Engineering Risk Management Special topic: Beer Game Copyright Old Dominion University, 2017 All rights reserved Revised Class Schedule Lac-Megantic Case Study Part 1: Timeline of events Part 2: Timeline + causal chain of events Part 3: Instructions Evaluate your causal-chain (network) Which are the root causes? Which events have the most causes? What are the relationship of the causes? Which causes have the most influence? Part 4: Instructions Consider these recommendations from TSB Which nodes in your causal chain will be addressed by which of these recommendations? Recap How would you summarize the steps in conducting post-event analysis of an accident? Beer Game Case Study The beer game was developed at MIT in the 1960s. It is an experiential learning business simulation game created by a group of professors at MIT Sloan School of Management in early 1960s to demonstrate a number of key principles of supply chain management. The game is played by teams of four players, often in heated competition, and takes at least one hour to complete.  Beer Game Case Study Beer Game Case Study A truck driver delivers beer once each week to the retailer. Then the retailer places an order with the trucker who returns the order to the wholesaler. There’s a four week lag between ordering and receiving the beer. The retailer and wholesaler do not communicate directly. The retailer sells hundreds of products and the wholesaler distributes many products to a large number of customers. Beer Game Case Study The Retailer Week 1: Lover’s Beer is not very popular but the retailer sells four cases per week on average. Because the lead time is four weeks, the retailer attempts to keep twelve cases in the store by ordering four cases each Monday when the trucker makes a delivery. Week 2: The retailer’s sales of Lover’s beer doubles to eight cases, so on Monday, he orders 8 cases. Week 3: The retailer sells 8 cases. The trucker delivers four cases. To be safe, the retailer decides to order 12 cases of Lover’s beer. Week 4: The retailer learns from some of his younger customers that a music video appearing on TV shows a group singing “I’ll take on last sip of Lover’s beer and run into the sun.” The retailer assumes that this explains the increased demand for the product. The trucker delivers 5 cases. The retailer is nearly sold out, so he orders 16 cases. Beer Game Case Study The Retailer Week 5: The retailer sells the last case, but receives 7 cases. All 7 cases are sold by the end of the week. So again on Monday the retailer orders 16 cases. Week 6: Customers are looking for Lover’s beer. Some put their names on a list to be called when the beer comes in. The trucker delivers only 6 cases and all are sold by the weekend. The retailer orders another 16 cases. Week 7: The trucker delivers 7 cases. The retailer is frustrated, but orders another 16 cases. Week 8: The trucker delivers 5 cases and tells the retailer the beer is backlogged. The retailer is really getting irritated with the wholesaler, but orders 24 cases. Beer Game Case Study The Wholesaler The wholesaler distributes many brands of beer to a large number of retailers, but he is the only distributor of Lover’s beer. The wholesaler orders 4 truckloads from the brewery truck driver each week and receives the beer after a 4 week lag. The wholesaler’s policy is to keep 12 truckloads in inventory on a continuous basis. Week 6: By week 6 the wholesaler is out of Lover’s beer and responds by ordering 30 truckloads from the brewery. Week 8: By the 8th week most stores are ordering 3 or 4 times more Lovers’ beer than their regular amounts. Week 9: The wholesaler orders more Lover’s beer, but gets only 6 truckloads. Week 10: Only 8 truckloads are delivered, so the wholesaler orders 40. Week 11: Only 12 truckloads are received, and there are 77 truckloads in backlog, so the wholesaler orders 40 more truckloads. Beer Game Case Study The Wholesaler Week 12: The wholesaler orders 60 more truckloads of Lover’s beer. It appears that the beer is becoming more popular from week to week. Week 13: There is still a huge backlog. Weeks 14-15: The wholesaler receives larger shipments from the brewery, but orders from retailers begin to drop off. Week 16: The trucker delivers 55 truckloads from the brewery, but the wholesaler gets zero orders from retailers. So he stops ordering from the brewery. Week 17: The wholesaler receives another 60 