### What is the initial-state matrix

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There are three different Redbox locations in the Grandville area located at Meijer, Walgreens, and CVS. If you rent a DVD from Redbox, you must return it to any Redbox location the next day.

If you get your DVD from the Meijer location, there is a 50% chance that it will be returned back to Meijer, 20% to Walgreens, and 30% to CVS.
If you get your DVD from the Walgreens location, there is a 40% chance that it will be returned to Meijer, 50% back to Walgreens, and 10% to CVS.
If you get your DVD from the CVS location, there is a 20% chance that it will be returned back to Meijer, 20% back to Walgreens, and 60% back to CVS.

Use Excel and create a transition matrix for this problem. Make sure to label the states.

Suppose that each of the 3 locations starts out with 100 DVD's. What is the initial-state matrix ?
Find and for this system.
How many DVD's will be at Walgreens after 3 days?

How many DVD's will be at CVS after 10 days?

In the long run, how many DVD's will be at each location? Use the equation to find the stationary matrix.
Suppose you are about to start working for Apple Corp. as an intern, so you do some research on the various job positions within the company. In your area of work, there are 4 basic positions that you can hold: intern, programmer, analyst, and project manager.

Each year there is a chance for promotion, and there is also a chance of "involuntary separation" from the company. (In other words, getting fired.)

If you are an intern, there is a 50% chance each year of being promoted to programmer, and a 10% of being fired.

If you are a programmer, there is a 12% chance each year of being promoted to analyst, and a 6% of being fired.

If you are an analyst, there is a 5% chance each year of being promoted to project manager, and a 3% of being fired.

If you are a project manager, you cannot be promoted but there is a 5% chance of being fired.

If you are not promoted or fired, then you stay in the same position. If you are fired, then you cannot be rehired.

What are the different states in this system?

Using excel, construct the transition matrix T for this system. Make sure to label the states.

Assuming you start working as an intern, what is the initial-state matrix ?
Find , , and for this system.

What is the probability that you will be an intern after 10 years?
What is the probability that you will be a project manger in 5 years?
What is the probability that you will be fired within 20 years?
Lake Michigan Credit Union finances auto loans for its members. Each loan can be classified into one of four categories: paid in full, good standing, poor standing (with one or more missing payments), and delinquent (where the debt is sold to a collection agency).

Past records indicate that each month 10% of the accounts in good standing are paid in full, 70% stay in good standing, and 20% become poor standing.
Also, 50% of the accounts in poor standing move to good standing, 40% remain in poor standing, and 10% are delinquent.

Using excel, construct the transition matrix T for this situation. Be sure to label the states.
Suppose Lake Michigan Credit Union currently has 600 outstanding auto loans, 400 of which are in good standing, and 200 of which are in poor standing. In the long run, approximate how many of these auto loans will be paid in full, and how many will end up as delinquent.

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