STAT*2060DE Questions for Assignment #2 Please submit your responses to this question on the D2L quiz labelled \"Assignment #2\". Please input your answer in decimal form, and not as a fraction (the program will not recognize 1/4 as 0.25). As always, give your responses to 3 decimal places where applicable. Laplace, one of the great early thinkers in Probability and Statistics, stated : \"La thorie e des probabilits n'est, au fond, que le bon sens rduit au calcul.\" This translates roughly e e as \"Probability is, at bottom, nothing but common sense reduced to calculation.\" Nevertheless, many people nd the probability section a little tough. And, in spite of Laplace's assertion, there are numerous examples of professionals such as doctors, lawyers, journalists etc making fairly fundamental errors in calculating probabilities. Even Professors of Statistics can struggle with these little beasts! So, although the questions in this assignment are fairly basic, you may need to spend some time on these questions. The rst four questions refer to the following information. If P (A) = 0.20, P (B) = 0.50, and the probability that both A and B occur is .12, what is: #1. P(A B)? #2. P(A|B)? #3. P((A B) Ac)? [Hint: This means nd the intersection of A and B, then the complement of A, and then nd the union of those two things. Draw a Venn diagram.] #4. A and B are: A) Mutually exclusive and independent. B) Mutually exclusive, but not independent. C) Not mutually exclusive and not independent. D) Not mutually exclusive, and independent. Suppose we roll an ordinary balanced six-sided die. Let A = {1, 2}, B = {2, 5}, C = {2, 4, 6} #5. What is P(B|A)? #6. Are A and B independent? #7. What is P(A B C)? A) Yes B) No The next 2 questions refer to the following information. 1 The following table, extracted from Railway Age (May 1999), lists the number of carloads of dierent types of commodities that were shipped by the major U.S. railroads during a week. Suppose the computer record for a carload shipped during the week is randomly selected from a master le of all carloads shipped that week and the commodity type shipped is identied. Type of Commodity Number of Carloads Agricultural products 41,690 Chemicals 38,331 Coal 124,595 Forest products 21,929 Metallic ores and minerals 34,521 Motor vehicles and equipment 22,906 Nonmetallic minerals and products 37,416 Other carloads 14,382 Total 335,770 #8. Suppose one of these 335,770 carloads is randomly selected. What is the probability it contains forest products or coal? [One single probability value here - P(A or B)] #9. Given a randomly selected carload contains either forest products or coal, what is the probability it contains coal? #10. Text 3.25 (Page 143). Suppose a handicapper estimates the odds against a horse winning are 4 to 1. Based on this handicapper's estimate, what is the probability of the horse winning? You have an expensive system that relies on an important component. This component has a probability of failure of .2. You feel this probability is much too high, and you have designed a system in which you can install as many independent components as you like. The system connects these devices in parallel, so if at least one of them works the system works. #11. If you connect 3 of these components in parallel, what is the probability of failure of the system? #12. What is the smallest number of these components that you would have to connect in parallel for the probability of failure to be less than .00001? #13. Suppose you have won the rst stage of a contest, which allows you to participate in the second stage. In the second stage, you can play a game in which you have a 10% chance of winning $1 Million, a 25% chance of winning $500,000, and a 65% chance of receiving nothing. Alternatively, you can take $200,000 and walk away without playing the game. If your only concern is maximizing your expected prot, what should you do? 2 A) Play the game. B) Take the guaranteed $200,000. The next question refers the following information. Most companies oer their employees a variety of health care plans to choose from - e.g. preferred provider organizations (PPOs) and health maintenance organizations (HMOs). A survey of 100 large, 100 medium, and 100 small companies that oer their employees HMOs, PPOs, and fee-for-service plans was conducted; each rm provided information on the plans chosen by their employees. These companies had a total employment of 833,303 people. A breakdown of the number of employees in each category by rm size and plan is provided in the table. Company Size Fee-for-Service PPO HMO Totals Small 1,808 1,757 1,456 5,021 Medium 8,953 6,491 6,938 22,382 Large 330,419 241,770 233,711 805,900 