truckloads. Retailers order zero. The wholesaler orders zero. The brewery keeps sending beer. Beer Game Case Study The Brewery The brewery is small but has a reputation for producing high quality beer. Lover’s beer is only one of several products produced at the brewery. Week 6: New orders come in for 40 gross. It takes two weeks to brew the beer. Week 14: Orders continue to come in and the brewery has not been able to catch up on the backlogged orders. The marketing manager begins to wonder how much bonus he will get for increasing sales so dramatically. Week 16: The brewery catches up on the backlog, but orders begin to drop off. Week 18: By week 18 there are no new orders for Lover’s beer. Week 19: The brewery has 100 gross of Lover’s beer in stock, but no orders. So the brewery stops producing Lover’s beer. Weeks 20-23. No orders. Beer Game Case Study At this point all the players blame each other for the excess inventory. Conversations with wholesale and retailer reveal an inventory of 93 cases at the retailer and 220 truckloads at the wholesaler. The marketing manager figures it will take the wholesaler a year to sell the Lover’s beer he has in stock. The retailers must be the problem. The retailer explains that demand increased from 4 cases per week to 8 cases. The wholesaler and marketing manager think demand mushroomed after that, and then fell off, but the retailer explains that didn’t happen. Demand stayed at 8 cases per week. Since he didn’t get the beer he ordered, he kept ordering more in an attempt to keep up with the demand. The marketing manager plans his resignation. Homework 4 Read the case and answer 1+6 questions. 0th What should go right? 1st What can go wrong? 2nd What are the causes and consequences? 3rd What is the likelihood of occurrence? 4rd What can be done to detect, control, and manage them? 5th What are the alternatives? 6th What are the effects beyond this particular time? Homework 4 In 500 words or less, summarize lessons learned in this beer game as it relates to supply chain risk management. Apply one of the tools (CCA, HAZOP, FMEA, etc.) to the case. Work individually and submit before Monday midnight (Feb. 20th). No class on Monday (Feb. 20th).

Engineering Risk Management Special topic: Beer Game Copyright Old Dominion University, 2017 All rights reserved Revised Class Schedule Lac-Megantic Case Study Part 1: Timeline of events Part 2: Timeline + causal chain of events Part 3: Instructions Evaluate your causal-chain (network) Which are the root causes? Which events have the most causes? What are the relationship of the causes? Which causes have the most influence? Part 4: Instructions Consider these recommendations from TSB Which nodes in your causal chain will be addressed by which of these recommendations? Recap How would you summarize the steps in conducting post-event analysis of an accident? Beer Game Case Study The beer game was developed at MIT in the 1960s. It is an experiential learning business simulation game created by a group of professors at MIT Sloan School of Management in early 1960s to demonstrate a number of key principles of supply chain management. The game is played by teams of four players, often in heated competition, and takes at least one hour to complete.  Beer Game Case Study Beer Game Case Study A truck driver delivers beer once each week to the retailer. Then the retailer places an order with the trucker who returns the order to the wholesaler. There’s a four week lag between ordering and receiving the beer. The retailer and wholesaler do not communicate directly. The retailer sells hundreds of products and the wholesaler distributes many products to a large number of customers. Beer Game Case Study The Retailer Week 1: Lover’s Beer is not very popular but the retailer sells four cases per week on average. Because the lead time is four weeks, the retailer attempts to keep twelve cases in the store by ordering four cases each Monday when the trucker makes a delivery. Week 2: The retailer’s sales of Lover’s beer doubles to eight cases, so on Monday, he orders 8 cases. Week 3: The retailer sells 8 cases. The trucker delivers four cases. To be safe, the retailer decides to order 12 cases of Lover’s beer. Week 4: The retailer learns from some of his younger customers that a music video appearing on TV shows a group singing “I’ll take on last sip of Lover’s beer and run into the sun.” The retailer