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:

Given Data

Let J = John Desk

M = Marys Desk

MB = Mailbox

                          J             M           MB

           J          0.3           0.6           0.1

          M         0.5            0.2          0.3

         MB          0               0             0

(a) Probability (John's desk will be paid within two days)

This can happen in three ways

John pay today (0.5) or leave today and pay on second day((0.5(0.3)) or he put it on mary desk today and pay it on second day(0.6(0.3))

=0.5 + ((0.5(0.3)) + (0.6(0.3))

=0.5 + 0.15 + 0.18

=0.83

(b)Probability (on johns desk will be on marys desk in three days later)

This can happen in three ways

=John leave day 1 and day 2 and put it in marys desk on day 3 (0.3 X 0.2 X 0.6) or john leave day 1 pass it on day 2 and may leave on day 3 (0.3 X 0.6 X 0.2). john pass on day 1 and mary leave it in day2 and 3 (0.6 X 0.3 X 0.2)

=(0.3 X 0.2 X 0.6) + (0.3 X 0.6 X 0.2) + (0.6 X 0.3 X 0.2)

=0.036 + 0.036 + 0.036

=0.108

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