2017/4/28 Section 6.3 (Part B) [74 points] WebAssign Section 6.3 (Part B) [74 points] (Homework) Current Score : - / 74 Huizhi Liu MA 114, section 601, Spring 2017 Instructor: Lavon Page Due : Sunday, April 30 2017 02:47 PM EDT 1. -/9 points Scenario: Kate keeps some of her money invested in technology and some invested in industrial stocks. At the end of a month during which the Dow Jones Industrial Average (a market indicator) goes up, she sells 10% of her industrial stocks and invests the money in technology stocks. At the end of a month during which the Dow goes down, she sells 20% of her technology stocks and invests the money in industrial stocks. Assume that each month the Dow goes up with probability 0.8 and that it goes down with probability 0.2, and that this is independent of what the Dow may have done in previous months. So, for a given share of industrial stock, the probability that it is sold at the end of a month and the money converted to technology stocks will be 0.8 .10 = 0.08 (the probability the Dow goes up multiplied times the percentage of industrial stock that is sold if the Dow goes up). Similarly, for a given share of technology stock, the probability that it is sold and the money reinvested in industrial stock is 0.2 .20 = 0.04 (the probability the Dow goes down times the 20% chance that this particular share will be sold if that happens). Based on this information, we can create a Markov Chain model of Kate's investments. This model is given below. (In other words, the following will be used as your transition matrix.) T I T 0.96 0.04 I 0.08 0.92 Note: We can look at this 2state Markov Chain as tracking a given dollar of Kate's investment. For example, a dollar invested in technology stocks has a 0.04 probability of being reinvested in industrial stocks the next month. Whereas a dollar invested in industrial stocks has a 0.08 probability of being shifted into technology stocks the next month. Based on this model, answer the following questions. (Give your answers correct to 3 decimal places.) (a) What is the probability that a dollar invested in technology stocks will be invested in technology stocks after 3 months? What is the probability that a dollar invested in industrial stocks will be invested in technology stocks after 3 months? (b) In the long run what fraction of Kate's money will be invested in technology stocks? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349099 1/6 2017/4/28 Section 6.3 (Part B) [74 points] 2. -/15 points Three FM radio stations (A, B, and C) are competing for customers in the same market area. Through an aggressive advertising campaign, each month station A is capturing 11% of station B's customers and 5% of station C's customers. (Hint: The phrase "station A is capturing 11% of station B's customers" means that 11% of the listeners are moving from B to A, not the other way around.) At the same time, station A is losing only 2% of its customers to B and 2% of its customers to C. Additionally, each month 1% of B's customers switch to C, and 1% of C's customers switch to B. Initially, A and B each have 30% of the listeners and C has 40%. Use this information to answer the following questions. (Give your answers correct to three decimal places. Note that you are asked to give decimal values and not percentages. For example, you would need to enter 0.628 as opposed to 62.800%.) (a) What fraction of the listeners will A have after 3 months? What fraction of the listeners will B have after 3 months? What fraction of the listeners will C have after 3 months? (b) In the long run, what fraction of the listeners will A have? In the long run, what fraction of the listeners will B have? In the long run, what fraction of the listeners will C have? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349099 2/6 2017/4/28 Section 6.3 (Part B) [74 points] 3. -/18 points RentAJalopy rents cars in NJ, NY, and PA. Some times cars are picked up and returned in the same state. Other times the cars are picked up in one state and returned in another. Of the cars rented in NJ, 50% are returned to NJ, 20% are returned to NY, and 30% are returned to locations in PA. Of the cars rented in NY, 20% are returned to NJ, 60% are returned to NY, and 20% are returned to locations in PA. Of the cars rented in PA, 10% are returned to NJ, 20% are returned to NY, and 70% are returned to locations in PA. Use the matrix tool to answer the following questions. (Give your answers correct to 3 decimal places.) (a) If a car is rented in NJ, what is the probability it will be in NY after it has been rented twice? (b) If a car is rented in NY, what is the probability it will be in PA. after it has been rented three times? (c) In the long run, what fraction of the time does a rental car spend in NJ? In the long run, what fraction of the time does a rental car spend in NY? In the long run, what fraction of the time does a rental car spend in PA? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349099 3/6 2017/4/28 Section 6.3 (Part B) [74 points] 4. -/18 points Martin commutes to work. Each day he either drives his car, takes the bus, or takes a taxi. He never drives his car two days in a row. There is a 13% chance that he will drive if he did NOT drive the previous day. He is always twice as likely to take the bus as he is to take a taxi. (Example: If the probability that he takes a taxi is 1/4, then the probability that he takes a bus is twice that: 2(1/4) = 2/4 = 1/2.) This gives you enough information to set up the transition matrix. (Hint: When creating transition matrices it is important not to round any of the numbers because when you start multiplying matrices the roundoff error will get compounded. To avoid this, use fractions instead of approximated decimals. For example, you would put a fraction such as 5/6 in your matrix as "5/6" instead of using the decimal approximation "0.83333333".) Use the matrix to answer the following questions. (Give your answers correct to three decimal places.) (a) If Martin drives on Monday, what is the probability he will take the bus on Wednesday? (b) If Martin drives on Monday, what is the probability he will take a taxi on Thursday? If Martin drives on Monday, what is the probability he will drive on Thursday? (c) In the long run, what fraction of the time does he drive? In the long run, what fraction of the time does he take the bus? In the long run, what fraction of the time does he take a taxi? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349099 4/6 2017/4/28 Section 6.3 (Part B) [74 points] 5. -/14 points In a 2state Markov Chain, suppose that the probability of going from state #1 to state #2 is 2/10, but that the probability of going from state #2 to state #1 is some unknown number p. Use this information to create a transition matrix, using the row labels as shown below. (Hint: If you are having trouble filling in the missing spots, make sure that each row adds up to exactly 1. Some of your answers will have p in them.) label label state #1 T = state #2 Assume that we want to find the longrun probabilities of being in each of these states. In order to do this, we'll need to find the steadystate probability distribution. Notice that since our T matrix contains the letter "p" we won't be able to use any of the tools (because the tools can't calculate using matrices containing letters). Therefore, all of the steps will have to be done by hand. The steps below will walk you through this process. Step 1: Letting s = [x, y] and using the T matrix you determined above, write out the matrix equation s*T = s. Use this matrix equation to write 2 equations. (Fill in the blanks below.) x + y = x x + y = y Step 2: Rewrite the equations so that each is equal to 0. Using the same equation order as above, put these new equations into a 2 3 matrix. (Fill in the blanks below.) Step 3: You need to perform a row operation which will make the value in R1C1 into a one. What row operation is needed? (Give your answer in a form such as R1 + (1/2)R2 or 2R1, etc.) Step 4: Perform the row operation you found in Step 3. Next, you'll need to perform a row operation which will turn R2C1 into a 0. What row operation is needed? Step 5: Perform the row operation you found in Step 4 and simplify each entry in your matrix as much as possible. What does your matrix look like now? (Note: Only fully simplified answers will be accepted.) http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349099 5/6 2017/4/28 Section 6.3 (Part B) [74 points] Step 6: Using row 1 of your matrix from Step 5, write an equation in terms of x and y. x + y = 0 Step 7: Solve your equation from Step 6 for x. x = y Step 8: Since x and y are probabilities which must add up to 100%, we know that x + y = 1. In Step 7 you found the value of x in terms of y. Plug in this value of x into your equation x + y = 1. (This eliminates all x terms from the equation.) Solve this equation for y. (Your answer will be in terms of p.) y = Step 9: Substitute the value of y you found in Step 8 into the value for x you found in Step 7. Solve for x. (Your answer will be in terms of p.) x = Step 10: Check your work. Add the y you found in Step 8 to the x you found in Step 9. Simplify as much as possible. What is the result? (Note: Only fully simplified answers will be graded as correct.) Use the work you did above to complete the following statements. Hint: You've already found the answers! No other calculations are necessary. (Note: Your answers will have p in them, but they should not contain the letters x or y.) (a) In the long run, the probability of being in state #1 is . (b) In the long run, the probability of being in state #2 is . http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349099 6/6 2017/4/28 Section 6.4 (Part B) [68 points] WebAssign Section 6.4 (Part B) [68 points] (Homework) Current Score : - / 68 Huizhi Liu MA 114, section 601, Spring 2017 Instructor: Lavon Page Due : Friday, April 28 2017 11:01 PM EDT 1. -/12 points T is the transition matrix for a 4state absorbing Markov Chain. State #1 and state #2 are absorbing states. T = 1 0 0 0 0 1 0 0 0 0.35 0.25 0.25 0 0.5 0.5 0.15 Use the standard methods for absorbing Markov Chains to find the matrices N = (I Q)1 and B = NR. Answer the following questions based on these matrices. (Give your answers correct to 2 decimal places.) (a) If you start in state #3, what is the expected number of steps needed to reach an absorbing state. (Your answer will come from the matrix N.) steps (b) If you start in state #4, what is the expected number of steps needed to reach an absorbing state. (Your answer here will come from the matrix N.) steps (c) If you start in state #3, what is the probability that you will eventually land in state #1? (Your answer will come from the matrix B.) (d) If you start in state #4, what is the probability that you will eventually land in state #1? (Your answer will come from the matrix B.) http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349101 1/5 2017/4/28 Section 6.4 (Part B) [68 points] 2. -/12 points An absorbing Markov Chain has 5 states where states #1 and #2 are absorbing states and the following transition