assumes that this explains the increased demand for the product. The trucker delivers 5 cases. The retailer is nearly sold out, so he orders 16 cases. Beer Game Case Study The Retailer Week 5: The retailer sells the last case, but receives 7 cases. All 7 cases are sold by the end of the week. So again on Monday the retailer orders 16 cases. Week 6: Customers are looking for Lover’s beer. Some put their names on a list to be called when the beer comes in. The trucker delivers only 6 cases and all are sold by the weekend. The retailer orders another 16 cases. Week 7: The trucker delivers 7 cases. The retailer is frustrated, but orders another 16 cases. Week 8: The trucker delivers 5 cases and tells the retailer the beer is backlogged. The retailer is really getting irritated with the wholesaler, but orders 24 cases. Beer Game Case Study The Wholesaler The wholesaler distributes many brands of beer to a large number of retailers, but he is the only distributor of Lover’s beer. The wholesaler orders 4 truckloads from the brewery truck driver each week and receives the beer after a 4 week lag. The wholesaler’s policy is to keep 12 truckloads in inventory on a continuous basis. Week 6: By week 6 the wholesaler is out of Lover’s beer and responds by ordering 30 truckloads from the brewery. Week 8: By the 8th week most stores are ordering 3 or 4 times more Lovers’ beer than their regular amounts. Week 9: The wholesaler orders more Lover’s beer, but gets only 6 truckloads. Week 10: Only 8 truckloads are delivered, so the wholesaler orders 40. Week 11: Only 12 truckloads are received, and there are 77 truckloads in backlog, so the wholesaler orders 40 more truckloads. Beer Game Case Study The Wholesaler Week 12: The wholesaler orders 60 more truckloads of Lover’s beer. It appears that the beer is becoming more popular from week to week. Week 13: There is still a huge backlog. Weeks 14-15: The wholesaler receives larger shipments from the brewery, but orders from retailers begin to drop off. Week 16: The trucker delivers 55 truckloads from the brewery, but the wholesaler gets zero orders from retailers. So he stops ordering from the brewery. Week 17: The wholesaler receives another 60 truckloads. Retailers order zero. The wholesaler orders zero. The brewery keeps sending beer. Beer Game Case Study The Brewery The brewery is small but has a reputation for producing high quality beer. Lover’s beer is only one of several products produced at the brewery. Week 6: New orders come in for 40 gross. It takes two weeks to brew the beer. Week 14: Orders continue to come in and the brewery has not been able to catch up on the backlogged orders. The marketing manager begins to wonder how much bonus he will get for increasing sales so dramatically. Week 16: The brewery catches up on the backlog, but orders begin to drop off. Week 18: By week 18 there are no new orders for Lover’s beer. Week 19: The brewery has 100 gross of Lover’s beer in stock, but no orders. So the brewery stops producing Lover’s beer. Weeks 20-23. No orders. Beer Game Case Study At this point all the players blame each other for the excess inventory. Conversations with wholesale and retailer reveal an inventory of 93 cases at the retailer and 220 truckloads at the wholesaler. The marketing manager figures it will take the wholesaler a year to sell the Lover’s beer he has in stock. The retailers must be the problem. The retailer explains that demand increased from 4 cases per week to 8 cases. The wholesaler and marketing manager think demand mushroomed after that, and then fell off, but the retailer explains that didn’t happen. Demand stayed at 8 cases per week. Since he didn’t get the beer he ordered, he kept ordering more in an attempt to keep up with the demand. The marketing manager plans his resignation. Homework 4 Read the case and answer 1+6 questions. 0th What should go right? 1st What can go wrong? 2nd What are the causes and consequences? 3rd What is the likelihood of occurrence? 4rd What can be done to detect, control, and manage them? 5th What are the alternatives? 6th What are the effects beyond this particular time? Homework 4 In 500 words or less, summarize lessons learned in this beer game as it relates to supply chain risk management. Apply one of the tools (CCA, HAZOP, FMEA, etc.) to the case. Work individually and submit before Monday midnight (Feb. 20th). No class on Monday (Feb. 20th).

