John finds a bill on his desk. He has three options: ignore it and leave it on h

Question

John finds a bill on his desk. He has three options: ignore it and leave it on his own desk, move the bill over to his wife Mary's desk, or pay the bill immediately. The probability that he leaves it on his own desk is 0.3. The probability that he moves it to Mary's desk is 0.6. The probability that he pays the bill immediately is 0.1. Similarly, if Mary finds a bill on her desk she can choose to leave it on her own desk, put it on John's desk, or pay it immediately. The probability that it remains on her desk is 0.2. The probability she moves it to John's desk is 0.5. The probability she pays it immediately is 0.3. Once a bill is paid it will not return to either of the desks. Assume this is a Markov Chain process. Set up the transition matrix and use it to answer the following questions. (Give your answers correct to three decimal places.) (a) Find the probability a bill now on John's desk will be paid within two days. (b) What is the probability a bill now on John's desk will be on Mary's desk 3 days later? (c) On average, how many days will pass before a bill placed on John's desk is paid?

Explanation / Answer

Let J = On John's Desk, M = On Mary's Desk, MB = Mailbox
Now the table

a) P(On John's desk will be in MB within 2 days) = This can happen in three ways: John pays it today (0.1), or he leaves it on his desk today and pays it the second day (0.6(0.1)), or he puts it in Mary's desk today and she pays it the second day (0.3(0.3))

0.1 + 0.6(0.1) + 0.3(0.3) = 0.25

b) Now for b Use the table I made. Start at J (on the row) and find out how many ways you can get to M in 3 moves. Remember that when you make the first move, say J to M, you'd then start at M (on the row) and go from there. In other words each new day you start at the appropriate row.

Thank you.hope u got the answer,if u have any problem comment in here i will reply.

J M MB J 0.6 0.3 0.1 M 0.3 0.4 0.3 MB 0 0 0

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