A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Chapter 9, Problem 9.6PTE
Compute the limiting
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Consider there are only two computer companies in a country. The companies are named “Dude” and “Imac”. Each year, company Dude keeps 1/5th of its customers, while the rest switch to Imac. Each year, Imac keeps 1/3rd of its customers, while the rest switch to Dude. If in 2002, Dude has 1/6th of the market and Imac has 5/6th of the market, what is the distribution of the market share between the two companies in 2003. (Hint: you can cast this as a matrix multiplication problem)
Chapter 9 Solutions
A First Course in Probability
Ch. 9 - Customers arrive at a bank at a Poisson rate ....Ch. 9 - Cars cross a certain point in the highway in...Ch. 9 - Suppose that in Problem 9.2, AI is agile enough to...Ch. 9 - Suppose that 3 white and 3 black balls are...Ch. 9 - Consider Example 2a. If there is a 50-50 chance of...Ch. 9 - Compute the limiting probabilities for the model...Ch. 9 - A transition probability matrix is said to be...Ch. 9 - On any given day, Buffy is either cheerful (c),...Ch. 9 - Suppose that whether it rains tomorrow depends on...Ch. 9 - A certain person goes for a run each morning. When...
Ch. 9 - Prob. 9.11PTECh. 9 - Determine the entropy of the sum that is obtained...Ch. 9 - Prove that if X can take on any of n possible...Ch. 9 - A pair of fair dice is rolled....Ch. 9 - A coin having probability p=23 of coming up heads...Ch. 9 - Prob. 9.16PTECh. 9 - Show that for any discrete random variable X and...Ch. 9 - Prob. 9.18PTECh. 9 - Events occur according to a Poisson process with...Ch. 9 - Prob. 9.2STPECh. 9 - Prob. 9.3STPECh. 9 - Prob. 9.4STPECh. 9 - Prob. 9.5STPE
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- Suppose you are a production manager for a small firm that manufactures designer phone cases. Your production facility utilizes three machines. Quality is binary — each phone case is either defective or it is not defective. Each hour, Machines 1, 2, and 3 produce 10, 10 and 20 phone cases, respectively. But because the machines differ slightly in their technology, data on past production has shown that they differ in terms of their rates of defective output — specifically, Machines 1, 2, and 3 have rates of defective production of 2%, 4%, and 5%, respectively. If one phone case is selected randomly from one hour’s total output, and that phone case is found to be defective, what is the probability that it was produced by Machine 1? Machine 2? Machine 3?arrow_forward1. A researcher observed a rat respond for a food reward by pressing one of the three levers in a cage. Pressing the lever to the right (R) produced no food reward. pressing the lever to the left (L) produced a single food pellet, and pressing the lever at the center (C) produced two food pellets. Because the center level produced the largest reward, the researcher hypothesized that the rat would press this lever most often. Each trial ended when the rat produced a level. The researcher recorded lever pressing for 30 trials. L, L, R, L, R, C, R, L, C, L, L, C, C, C, R, C, R, C, L, C, C, L, C, C, C, L, C, C, C, C, C - Create the appropiate graph for this data - Do these data support the hypothesis? Explain. 2. Which scales of measurement are assumed to be discrete? What does this mean? Which scales of measurement are assumed to be continuous? What does this mean? 3. What type of graph should you create to visualize the following frequency data? Explain. -…arrow_forwardOne major factor in low productivity is the amount of time wasted by workers. Wasted time could include things such as time spent waiting for more material and equipment, cleaning up mistakes, and performing any other activity not related to production. In a project designed to examine this problem, a consultant took a survey of 200 workers in companies that were classified as successful (on the basis of their latest annual profits) and another 200 workers from unsuccessful companies. The amount of time (in hours) wasted during a standard 40-hour workweek was recorded for each worker. Do these data provide enough evidence at the 5% significance level to infer that the amount of time wasted in unsuccessful firms exceeds that of successful ones? To answer this question, the consultant ran several analyses and provided the out below. Note that none of these analyses might be correct. A. H0: (μ1 - μ2) = 0 H1: (μ1 - μ2) > 0 B. H0: μD = 0 H1: μD > 0 C. H0: μD = 0…arrow_forward