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Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

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Book review The Shareholder Value Myth: How Putting Shareholders First Harms Investors, Corporations, and the Public by Lynn Stout Provide 1) a 900 word review of this book (word range 900-1,200) and 2) a 350 word reflection where you force yourself to relate the message of the book . As per the format of the review, I like the ones done by the folks of the WSJ. This is an example: http://forums.delphiforums.com/diversecity/messages?msg=17531.1264 or http://www.wsj.com/articles/book-review-how-adam-smith-can-change-your-life-by-russ-roberts-1413846808?KEYWORDS=book+reviews

Book review The Shareholder Value Myth: How Putting Shareholders First Harms Investors, Corporations, and the Public by Lynn Stout Provide 1) a 900 word review of this book (word range 900-1,200) and 2) a 350 word reflection where you force yourself to relate the message of the book . As per the format of the review, I like the ones done by the folks of the WSJ. This is an example: http://forums.delphiforums.com/diversecity/messages?msg=17531.1264 or http://www.wsj.com/articles/book-review-how-adam-smith-can-change-your-life-by-russ-roberts-1413846808?KEYWORDS=book+reviews

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Lab #02 Relationship between distance & illumination As engineers, we deal with the effects of light on many projects. The first key to working with light is understanding how the light waves propagate. Once we understand light waves, we will test a manufacturers claim that lower wattage fluorescent bulbs output the same quantity of light as incandescent bulbs. This experiment is designed for you to work as a class to collect data regarding a given light source and then, working within your individual group, attempt to determine the re-lationship(s) between the measured parameter (lux) and the distance (meter) from the source. Measure and record data, in the manner described below, as a class. Work on your so-lutions as a group of 2-3. Your first task is to develop a mathematical formula, or a simple relationship that predicts the amount of lux that can be expected at a given distance from the light source. Purpose: The purpose of this assignment is to accomplish the following goals: • Gain experience collecting data in a controlled, systematic fashion. • Practice working as a group to infer relationships between variables from your collected data. • Use the data you collect to draw conclusions. In this case, to evaluate the hypothesis that the fluorescent and incandescent bulb output the same quantity of light. • Become accustomed to working in teams (note, teamwork often requires individual work as well). • Learn to balance workload across your team. (Individuals will be responsible for certain tasks, and ensure they are performed on time and to the desired quality level. • Demonstrate to me what your group’s attention to detail is, as well as your ability to construct a written explanation of work. Problem: What effect does distance have on the lux, intensity, emitted from a light source and are the 5 light bulbs producing the same intensity light? Use the rough protocol listed below and the data sheet provided to collect your data, then complete the assignment outlined below. 1. Set up a light source on one of the lab tables. 2. Using the illumination meter, measure the lux at 0.5 meter increments from the source back to 3 meters from the source. • Be sure the keep the meter perpendicular to the horizontal line from the source at all times! 3. Record your measurements on your data sheets. 4. Measurements should be taken in a random order 5. Repeat the experiment 3 times, using different people and a different order of collection and different colors. Assignment Requirements: 1. Create the appropriate graph(s) to express the data you have collected. Your report must, at the minimum, contain the following: a. An X-Y Scatter plot showing the data from both bulbs. The chart should follow all conventions taught in lecture, and display the equation for the trend-line you choose. b. A column or bar chart of your choosing showing the difference, if any, between the two bulbs. 2. Write an introduction, briefly explaining what you are accomplishing with this exper-iment. 3. Create a hierarchal outline that states, step by step, each activity that was performed to conduct the experiment and analyze the experimental data. 4. Anova analysis for data collected 5. Write a verbal explanation of what each of the charts from requirement #1 are showing. 6. Include, at the end of the document, a summary of all the tasks required to complete the assignment, including the 5 listed above, and which member or members of the group were principally responsible for completing those tasks. This should be in the form of a simple list. 7. Write at least 3 possible applications of the experiment with detailed explanation. DUE DATE: This assignment is to be completed and turned in at the beginning of your laboratory meeting during the week of 18th February Microsoft office package: Excel: Insert, page layout tab functions, Mean, standard deviation, graph functions

