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- 6. Suppose in small town there are three places to eat, one Chinese restaurant, one Mexican restaurant, and a pizza place. Everyone in town eats dinner in one of these places or has dinner at home. Assume that 20% of those who eat in Chinese restaurant go to Mexican, 20% eat at home, and 30% go to pizza place next time. From those who eat in Mexican restaurant, 10% go to pizza place, 25% go to Chinese restaurant, and 25% eats at home next time. From those who eat at pizza place 30% eat at home, 30% eat at Chinese and 10% eat at Mexican restaurant next time. Those who eat at home 20% go to Chinese, 25% go to Mexican place, and 30% to pizza place next time. (a) Construct the (one-step) transition matrix? (b) Does this Markov chain have steady-state?2.A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different cooling system, a garage, and a patio porch. How many different plans are available to a buyer?2. You manage a company that purchases movie theaters. You have a spending goal of $4, 300 million for the purchase of new theaters to seat a total of 4,980 guests. You are given the following data: ABC theaters seat a total of 330 guests and cost $250 million each, CineMovie theaters seat 270- guests and cost $200 million each, and Movies 'n' Stuff theaters seat 240 guests and cost $250 million each. Your company wants you to buy half as many ABC theaters as the combined total of CineMovie and Movies 'n' Stuff theaters. You are tasked with determining how many of each theater your company should purchase. (a) Define variables and determine the constraints associated with this problem. Use them to set up a system of linear equations that models this scenario. (b) After applying Gauss-Jordan reduction, you arrive at the following augmented matrix: T10 0 6 Io 104 0 018 How many of each theater should your company purchase?
- 14. Suppose the Toronto Maple Leafs and the Edmonton Oilers make it to the 2021-2022 Stanley Cup Championship and are equally matched. The Stanley Cup is awarded to the first of the two hockey teams to win four games against the other (i.e. the teams compete in a best-of-seven series of games). Games 1 and 2 are played in Toronto while games 3 and 4 are played in Edmonton. If necessary, games 5 and 7 would be played in Toronto while game 6 would be played in Edmonton. Games are independent of one another, and the probability that a team wins any given game is 3/5 when it plays on home ice. What is the approximate probability that the Toronto Maple Leafs win the Stanley Cup if they lose games 1 and 2? а. 15.47% b. 16.47% с. 17.47% d. 18.47%6. Among the Grade 11 students in a certain school, - belong to the Science Club, 15% belong to the Music Club, - belong to the Mathematics Club, - belong to the English Club, 5% are members of the swimming team, while the remaining 24 do not belong to any of the above co-curricular organizations. If no student belongs to more than one organization, determine the number of students who belong to each club or organization.4. A veterinarian with a large practice was interested in whether to concentrate on only purebred dogs or to continue to treat mixed breeds as well. The table below summarizes the spending habits of his clients for the last year. Туре of Dog Pruebred Mixed Breed Total Less than S100 191 213 404 Нealth Care S100 or more 354 152 506 Spending Total 545 365 910 (a) If one of these dogs was selected at random, what is the probability that $100 or more was spent on the dog's health care last year? Given that one of these dog owners spent less than $100, find the probability that the dog was a mixed breed? (b) (c) Given that a randomly selected dog is a mixed breed, what is the probability that less than $100 was spent on its health care less year? Are the events “spending $100 on health care" and “have a mixed breed dog" independent of dependent? Explain. (d)