This is a MATH project - (Linear Algebra) Project Assignment In the early part of the 20th Century, Andrei Markov used matrices to model probabilities of moving from one state of being to another. These matrices are usually called stochastic matrices. Our textbook uses left stochastic matrices, which are matrices with each column summing to 1. Other textbooks use right stochastic matrices, which are matrices with each row summing to 1. To prevent confusion, please use left stochastic matrices as those are the matrices used in the text, power points, and notes. This part of the portfolio contains two scenarios that you will analyze as Markov chains using matrices. 1. Fran’s Fine Family Fotos is a chain of professional photography studios specializing in family portraits. They offer two kinds of picture packages: regular and deluxe. Market research shows that, if a family in the area around the studio did not have a portrait done last year, then there is a 10% chance they will by a regular package this year, and a 5% chance they will buy a deluxe package. Also, if a family purchased a regular package last year, then there is a 30% chance they will not buy a package this year, but a 10% chance they will purchase a deluxe package. And, if a family purchased a deluxe package last year, there is a 10% chance they will not buy a package this year, but a 20% chance they will buy a regular package. (a) Construct a matrix of transition probabilities for this scenario. (b) Suppose a studio has a grand opening in a community, i.e. no family used the studio last year. What percentages of families this year will not have a portrait made, will buy a regular package, will buy a deluxe package? Answer the question using full sentence(s). (c) What will the percentages be in the studio’s fifth year of business? Round your percents to two decimal places (after the decimal). (d) Compute the steady state solution. Round your percents to two decimal places. Interpret the solution using complete sentence(s). (e) Suppose in the fifth year, the studio starts to give discounts to families that use the studio and that this changes only the percentages such that: if a family purchased a regular package, then there is a 20% chance that they will not return at all, but a 15% chance that they will buy a deluxe package. So, the numbers will work according to the original percentages up to and including the fifth year, but then will work according to these new percentages from the sixth year onwards. How will this affect the steady state solution? Round your percents to two decimal places. Interpret the solution using complete sentence(s). 2. Suppose that in a different city “Fran’s” opens a new studio, and this studio offers a “Lifetime” package. The “Lifetime” package means that once a family buys this package, they are permanently in this “state”. Market research shows that the only families that will buy the “Lifetime” package are ones that bought the deluxe package the year before, but only 2% of them. Research also shows that this is a tougher market than the first one. If a family did not purchase a portrait package last year, then there is an 8% chance that they will buy a regular package, and a 2% chance that they will buy a deluxe package. If a family purchased a regular package last year, then there is a 50% chance that they will not purchase a package this year, but a 10% chance that they will buy a deluxe package. If a family purchased a deluxe package last year, there is a 20% chance that they will not buy a package this year, a 28% chance that they will buy a regular package, and we already mentioned that there is a 2% chance that they will buy the “Lifetime” package. (a) Construct a matrix of transition probabilities for this scenario. (b) Suppose the studio has a grand opening this year. What will the percentages of the customers in different “states” be in the 7th year be? Round the percents to two decimal places. Interpret your answers using complete sentence(s). (c) What is the steady state solution? Is this reasonable for this scenario? Explain using complete sentence(s).