Lab #02 Relationship between distance & illumination As engineers, we deal with the effects of light on many projects. The first key to working with light is understanding how the light waves propagate. Once we understand light waves, we will test a manufacturers claim that lower wattage fluorescent bulbs output the same quantity of light as incandescent bulbs. This experiment is designed for you to work as a class to collect data regarding a given light source and then, working within your individual group, attempt to determine the re-lationship(s) between the measured parameter (lux) and the distance (meter) from the source. Measure and record data, in the manner described below, as a class. Work on your so-lutions as a group of 2-3. Your first task is to develop a mathematical formula, or a simple relationship that predicts the amount of lux that can be expected at a given distance from the light source. Purpose: The purpose of this assignment is to accomplish the following goals: • Gain experience collecting data in a controlled, systematic fashion. • Practice working as a group to infer relationships between variables from your collected data. • Use the data you collect to draw conclusions. In this case, to evaluate the hypothesis that the fluorescent and incandescent bulb output the same quantity of light. • Become accustomed to working in teams (note, teamwork often requires individual work as well). • Learn to balance workload across your team. (Individuals will be responsible for certain tasks, and ensure they are performed on time and to the desired quality level. • Demonstrate to me what your group’s attention to detail is, as well as your ability to construct a written explanation of work. Problem: What effect does distance have on the lux, intensity, emitted from a light source and are the 5 light bulbs producing the same intensity light? Use the rough protocol listed below and the data sheet provided to collect your data, then complete the assignment outlined below. 1. Set up a light source on one of the lab tables. 2. Using the illumination meter, measure the lux at 0.5 meter increments from the source back to 3 meters from the source. • Be sure the keep the meter perpendicular to the horizontal line from the source at all times! 3. Record your measurements on your data sheets. 4. Measurements should be taken in a random order 5. Repeat the experiment 3 times, using different people and a different order of collection and different colors. Assignment Requirements: 1. Create the appropriate graph(s) to express the data you have collected. Your report must, at the minimum, contain the following: a. An X-Y Scatter plot showing the data from both bulbs. The chart should follow all conventions taught in lecture, and display the equation for the trend-line you choose. b. A column or bar chart of your choosing showing the difference, if any, between the two bulbs. 2. Write an introduction, briefly explaining what you are accomplishing with this exper-iment. 3. Create a hierarchal outline that states, step by step, each activity that was performed to conduct the experiment and analyze the experimental data. 4. Anova analysis for data collected 5. Write a verbal explanation of what each of the charts from requirement #1 are showing. 6. Include, at the end of the document, a summary of all the tasks required to complete the assignment, including the 5 listed above, and which member or members of the group were principally responsible for completing those tasks. This should be in the form of a simple list. 7. Write at least 3 possible applications of the experiment with detailed explanation. DUE DATE: This assignment is to be completed and turned in at the beginning of your laboratory meeting during the week of 18th February Microsoft office package: Excel: Insert, page layout tab functions, Mean, standard deviation, graph functions

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Assignment 2 Conditional Probability, Bayes Theorem, and Random Variables Conditional Probability and Bayes’ Theorem Problems 1-14 from Problem Set on Conditional Probability and Bayes’ Theorem I am including all the question here so that there is no confusion. Q1. Pair of six sided dices are rolled and the outcome is noted: What is the sample space? What is the size of the sample space? Suppose all we are interested in is the sum of the two outcomes. What is the probability that the sum of the two is 6? 7? 8? (Note: This can be solved using both enumeration and conditional probability method). Here, it makes more sense to use the enumeration approach than conditional probability. It is, however, listed here to set the stage for Q5. What is the probability that the sum of the two is above 5 and the two numbers are equal? Express this question in terms of events A, B, and set operators. What is the probability that the sum of the two is above 5 or the two numbers are equal? Express this question in terms of events A, B, and set operators. Q2. If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3 What is the value of (a) P(A|B) and (b) P(B|A) Q3. At a fair, a vendor has 25 helium balloons on strings: 10 balloons are yellow, 8 are red, and 7 are green. A balloon is selected at random and sold. Given that the balloon sold is yellow, what is the probability that the next balloon selected at random is also yellow? Q4. A bowl contains seven blue chips and three red chips. Two chips are to be drawn at random and without replacement. What is the probability that the fist chip is a red chip and the second a blue? Express this question in terms of events A, B, and set operators and use conditional probability. Q5. Three six sided dices are rolled and the outcome is noted: What is the size of the sample space? What is the probability that the sum of the three numbers is 6? 13? 18? Solve using conditional probability How does the concept of conditional probability help? Q6. A grade school boy has 5 blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left pocket to his right pocket, what is the probability of his then drawing a blue marble from his right pocket? Q7. In a certain factory, machine I, II, and III are all producing springs of the same length. Of their production, machines I, II, and III produce 2%, 1%, and 3% defective springs respectively. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25%, and machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? Given that the selected spring is defective, what is the probability that it was produced on machine III? Q8. Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and Bowl B4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl B1, B2, B3, and B4 are 1/2, 1/4, 1/8, and 1/8 respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find P(W), the probability of drawing a white chip P(B1|W): the probability that bowl B1 was selected, given that a white chip was drawn. Q9. A pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negative. For women without cervical cancer, there are about 19% false positive. In the US, there are about 8 women in 100,000 who have this cancer. What is the probability that a woman who has been tested positive actually has cervical cancer? Q10. There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population and the test indicates that this person has the disease. What are the conditional probabilities that The person has the disease The person does not have the disease Q11. Consider two urns: the first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was a head or a tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected? Q12. In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer. Assume that a student who guesses at the answer will be correct with probability 1/m where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer given that she answered it correctly? Q13. A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested (i.e., if a healthy person is tested, then, with probability 0.01, the test result will imply that he has the disease.). If 0.5% of the population actually have the disease, what is the probability a person has the disease given that his test results are positive? Q14. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn, c additional balls of the same color are put in it with it. Now suppose that we draw another ball. What is the probability that the first ball drawn was black given that the second ball drawn was red? Random Variables Q15. Suppose an experiment consists of tossing two six sided fair dice and observing the outcomes. What is the sample space? Let Y denote the sum of the two numbers that appear on the dice. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=7? What does it mean if I say that Y<7? Q16. Suppose an experiment consists of picking a sample of size n from a population of size N. Assume that n≪N. Also, assume that the population contains D defective parts and N-D non defective parts, where n<D≪N. What is the sample space? If we are interested in knowing the number (count) of defective parts in the sample space, describe how, the concept of a random variable could help. Define a random variable Y and describe what values the random variable Y can take? What does it mean if we say Y=5? Q17. Suppose an experiment consists of tossing two fair coins. Let Y denote the number of heads appearing. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=1? What are the probabilities associated with each outcome? What is the sum of the probabilities associated with all possible values that Y can take? Q18. A lot, consisting of 100 fuses, is inspected by the following procedure. Five fuses are chosen at random and tested: if all 5 fuses pass the inspection, the lot is accepted. Suppose that the lot contains 20 defective fuses. What is the probability of accepting the lot? Define the random variable, its purpose, and the formula/concept that you would use. Q19. In a small pond there are 50 fish, 10 of which have been tagged. If a fisherman’s catch consists of 7 fish, selected at random and without replacement. Give an example of a random variable that can be defined if we are interested in knowing the number of tagged fish that are caught? What is the probability that exactly 2 tagged fish are caught? Define the random variable, its purpose, and the formula/concept that you would use. Applied to Quality Control Q20. My manufacturing firm makes 100 cars every day out of which 10 are defective; the quality control inspector tests drives 5 different cars. Based on the sample, the quality control inspector will make a generalization about the whole batch of 100 cars that I have on that day. Let d denote the number of defective cars in the sample What are the values that d can take (given the information provided above)? What is the probability that the quality control inspector will conclude that: (a) 0% of the cars are defective- call this P(d=0); (b) 20% of the cars are defective- call this P(d=1); (c) 40% of the cars are defective- call this P(d=2); (d) 60% of the cars are defective- call this P(d=3),(e) 80% of the cars are defective- call this P(d=4), and (f) 100% of the cars are defective- call this P(d=5) What is P(d=0)+ P(d=1)+ P(d=2)+ P(d=3)+ P(d=4)+ P(d=5) Let’s assume that the quality control inspector has been doing the testing for a while (say for the past 1000 days). What is the average # of defective cars that he found? Q21. Assume that the quality control inspector is selecting 1 car at a time and the car that he tested is put back in the pool of possible cars that he can test (sample with replacement). Let d denote the number of defective cars in the sample (n) What are the values that d can take (given the information provided above)? What is the probability that the quality control inspector will conclude that: (a) 0% of the cars are defective, (b) 20% of the cars are defective, (c) 40% of the cars are defective, (d) 60% of the cars are defective, (e) 80% of the cars are defective, and (f) 100% of the cars are defective. Let’s call these P(d=0)….P(d=5) What is P(d=0)+ P(d=1)+ P(d=2)+ P(d=3)+ P(d=4)+ P(d=5) Let’s assume that the quality control inspector has been doing the testing for a while (say for the past 1000 days). What is the average # of defective cars that he found? Interesting Problems Q22. A closet contains n pairs of shoes. If 2r shoes are chosen at random (2r<n), what is the probability that there will be no matching pair in the sample? Q23. In a draft lottery containing the 366 days of the leap year, what is the probability that the first 180 days drawn (without replacement) are evenly distributed among the 12 months? What is the probability that the first 30 days drawn contain none from September? Q25. You and I play a coin-tossing game. If the coin falls heads I score one, if tails, you score one. In the beginning, the score is zero. What is the probability that after 2n throws our scores are equal? What is the probability that after 2n+1 throws my score is three more than yours?

