Math 166 WIR, Fall 2011,cBenjamin AurispaMath 166 Week in Review 10Sections M.1 & M.21. Determine if the following matrices are stochastic. If stochastic, is it regular?(a)1 0.6 0.70 0.4 0.3(b)0.1 0.2 0.40.1 0.6 0.30.8 0.2 0.3(c)0.4 0.2 0.10.3 0.5 0.60.3 0.2 0.3(d)1 0.1 0.30 0.5 0.50 0.4 0.2(e)0.3 0.3 0.40.1 0.2 0.70.6 0.5 −0.1(f)0.8 1 0.40.1 0 0.30.1 0 0.31Math 166 WIR, Fall 2011,cBenjamin Aurispa2. (Modified from Finite Mathematics by Waner/Constenoble)At a certain college, 10% of all business majors switched to another major the next semester. Of allnonbusiness majors, 20% switched to a business major the following semester.(a) Determine the transition matrix for this Markov process.(b) If in a certain freshman class, 40% are business majors and 60% are not, what will be thedistribution a semester later? 3 semesters later?(c) What is the probability that a student who is initially a business major will not be a businessmajor during the first semester of their junior year (4 semesters later)?2Math 166 WIR, Fall 2011,cBenjamin Aurispa(d) Is this Markov process regular? If so, find the steady-state distribution.3. Data was collected about the types of toothpaste that were used among households in a certain city.It was found that if a household is currently using Crest, 40% will continue to use Crest the nextmonth whereas 10% will switch to Aquafresh and the remaining 50% will switch to Colgate Total. Ifa household is currently using Aquafresh, the next month 30% will continue to use Aquafresh, 50%will switch to Crest, and 20% will switch to Colgate Total. Finally, if a household is currently usingColgate Total, 30% will switch to Crest and 50% will switch to Aquafresh the next month.(a) If 35% of households are currently using Crest, 20% are using Aquafresh, and 45% are usingColgate Total, what is the household toothpaste distribution in this city 5 months later?3Math 166 WIR, Fall 2011,cBenjamin Aurispa(b) If a household is currently using Aquafresh, what is the probability that they will be using Crest2 months later?(c) If a household is currently using Colgate Total, what is the probability that they will be usingAquafresh 3 months later?(d) Is this Markov process regular? If so, find the steady-state
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