4.A parking garage at UNM has installed an automatic gate. Unfortunately, the drivers have aprobability of p of crashing into the gate, in which case it needs to be replaced. Also, if the gate has survivedm days without a crash, it still must be replaced because of wear-and-tear. What is the long-term expectedfrequency of gate replacementsHint: Let the states of the Markov chain denote the number of days the gate has survived. Therefore, thershould be a total of m+1 states. Finally, the long-term frequency of gate replacements is equal to the long-termexpected frequency of visits to state 0.

Answer: The drivers have the probability of crashing gate is p.

Answered: 4. A parking garage at UNM has…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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7. A student has three routes of getting to his school: A, B, and C. Each day he picks one, and his choice is influenced only by his previous day choice: – if he takes route A today, then he will take route A, B, or C next day with respective 1 1 probabilities, 244 - if he takes route B today, then he will take route A, B, or C next day with respective 1 probabilities, 0, 2 - if he takes route C today, then he will take route A, B, or C next day with respective 1 probabilities 3'4'12' (a) Formulate this as a Markov chain by defining its states and giving its (one-step) transition matrix. (b) Find the n-step transition matrix P(n)=2 and 20. (c) If he takes route A today, what is the probability that he will take route A in 2 the day after tomorrow? (d) The probability that he takes route A today is 0.4. Use the results from part (b) to determine the probability that he will take route B 20 days from now. i=0 (a) Use equations лP = π, Σ²0¹₁ = 1 to find the steady state probabilities. Are…
18) Mordor is considering invading and conquering Gondor. Gondor has an alliance with Rohan, however it is unclear whether or not Rohan will maintain the alliance. Assume that there is a 30% probability that Rohan is a loyal ally that would help Gondor, and a 70% probability that they would abandon Gondor. Neither Gondor nor Mordor know whether Rohan is a loyal ally, although of course Rohan knows their own type. After nature chooses whether Rohan is a loyal ally, Gondor can either surrender or not. If Gondor surrenders, the game ends. If Gondor does not surrender, Mordor can invade or not. If Mordor does not invade, the game ends. If Mordor does invade, Rohan can either ride to Gondor's aid or not. If Rohan aids Gondor, the alliance will prevail. If they do not, Mordor will prevail. Gondor gets -80 utility for surrendering, -100 utility from being conquered, and 30 utility for winning the war, and 70 utility if Mordor does not invade. Mordor gets 100 utility if Gondor surrenders, 50…
4. Neo notices that a glitch in the matrix occurs an average of 3.3 times per week. He also sees that 31% of all cats are black. Last week, he saw 39 cats. Here, assume that a observing a glitch in the matrix and seeing a black cat are independent events. а. What is the probability of Neo seeing more than 3 black cats and noticing at most one glitch in the matrix last week? b. What is the probability of Neo seeing fewer than 22 black cats?
Suppose that a new detection tool is introduced starting in week 3 and that from week 3 onwards, each fake bill has a 1 in 3 chance of being detected at each transaction. In this case, what is the totalexpected number of times that the fake bills are successfully used in transactions without being detected?
1. A machine can be in one of four states: 'running smoothly' (state 1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and 'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. • If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. • If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine…
1. A machine can be in one of four states: 'running smoothly' (state 1), 'running but needs adjustment' (state 2), 'temporarily broken' (state 3), and 'destroyed' (state 4). Each morning the state of the machine is recorded. Suppose that the state of the machine tomorrow morning depends only on the state of the machine this morning subject to the following rules. • If the machine is running smoothly, there is 1% chance that by the next morning it will have exploded (this will destroy the machine), there is also a 9% chance that some part of the machine will break leading to it being temporarily broken. If neither of these things happen then the next morning there is an equal probability of it running smoothly or running but needing adjustment. • If the machine is temporarily broken in the morning then an engineer will attempt to repair the machine that day, there is an equal chance that they succeed and the machine is running smoothly by the next day or they fail and cause the machine…
Suppose that arrivals to an emergency room follow a Poisson process with meantwo patients every 5 minutes. Calculate the probabilities of (a) no customers in 2 minutes; (b) exactly 2 customers in a minute; (c) exactly 5 customers in 3 minutes;and (d) no more than 2 customers in 5 minutes.
1. Assume your computer network may have been hacked and computers may have 2 different types of viruses (virus type-A and virus type-B). The current situation shows that 15% of computers end up with type-A virus, 10% end up with type-B virus, and 10% of the computers infected with type-A or type-B virus have both viruses. 1.a) Compute the probability that a computer is infected with both viruses 1.b) Compute the probability that a computer has type-B virus given that it has type-A virus
A service company receives on average 4 service requests per day. The requests are received randomly according to Poisson process. The company has 2 service engineers and sends one engineer to attend each request. 1 An engineer needs an exponentially distributed service time with the mean of day(s). 2 The company's policy is to have maximum of 2 requests waiting in the queue If this number is reached, all incoming requests are rejected (sent to a competitor). Answer the following questions based on the information provide above: (a) Using the Kendall's notation, indicate what type of queueing system it is: (b) Compute the system state probabilities (provide at least 3 decimals): Po = P1= P2 = P3 = Pa = (c) Compute the expected total number of customer requests (waiting and served) in the system. ELL] = (d) Compute the expected number of accepted requests. Aaccepted = (e) Compute the expected total processing time (waiting + being served) for the accepted requests. E[Time] =
Assume that there are two types of borrowers high and low. The high type occurs with probability 0.75 and the low type occurs with probability 0.25. The high type succeeds and generates a gross return of 2 with probability 0.8 and fails and generates a gross return of 0 with probability 0.2. The low type succeeds and generates a gross return of 10 with probability 0.2 and fails and generates a gross return of 0 with probability 0.8. Assume limited liability. In addition, assume that the two types of potential borrowers are INDISTINGUISHABLE. What do you expect to happen if a commercial bank chooses to charge a gross interest rate of 1.5 to a potential borrower? O Both types will apply O Only the low type will apply O Neither type will apply O Only the high type will apply
7. Player A and player B have a payoff matrix shown below. If A (row player) has an optimal strategy of (424) and B has an optimal strategy of (1/3 1/3 1/3), what is the expected value of the game? P = 2 5 -2 -3 1 2-2 4 3
5) A Professor conducts an exam in a class with 24 boys and 16 girls. 6 boys and 4 girls make an A in the course. a) Write the joint pohahility matrix of the gender of the student and his/her grade (A or no A). b) Find the marignal probabilities of a student being a boy or a girl and the marginal probabilities of a student making or not making an A in the course. Show that the gender of a student and whether he/she makes an A in the course, are independent.

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