Assignment 2 Conditional Probability, Bayes Theorem, and Random Variables Conditional Probability and Bayes’ Theorem Problems 1-14 from Problem Set on Conditional Probability and Bayes’ Theorem I am including all the question here so that there is no confusion. Q1. Pair of six sided dices are rolled and the outcome is noted: What is the sample space? What is the size of the sample space? Suppose all we are interested in is the sum of the two outcomes. What is the probability that the sum of the two is 6? 7? 8? (Note: This can be solved using both enumeration and conditional probability method). Here, it makes more sense to use the enumeration approach than conditional probability. It is, however, listed here to set the stage for Q5. What is the probability that the sum of the two is above 5 and the two numbers are equal? Express this question in terms of events A, B, and set operators. What is the probability that the sum of the two is above 5 or the two numbers are equal? Express this question in terms of events A, B, and set operators. Q2. If P(A)=0.4, P(B)=0.5 and P(A∩B)=0.3 What is the value of (a) P(A|B) and (b) P(B|A) Q3. At a fair, a vendor has 25 helium balloons on strings: 10 balloons are yellow, 8 are red, and 7 are green. A balloon is selected at random and sold. Given that the balloon sold is yellow, what is the probability that the next balloon selected at random is also yellow? Q4. A bowl contains seven blue chips and three red chips. Two chips are to be drawn at random and without replacement. What is the probability that the fist chip is a red chip and the second a blue? Express this question in terms of events A, B, and set operators and use conditional probability. Q5. Three six sided dices are rolled and the outcome is noted: What is the size of the sample space? What is the probability that the sum of the three numbers is 6? 13? 18? Solve using conditional probability How does the concept of conditional probability help? Q6. A grade school boy has 5 blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left pocket to his right pocket, what is the probability of his then drawing a blue marble from his right pocket? Q7. In a certain factory, machine I, II, and III are all producing springs of the same length. Of their production, machines I, II, and III produce 2%, 1%, and 3% defective springs respectively. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25%, and machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? Given that the selected spring is defective, what is the probability that it was produced on machine III? Q8. Bowl B1 contains 2 white chips, bowl B2 contains 2 red chips, bowl B3 contains 2 white and 2 red chips, and Bowl B4 contains 3 white chips and 1 red chip. The probabilities of selecting bowl B1, B2, B3, and B4 are 1/2, 1/4, 1/8, and 1/8 respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find P(W), the probability of drawing a white chip P(B1|W): the probability that bowl B1 was selected, given that a white chip was drawn. Q9. A pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negative. For women without cervical cancer, there are about 19% false positive. In the US, there are about 8 women in 100,000 who have this cancer. What is the probability that a woman who has been tested positive actually has cervical cancer? Q10. There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population and the test indicates that this person has the disease. What are the conditional probabilities that The person has the disease The person does not have the disease Q11. Consider two urns: the first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was a head or a tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected? Q12. In answering a question on a multiple-choice test a student either knows the answer or guesses. Let p be the probability that she knows the answer. Assume that a student who guesses at the answer will be correct with probability 1/m where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer given that she answered it correctly? Q13. A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1% of the healthy persons tested (i.e., if a healthy person is tested, then, with probability 0.01, the test result will imply that he has the disease.). If 0.5% of the population actually have the disease, what is the probability a person has the disease given that his test results are positive? Q14. An urn contains b black balls and r red balls. One of the balls is drawn at random, but when it is put back in the urn, c additional balls of the same color are put in it with it. Now suppose that we draw another ball. What is the probability that the first ball drawn was black given that the second ball drawn was red? Random Variables Q15. Suppose an experiment consists of tossing two six sided fair dice and observing the outcomes. What is the sample space? Let Y denote the sum of the two numbers that appear on the dice. Define Y to be a random variable. What are the values that the random variable Y can take? What does it mean if we say Y=7? What does it mean if I say that Y<7? Q16. Suppose an experiment consists of picking a sample of size n from a population of size N. Assume that n≪N. Also, assume that the population contains D defective parts and N-D non defective parts, where n

