This is what I got for the transition matrix, is there anything wrong with it:
P[00] = q P[01] = p P[02] = p^2 P[03] = p^3
P[10] = q P[11] = p P[12] = p^2 P[13] = P^2
P[20] = q^2 P[21] = (p*q)*2 P[22] = p^2 P[23] = p^3
P[30] = q^3 P[31] = p*q^2 P[32] = q*p^2 P[33] = p^3So the reasoning for the values that I got are as follows:
P[00] = q: It means that no working machine exists by the end of one day, then a repairman repairs one machine and then it fails by the end of the next day. Thus, the probability that the one working machine at the start of the day will fail by the end of the day is q.
P[01] = p: It means that no working machine exists by the end of one day, then a repairman repairs one machine and then it remains working by the end of the next day. Thus, the probability that the one working machine remains working is p.
P[02] = p^2: It means that no working machine exists by the end of one day, then a repairman repairs one machine and then it remains working by the end of the next day, but also an extra working machine is added to the production line. Thus, the probability that both machines remain working during the day is p^2.
P[03] = p^3: It means that no working machine exists by the end of one day, then a repairman repairs one machine and then it remains working by the end of the next day, but also 2 extra working machines are added to the production line. Thus, the probability that both machines remain working during the day is p^3.
P[10] = q: It means that one working machine exists by the end of one day. Thus, the probability that the machine breaks down by the end of the day is q (Sorry I typed p, when it should be q, right?).
P[11] = p: It means that one working machine exists by the end of one day. Thus, the probability that the machine remains working by the end of the day is p (sorry I thought it is p*q but now I think it is just p).
P[12] = p^2: It means that one working machine exists by the end of one day. Thus, the probability that the machine remains working by the end of the day plus an extra working machine is added, is p^2 (or is it just 0, because the probability of adding an extra machine is not specified in the question which means it is zero?)
P[13] = p^2: It means that one working machine exists by the end of one day. Thus, the probability that the machine remains working by the end of the day plus 2 extra working machines are added, is p^2 (or is it just 0, because the probability of adding an extra machine is not specified in the question which means it is zero?)
P[20] = q^2: It means two working machines exist by the end of one day. Thus, the probability that both of the machines break down by the end of the next day is q^2
P[21] = (p*q)*2: It means two working machines exist by the end of one day. Thus, the probability that one of the machines breaks down by the end of the next day while the other remains working is (p*q)*2 (the reason it is multiplied by 2 is because it could be machine 1 or machine 2 that breaks down or remains work).
P[22] = p^2: It means two working machines exist by the end of one day. Thus, the probability that both remain working by the end of the next day is p^2
P[23] = p^3: It means two working machines exist by the end of one day. Thus, the probability that both remain working by the end of the next day plus an extra machine is added, is p^3 (or is it just 0, because the probability of adding an extra machine is not specified in the question which means it is zero?)
P[30] = q^3: It means 3 working machines exist by the end of one day. Thus, the probability that all 3 machines will be broken down by the end of the next day is q^3
P[31] = p*q^2: It means 3 working machines exist by the end of one day. Thus, the probability that 2 machines will be broken down by the end of the next day and 1 machine remains working by the end of the next day, is p*q^2
P[32] = q*p^2: It means 3 working machines exist by the end of one day. Thus, the probability that 1 machine will be broken down by the end of the next day and 2 machines remain working by the end of the next day, is q*p^2
P[33] = p^3: It means 3 working machines exist by the end of one day. Thus, the probability that all 3 machines remain working by the end of the next day, is p^3.
Please correct me as soon as you can and let me know what mistakes did I make?
Thanks for your help, I truly appreciate it everyone!