probabilities are known: p3,2=0.55, p3, 3=0.25, p3,5=0.2 p4,1=0.55, p4,3=0.2, p4,4=0.25 p5,1=0.3, p5,2=0.2, p5,4=0.3, p5,5 = 0.2 (a) Let T denote the transition matrix. Compute T3. Find the probability that if you start in state #3 you will be in state #5 after 3 steps. (b) Compute the matrix N = (I Q)1. Find the expected value for the number of steps prior to hitting an absorbing state if you start in state #3. (Hint: This will be the sum of one of the rows of N.) steps (c) Compute the matrix B = NR. Determine the probability that you eventually wind up in state #1 if you start in state #4. http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349101 2/5 2017/4/28 Section 6.4 (Part B) [68 points] 3. -/18 points The victims of a certain disease being treated at Wake Medical Center are classified annually as follows: cured, in temporary remission, sick, or dead from the disease. Once a patient is cured, he is permanently immune. Each year, those in remission get sick again with probability 0.12, are cured with probability 0.23, die with probability 0.05, and stay in remission with probability 0.6. Those who are sick are cured with probability 0.05, die with probability 0.2, go into remission with probability 0.4, and remain sick with probability 0.35. Find the transition matrix and do the calculations necessary to answer the following questions. (Give your answers correct to three decimal places.) (a) If a patient is now in remission, what is the probability he is still alive in two years? (Hint: In which states is a patient alive?) (b) If a patient is now in remission, what is the probability he dies within three years? (c) On average, how many years will a patient in remission live before being cured or dying from the disease? years (d) If a patient is presently sick, what is the expected number of years before the patient is cured or dies? years (e) What is the probability that someone who is currently in remission will eventually be cured? (f) What is the probability that someone who is currently sick will eventually be cured? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349101 3/5 2017/4/28 Section 6.4 (Part B) [68 points] 4. -/10 points Here is data on the flow of students through a school. 70% of freshmen pass and become sophomores, 20% fail and repeat as freshmen, 10% drop out 80% of sophomores pass and become juniors, 10% fail and repeat as sophomores, 10% drop out 80% of juniors pass and become seniors, 10% fail and repeat as juniors, 10% drop out 86% of seniors pass and graduate, 4% fail and repeat as seniors, 10% drop out Treat this as a 6state Markov chain with "graduated" and "dropped out" being absorbing states. Use an appropriate matrix tool to answer the following questions. (Give your answers correct to 3 decimal places.) (a) What fraction of freshmen graduate within 4 years? (b) What fraction of freshmen graduate within 5 years? (c) What fraction of freshmen eventually graduate? (d) What fraction of juniors eventually graduate? (e) Assume that a student enters the school as a senior. What is the expected value for the number of years that person will spend in school before either graduating or dropping out? years http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349101 4/5 2017/4/28 Section 6.4 (Part B) [68 points] 5. -/16 points Roulette is one of the most common games played in gambling casinos in Las Vegas and elsewhere. An American roulette wheel has slots marked with the numbers from 1 to 36 as well as 0 and 00 (the latter is called "double zero"). Half of the slots marked 1 to 36 are colored red and the other half are black. (The 0 and 00 are colored green.) With each spin of the wheel, the ball lands in one of these 38 slots. One of the many possible roulette bets is to bet on the color of the slot that the ball will land on (red or black). If a player bets on red, he wins if the outcome is one of the 18 red outcomes, and he loses if the outcome is one of the 18 black outcomes or is 0 or 00. So, when betting on red, there are 18 outcomes in which the player wins and 20 outcomes in which the player loses. Therefore, when betting on red, the probability of winning is 18/38 and the probability of losing is 20/38. When betting on red, the payout for a win is "1 to 1". This means that the player gets their original bet back PLUS and additional amount equaling their bet. In other words, they double their money. (Note: If the player loses they lose whatever amount of money they bet.) Scenario 1: Mike goes to the casino with $300. His plan is to bet $100 on red 7 consecutive times or until he either has increased his total to $500 or has lost all of his money. What is the probability he will go bankrupt (i.e. end up with $0)? (Give your answer correct to four decimal places.) (Hint: Set this problem up as an absorbing Markov Chain with 6 states where the states keep track of his current amount of money. The amount of money he has will always be $0, $100, $200, $300, $400, or $500. The states where he has $0 or $500 are absorbing states since he quits playing whenever one of these states is reached. Scenario 2: Mike goes to the casino with $300 and will still be betting on red. As in Scenario 1, he will bet $100 each time he places a bet. However, instead of limiting himself to a maximum of 7 bets he decides to play indefinitely until he has reached $500 or goes bankrupt. If he uses this betting method, what is the probability he will eventually go bankrupt? (Give your answer correct to four decimal places.) http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349101 5/5