xposure Evaluation – Single substance, different exposure time, different concentrations: 3- A person is working in a factory producing. This person is exposure to different concentrations of Toluene with different exposures time. The results of a personal sampling in an 8-hour shift is shown here: Exposure Time Concentration 3 hr. 35 min 790 mg/m^3 43 min 27 ppm 3.70 hr. 800 mg/m^3 What is this worker’s time weighted average of exposure in mg/m^3? Is the company in compliance with the OSHA requirement? 6- Phenyl ether can be used in soap factories as fragrance. A worker is exposed to this material during 9-hour shift and the exposure information is given in the following table: Exposure Time Concentration 1 hr. 45 min 4×〖10〗^(-6 ) mg/〖cm〗^3 2 hr. 5 min 7×〖10〗^(-6 ) mg/〖cm〗^3 65 min 3×〖10〗^(-3 ) mg/L Remaining Time 7.5 mg/m^3 What is this worker’s time weighted average of exposure in mg/m^3? Is the company in compliance with the OSHA requirement? 7- One of the major ingredients of insect repellents is Naphthalene. Consider a situation in which a worker is exposed to this material. The exposure time and concentration is given in a table below: Exposure Time Concentration 275 min 12 ppm 40 min 5 ppm 165 min 10 ppm What is this worker’s time weighted average of exposure? Is the condition hazardous? Exposure Evaluation – Multiple substance, equal exposure time, constant concentrations: 1- A person is exposed to the vapors of Benzene and Ethyl alcohol. Tests show that the concentration of Benzene is 1 ppm and Ethyl alcohol is 450 ppm. What is the threshold limit value of the mix? Is this person at risk? 6- Several workers at a rubber and leather manufacturing company are exposed to vapors of Vinyl chloride, Toluene and Xylene with concentration of 0.2 ppm, 135 ppm 200 mg/m^3 respectively. What is the threshold limit value of the mix? Are the employees at risk? 7- Several workers exposed to vapors of Ammonia, Arsine, Chloroform and Acetone with concentration of 12 ppm, 0.04 mg/m^3, 15 ppm, and 570 mg/m^3 respectively. What is the threshold limit value of the mix? Are the employees at risk?

xposure Evaluation – Single substance, different exposure time, different concentrations: 3- A person is working in a factory producing. This person is exposure to different concentrations of Toluene with different exposures time. The results of a personal sampling in an 8-hour shift is shown here: Exposure Time Concentration 3 hr. 35 min 790 mg/m^3 43 min 27 ppm 3.70 hr. 800 mg/m^3 What is this worker’s time weighted average of exposure in mg/m^3? Is the company in compliance with the OSHA requirement? 6- Phenyl ether can be used in soap factories as fragrance. A worker is exposed to this material during 9-hour shift and the exposure information is given in the following table: Exposure Time Concentration 1 hr. 45 min 4×〖10〗^(-6 ) mg/〖cm〗^3 2 hr. 5 min 7×〖10〗^(-6 ) mg/〖cm〗^3 65 min 3×〖10〗^(-3 ) mg/L Remaining Time 7.5 mg/m^3 What is this worker’s time weighted average of exposure in mg/m^3? Is the company in compliance with the OSHA requirement? 7- One of the major ingredients of insect repellents is Naphthalene. Consider a situation in which a worker is exposed to this material. The exposure time and concentration is given in a table below: Exposure Time Concentration 275 min 12 ppm 40 min 5 ppm 165 min 10 ppm What is this worker’s time weighted average of exposure? Is the condition hazardous? Exposure Evaluation – Multiple substance, equal exposure time, constant concentrations: 1- A person is exposed to the vapors of Benzene and Ethyl alcohol. Tests show that the concentration of Benzene is 1 ppm and Ethyl alcohol is 450 ppm. What is the threshold limit value of the mix? Is this person at risk? 6- Several workers at a rubber and leather manufacturing company are exposed to vapors of Vinyl chloride, Toluene and Xylene with concentration of 0.2 ppm, 135 ppm 200 mg/m^3 respectively. What is the threshold limit value of the mix? Are the employees at risk? 7- Several workers exposed to vapors of Ammonia, Arsine, Chloroform and Acetone with concentration of 12 ppm, 0.04 mg/m^3, 15 ppm, and 570 mg/m^3 respectively. What is the threshold limit value of the mix? Are the employees at risk?

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Which statement regarding science technology is false? Select one: Science technology has brought about life-improving discoveries, such as antibiotics. Science technology helps us to understand the causes of cancer. Science technology is a basis for all ethical or moral decisions. Science technology may ease the feeding of the world population by producing new plant strains. Scientific experimentation may make use of a model instead of an actual subject.

Which statement regarding science technology is false? Select one: Science technology has brought about life-improving discoveries, such as antibiotics. Science technology helps us to understand the causes of cancer. Science technology is a basis for all ethical or moral decisions. Science technology may ease the feeding of the world population by producing new plant strains. Scientific experimentation may make use of a model instead of an actual subject.

Which statement regarding science technology is false? Select one: Science … Read More...