In class exam, do for problem la, 1b, 2a,2b,3c, 8a and the others do at home until Monday December 11, 2017 at 8 am. Problem 1 Customer in a certain city are continually switching the brand of soap they buy. If a customer is now using brand A, the probability he will use brand A next week is 0.5, that he switch to brand B is 0.2, and switches to brand C is 0.3. If he now use brand B, the probability he uses brand B next week is 0.6, and the probability that he switches to brand C is 0.4. If he now uses brand C, the probability he uses C next week is 0.4, and that he switches to A is 0.2, and to B is 0.4. Now as initial condition of probability, the market share is described as: the percentage of customers now using brand A is 30 %, the percentage using brand B is 20 %, and the percentage using brand C is 50 % a) Develop Markov Chain model for this problem in state and write the Markov Chain b) Using state space analysis, find the probability of each brand as function of time (in o) Find the probability a customer now using brand A will be using brand B in two d) Find the steady state the probability of market share of the product A,B and C (hint model. weeks) (hint : in Laplace transform y'(t) ←→ sF(s)-f(0)) weeks. zero value of the change of probability over time is called the steady state condition), the B in 6 e) Find also the probability a customer now using brand A will be using brand weeks