0.9 and the probability that a broken machine the repairman works on is fixed is 0.5. The turn out is 0.45. For state 1 to state 1, note that 1 - (0.05+0.45)=0.5.
However, I fail to understand the reasoning given for the state 2 to 3 and 2 to 1. It doesn't make one bit of sense to me. What is the physical reasoning behind 0.2 and 0.8 in the below excerpt. I would also like to request for an explanation of the probability transition in row 3.
Any help is appreciated.
To move from state 2 to state 1, we need two things to happen:
So starting in state 2 the probability of moving to state 1 is $(.2)(.5) = .1$.
To move from state 2 to state 3, we need two things to happen:
So starting in state 2 the probability of moving to state 3 is $(.8)(.5) = .4$.