In a small shop at the beginning of each day, a machine is in good, fair, or bro
ID: 3240638 • Letter: I
Question
In a small shop at the beginning of each day, a machine is in good, fair, or broken-down condition. A good machine will be good at the beginning of the next day with probability 0 85, fair with probability 0.10, or broken-down with probability 0.05. A fair machine will be fair at the beginning of the next day with probability 0.7 or broken-down with probability 0.3. A broken down-machine cannot be fixed. a. What is the average number of days a good machine works before it gets broken down? b. What is the probability that a good machine will eventually break down?Explanation / Answer
Ans: This is DTMC with three states Good,fair,Broken.
Transition Probability matrix P:
a) it can go from good state to broken in average number of days=
(1*0.05+2*0.05*0.85+3*0.05*0.852+4*0.05*0.853+................)+(2*0.1*0.3+3*0.1*0.3*0.85+4*0.1*0.3*0.852+5*0.1*0.3*0.853+................)
=0.05*(1+2*0.85+3*0.852+4*0.853+................)+0.1*0.3(2+3*0.85+4*0.852+5*0.853+................)
b)Probability of a good machine eventually breaking=0.05+0.05*0.85+0.05*0.852+0.05*0.853+................
=0.05(1/(1-0.85))=0.05/0.15=1/3=0.33
Good fair Broken Good 0.85 0.1 0.05 fair 0 0.7 0.3 Broken 0 0 1