- 2. Baldur is a big fan of salaries. He collects data on wages and the length of schooling the people had from 100 employees in a large company with the aim of creating a model that can predict wages by level of education (measured in the number of years in school after primary school). The employees with the lowest level of education in Baldur's data immediately entered the labor market after compulsory school leaving, while those with the highest level of education have completed 9 years of schooling after leaving compulsory school leaving. Baldur started by drawing the data and saw that the relationship is linear. He also calculated some fish sizes that can be seen below: standard Change average deviation salary 660020 150100 Education (year from primary school diploma) 5.5 2.2 He also calculated the correlation coefficient between salary and level of education (measured in the number of years in school from primary school exam) and obtained r = 0.55. a.) Find the equation regression…arrow_forwardAnita's, a fast-food chain specializing in hot dogs and garlic fries, keeps track of the proportion of its customers who decide to eat in Es the restaurant (as opposed to ordering the food "to go") so it can make decisions regarding the possible construction of in-store play areas, the attendance of its mascot Sammy at the franchise locations, and so on. Anita's reports that 45% of its customers order their food to go. Suppose that this proportion is correct and that a random sample of 40 individual customers is taken. Answer the following. (If necessary, consult a list of formulas.) (a) Estimate the number of customers in the sample who order their food to go by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response. (b) Quantify the uncertainty of your estimațe by giving the standard deviation of the distribution. Round your response to at least three decimal places. Eplanation Check © 2022 McGraw Hill LLC.…arrow_forward1.9 A local television station plans to drop three Friday evening programs at the end of the season. Steve Botuchis, the station manager, developed a list of three potential replacement programs. Estimates of the advertising revenue (in dollars) that can be expected for each of the new programs in the three vacated time slots are as in Table 3. Mr. Botuchis asked you to find the assignment of programs to time slots that will maximize total advertising revenue.arrow_forward
- Problem: A bank rewards its employees by giving awards to any employee who is cited by a customer for giving special service. Each award consists of two gift certificates contained in a sealed envelope. Each envelope contains certificates for one of the five following combina- tions of items:arrow_forwardExample 4: The number of defects on 20 items are given below Item No. 1 2 3 4 5 6 78 9 10 11 12 13 14 15 16 17 18 19 20 No. of defects 2 0 4 1 0 8 0 1 2 0 6 0 2 1 0 3 2 1 0 2 Devise a suitable control scheme for the future.arrow_forwardA circus is scheduled to appear in a city on a given date. The profits obtained are heavily dependent on the weather which can be classified as “good” or” bad”. The circus owners may choose to setup operations in a large open field that is centrally located or rent a small building to stage a small version of the circus. The small building is not expected to be adversely affected by bad weather – thus will not affect the circus for it is well secure and has covered parking for the guests. The following shows the profits of the options and states of nature: States of nature Decision alternatives good bad Set up in field $14,500 -$15,000 Rent small building $5000 $4,000 Probability P(G)=0.5 P(B)=0.5 The circus owners may choose to delay the decision until the day before the event is due. At this time they can obtain the one-day weather report (free) which is usually reliable. This delay will however increase their set up cost by $1000.00 or if they choose to rent,…arrow_forward
- A circus is scheduled to appear in a city on a given date. The profits obtained are heavily dependent on the weather which can be classified as “good” or” bad”. The circus owners may choose to setup operations in a large open field that is centrally located or rent a small building to stage a small version of the circus. The small building is not expected to be adversely affected by bad weather – thus will not affect the circus for it is well secure and has covered parking for the guests. The following shows the profits of the options and states of nature: States of nature Decision alternatives Good Bad Set up in field $14,500 -$15,000 Rent small building $5,000 $4,000 Probability P(G)=0.5 P(B)=0.5 The circus owners may choose to delay the decision until the day before the event is due. At this time they can obtain the one-day weather report (free) which is usually reliable. This delay will however increase their set up cost by $1000.00 or if they choose to rent, the rental cost will…arrow_forward(Use an excel file to show formulas used to answer the questions)arrow_forwardSuppose that every year, only 10% of the fish in a lake have surviving offspring. If there were 100 fish in the lake last year, there would now be 110 fish. If there were 1000 fish in the lake last year, there would now be 1100 fish. Absent any inhibiting factors, populations of people and animals tend to grow by a percent of the existing population each year. Suppose our lake began with 1000 fish, and 10% of the fish have surviving offspring each year. Since we start with 1000 fish, Po 1000. How do we calculate P₁? The new population will be the old population, plus an additional 10%. Symbolically: = P1 = Po + 0.10Po Notice this could be condensed to a shorter form by factoring: P₁= Po+0.10Po = 1Po + 0.10Po= (1+0.10)Po= 1.10Po While 10% is the growth rate, 1.10 is the growth multiplier. Notice that 1.10 can be thought of as "the original 100% plus an additional 10%" For our fish population, P₁ = 1.10(1000) = 1100 We could then calculate the population in later years. Fill in the…arrow_forward
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