Totals 341,180 250,018 242,105 833,303 #14. Suppose one of these 833,303 people is randomly selected. What is the probability they chose an HMO, given they are from a medium sized company? The next 3 questions refer to the following information. A meat inspector is interviewing for a new job. One component of the interview involves him inspecting 100 sides of beef, and giving each one a pass or fail grade. Later that same day, a committee of experts inspects the same 100 sides of beef, and also passes or fails them. The results are: For the same 100 sides of beef: Inspector passes Inspector fails Committee passes 74 8 Committee fails 2 16 If we randomly pick one of these sides of beef: #15. What is the probability the inspector passed it? #16. Given the inspector passed it, what is the probability the committee passed it? #17. Are the following events independent? Let A: The committee passes the meat B: The inspector passes the meat. A) Yes, they are independent. B) No, they are not independent. The next 2 questions refer to the following information. You need to buy an expensive part from a supplier. He has 2 sources for parts, and you know that source A gives a faulty part with probability .16. For Source B the probability is 3 only .03. The supplier promises you he will get the part from Source B. Given the current information, you feel there is an 65% chance he's telling you the truth (there is an 65% chance the part comes from Source B). #18. What is the probability you get a faulty part? #19. Given the part you got is faulty, what should be your estimate of the probability your supplier lied to you? [Hint: This is P(lie|faulty) = P(lie faulty)/P(faulty)]. #20. You later use a dierent supplier, who assures you that if you order from her, there is only a 1.5% chance that any given part will be faulty. You order 4 parts from her, and nd that one of them is faulty. Assuming her claim is correct, what is the probability of you getting at least one faulty part in your order of 4? It may not be realistic depending on the situation, but assume independence between parts. #21. Which one of the following statements is true? (Assume for the purposes of this question that the events do not have probability exactly equal to 0 or 1.) A) Independent events are never mutually exclusive. B) Mutually exclusive events can be independent. C) Mutually exclusive events are always independent. D) Independent events are always mutually exclusive. The next three questions refer to the following information. Suppose in a certain restaurant the probability that a customer orders shrimp is .050, the probability they get food poisoning is .004, and the probability they order shrimp or get food poisoning is .051. #22. What is the probability they order shrimp and get food poisoning? #23. What is the conditional probability they ordered shrimp, given they got food poisoning? #24. Are the events, \"The customer orders shrimp\" and \"The customer gets food poisoning\" independent? A) Yes B) No Consider the following discrete probability distribution: X Probability -.50 0.1 0 0.2 0.50 0.3 #25. What is the probability X is at least 0? #26. What is P(X < .25|X < 1.25)? 4 1 0.2 1.5 0.2 #27. What is the standard deviation of X? [N.B. The above is not sample data, but a discrete probability distribution. You have to calculate the standard deviation for a discrete probability distribution. ] Consider the following discrete probability distribution: X Probability -.50 0.1 0 0.1 0.50 0.2 1 c Find the value of c that makes this a legitimate discrete probability distribution, then calculate the mean (expected value) of the random variable X. #28. What is the mean (Expected Value) of X? I'm looking for a numeric response. You work at a restaurant that has a promotional wheel. If a customer is there on their birthday they can spin the wheel and possibly win a prize. The wheel is set up such that there approximately a .125 probability the customer wins a free meal. Suppose that in a given week 25 customers spin the wheel. #29. #30. #31. #32. What What What What is is is is the the the the probability that exactly 3 of them win a free meal? probability that at least 2 win a free meal? mean number of customers that will win the free meal? variance of the number of customers that will win the free meal? The next question refers to the following information. A study of gambling activity at the University of West Georgia (UWG) discovered that 60% of the male students wagered on sports in the past year (The Sport Journal, Fall 2006). #33. If 6 male students from this university are randomly selected, what is the probability that at least two of them wagered on sports in the past year? 5