Romeo and Juliet wants to meet in secret, but they cannot see each other and cannot tell where to…

Romeo and Juliet wants to meet in secret, but they cannot see each other and cannot tell where to…

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Romeo and Juliet wants to meet in secret, but they cannot see each other and cannot tell where to meet. There are two
places they can meet. Every day, each one randomly and independently picks one of those two locations and goes to that
location hoping to meet the other. Picks are also random and independent from the other days. Romeo starts at location 1
and picks a location according to the following transition matrix:
12
PR 1 1/3 2/3]
2 4/5 1/5
Unfortunately, Juliet starts with location 2 and picks a location according to the following transition probability matrix:
2
12
1/2 1/2]
2/3 1/3]
The problem ends when they meet (i.e., if and when they meet, they can arrange their next meeting (hmmm… maybe not)).
We would like to formulate an MC that will help us determine the probability that they will meet, when they meet etc.
(a) (70 points) Write the state definition and derive the transition probability matrix for the corresponding MC. Note: Do
not subtract sum of other elements of a row to find the remaining probability, derive all probabilities one by one in the
transition matrix.
(b) (15 points) Draw the state transition diagram.
(c) (15 points) Calculate the probability that they will meet three days after the initial day.

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