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TAMU MATH 166 - 166wir11ws_11c

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Math 166 WIR, Fall 2011,cBenjamin AurispaMath 166 Final Exam ReviewNote: This review does not cover every concept that could be tested on a final. Please also take a look atprevious Week in Reviews for more practice problems. Every in structor makes up their own final, so itwould be beneficial for you to also look over your old tests, quizzes, h omework, and class notes.1. Determine whether each of the following stochastic matrices is a transition matrix for a Markov processthat is regular, absorbing, or neither.(a)0.8 1 0.40.1 0 0.30.1 0 0.3(b)1 0.5 00 0.3 0.60 0.2 0.4(c)0.7 0.2 00.3 0.8 00 0 12. Suppose that in a s tudy of Coke, Pepsi, and Dr. Pep per consumers in a certain city, it was found that65% of those who buy Coke one month will also buy Coke the next month, while 28% will switch toPepsi, and 7% will switch to Dr. Pepper. Further, 54% of those wh o buy Pepsi one month will buyit the next month, while 35% will switch to Coke, and 11% will switch to Dr. Pepper. Finally, if aconsumer buys Dr. Pepper one month, data show s that 70% will buy it the next month, 20% willswitch to Coke, and 10% will switch to Pepsi.(a) If in a given month, 40% bought Coke, 20% bought Pepsi, and 40% bought Dr. Pepper, whatpercentage of these consumers will buy each of these brands 4 months later?1Math 166 WIR, Fall 2011,cBenjamin Aurispa(b) What is the steady-state distribution for this Markov p rocess, if it exists?3. The following matrix is a transition matrix for an absorbing Markov process with four states A, B, C,and D.T =1 0 0.3 0.20 1 0.4 00 0 0.3 0.50 0 0 0.3(a) Find the limiting m atrix and interpret.(b) If something starts in State D, how many times can you expect it to be in States C and D beforebeing absorbed by A or B?2Math 166 WIR, Fall 2011,cBenjamin Aurispa4. Solve the following system of equations.5x + y = 2z − 133x + 3z = −18 + 6y4y − 2z = 2x + 125. Solve the following matrix equ ation for the variables a, b, c, and d.2 a3 bT+1 −2 c0 5 86 d−4 1−1 0= −2−6 −51 46. A house costs $189,000. Ben makes a down payment of $12,000 and secures a loan for the remainingbalance. The loan is to be paid off over 25 years at an interest rate of 6%/yr compounded monthly.(a) What is the required monthly payment?N= PMT=I%= FV=PV= P/Y=C/Y=(b) How much total interest will be paid on this loan?3Math 166 WIR, Fall 2011,cBenjamin Aurispa(c) How much of the fir st payment goes towards the principal of the loan?(d) What is his outstanding principal after 20 years of payments? What is his equity?N= PMT=I%= FV=PV= P/Y=C/Y=7. A survey was taken of 300 A&M students asking what video game systems out of Atari (A), Nintendo(N), and Gameboy (G) they had growing up. The following data was found.45 students only had Nintendo.156 students had a Gameboy.73 students had all 3 systems.11 students had an Atari and a Gameboy but not a Nintendo.115 students had a Nintendo and a Gameboy.100 students had exactly one of these sy stems.101 students had a Nintendo or an Atari, but not a Gameboy.abcdefghA NG(a) How many students did not have an Atari?(b) What is n((N ∩ A) ∪ (G ∩ Nc))?(c) What is the probability that a s tudent in this group had exactly 2 of these systems?4Math 166 WIR, Fall 2011,cBenjamin Aurispa8. The odds that an event E occurs are 9 to 11, the odd s that an event F occurs are 7 to 5, and the oddsthat E occurs but not F are 1 to 4 . What is the prob ab ility that neither E nor F occurs?9. I have a bag of red, yellow, and green Skittles and M&M’s. The number of each is given in the tablebelow. An experiment consists of reaching into the bag and pulling out a piece of cand y.Skittles M&M’s TotalRed 3 4 7Yellow 8 14 22Green 4 2 6Total 15 20 35(a) What is the probability that the ch osen piece is yellow or a Skittle?(b) What are the odds that a green piece is not chosen?(c) I f an M&M is chosen, what is the probability it is red?(d) What is the probability that a yellow piece of candy is a Skittle?Let E be the event that a red candy is dr awn. Let F be the event that an M&M is drawn.(e) Are E and F mutually exclusive?(f) Are E and F independent?5Math 166 WIR, Fall 2011,cBenjamin Aurispa10. A bag of Hershey’s miniatures contains 10 milk chocolates, 9 Mr. Goodbar’s, 7 Krackels, and 8 darkchocolates. A s ample of 8 chocolates is taken from the bag.(a) How many samples contain exactly 3 milk chocolates or exactly 4 dark chocolates?(b) What is the probability that the sample contains at least 2 Krackels?11. In a certain group of students, it is known that 21% live in the dorms . Further, 67% of those w ho livein the dorms are freshmen whereas 39% of those who do not live in the dorms are freshmen.(a) What is the probabality that a student in this group who is not a freshman lives in the dorms?(b) Ar e “living in the dorms” and “being a freshman” independent events for this group?6Math 166 WIR, Fall 2011,cBenjamin Aurispa12. A game consists of rolling a pair of fair 6-sided dice. The game costs $2 to play. If a double is rolled,you win $4. If the sum of the dice is 9, you win $7. If exactly one two is rolled, you win $1. Otherw ise,you win nothing. Let X be the net winnings of a person who plays this game. Find the expected valueof X.13. T he probability that a battery produced at a certain factory lasts for more than 5 hours is 0.31. If afactory produces 400 batteries, what is the probability that(a) at least 150 of them will last for more than 5 hour s?(b) m ore than 100 but fewer than 140 will last for more than 5 hours?14. C on sider the follow ing propositions (statements).p: The Aggies are going to the Cotton Bowl.q: The Aggies beat Oklahoma.r: The Aggies beat Kansas.(a) Ex press the following in words: (p ∧ r)∨ ∼ q(b) Ex press the following using logic symbols: “Either the Aggies beat Kansas but are not going tothe Cotton Bowl or the Aggies beat Oklahoma and are going to the Cotton Bowl.”7Math 166 WIR, Fall 2011,cBenjamin Aurispa15. Ben wants to have $9000 so that he can go on a vacation. He deposits $400 ever y quarter in a savingsaccount that has an interest rate of 4.9%/yr compounded quarterly.(a) How long will it take for him to achieve his goal?N= PMT=I%= FV=PV= P/Y=C/Y=(b) How much interest is earned on the account in the 3rd quarter of the 3rd year?(c) Wh at would the q uarterly deposit need to be to have the $9000 in only 3 years?N=


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