SM230 Syllabus and Homework / Spring 2002
Text: Walpole, Myers, and Myers , Probability and Statistics for Engineers and Scientists

All problems are from [WMM], except for those marked [T], which are from Turner [T].


Lecture Text Section Homework Problems
1. Three Basic Ingredients of Probability Theory 2.1  
2. Sets and Set Operations; Events 2.2 2.2: 1,2,4,5,7,8,15,20
3. Probabilities of Events; Venn Diagrams 2.4, 2.5 2.5: 1,2,5,6,8,10
4. Venn Tables, Venn Problems 2.5 [T], §1.17: 2,3,11,4,5
5. Conditional Probabilities and Bayes' Theorem 2.6, 2.7 2.7: 1,3,8,11,14
6. Independence and Correlation 2.7, 2.8 2.7: 17; 2.8: 1,5; [T], §1.17: 12,13,24,31
7. Bayesian (Simple) Search handout [T], §1.17: 40b-e, 35a,c,d
8. Single Sector Search " [T], §1.17: 35a,b,c,f, 34, 39a-d
9. Basic Counting Principle;
      Permutations, Combinations		 2.3 2.3: 3,6,10,11,12,14,18,20,25,28
10. Coin flipping: 3 Basic Ingredients Revisited handout page 49, review exercises 2,6,9
11. Review   page 49, review exercises 1,5,7,11,12
12. Exam 1    
13. Discrete R.V.s and their PDF's 3.1,3.2 3.3: 3,4,5,8,10,11,25
14. Discrete R.V.s and their CDF's 3.2 3.3: 13,15,23,26
15. Binomial Distribution 5.3 (pp. 116-120) 5.3: 4,5,10,16
16. Binomial Problems 5.3 [T], §2.13: 1,2,3,4,6,8,9
17. Negative Binomial Distribution 5.5 (omit Theorem 5.4) 5.6: 1,7; [T], §2.13: 16,17,19,20
18. Hypergeometric Distribution 5.4 5.4: 1,4,7,8,12,16
19. Poisson Distribution 5.6 5.6: 3, 8; [T], §3.13: 2,4,8,9,12
20. Prediction Intervals for Discrete R.V.s handout [T], §2.13: 12, 32
21. Confidence Intervals for Discrete R.V.s " [T], §3.13: 20,21,22
22. Review    
23. Exam 2    
24. Expectation Value of Discrete R.V.s 4.1 4.1: 2,4,5,6,7,8,9,11
25. Variance of Discrete R.V.s 4.2 4.2: 1,2,3,4
26. Continuous R.V.s: PDF's 3.3 3.3: 1,6,7,9,14
27. Continuous R.V.s: CDF's 3.3 3.3: 17, 18, 19 20, 21
28. Uniform Distribution 6.1 6.1: 21, 22; [T], §5.8, 1,2a-f
29. Exponential Distribution 6.6 6.6:7,8,15; review exercises: 2,6
30. Expectation Value of Continuous R.V.s 4.1, 4.2 4.1: 12,13,14,15,21
31. Variance of Continuous R.V.s 4.1, 4.2 4.2: 5,6,7,9,11
32. Normal Distribution:
        Introduction; Expectation Value; Variance 		6.2-6.4  
33. Review    
34. Exam 3    
35. Normal Distribution: Computing Probabilities 6.4 6.4: 1,3,4,5,7,9,10,11; [T], §4.17: 16
36. Sums of Discrete R.V.s: 2 Dice, 2 Coins 3.5 pages 69-71 3.5: 1,2,3,15,16
37. Sums of Continuous R.V.s:
         2 Uniform, 2 Normal		 3.5 pages 71-72 3.5: 5,7,8; [T] §4.17: 10(g)-(i)
38. Sums of Many IID R.V.s:
                Computer Simulation 		3.5 pages 72-77  
39. Central Limit Theorem handout  
40. Central Limit Theorem Problems   [T] §4.17: 23,24,29,30,33,34
41. Review    
42. Review    
43. Review    




Problems from Turner [T]
  1. §1.17, 2: Suppose that 82% of all butterflies are spotted and 37% of all butterflies prefer yellow flowers. If 19% of butterflies are spotted and prefer yellow flowers, what percentage are neither spotted nor prefer yellow flowers?

  2. §1.17, 3: Suppose 70% of all students in high school detention have academic problems and 60% have behavior problems.

    (a) What is the maximum percentage which have both problems?

    (b) If 40% have both, what percentage have neither problem?

    (c) If 40% have both, what percentage have exactly one problem?

  3. §1.17, 4: During a snow emergency, a car must have snow tires or chains or both. A survey showed 70% of all cars had snow tires and 25% had chains.

    (a) If 5% of all cars has both, what percentage will be legal during a snow emergency?

    (b) If 5% of all cars has both, what percentage will have exactly one type?

    (c) What is the minimum percentage that will not be legal during a snow emergency?

    (d) What is the maximum percentage that will not be legal during a snow emergency?

  4. §1.17, 5: For a field trip, students were told to wear either a jacket or sweater (or both). Suppose 55% wore a jacket and 70% wore a sweater.

    (a) What is the minimum percentage that wore both?

    (b) What is the maximum percentage that wore both?

    (c) If 35% wore both, what percentage wore neither?

    (d) If 35% wore both, what percentage wore exactly one or the other?

  5. §1.17, 11: Suppose that 45% of flights are non-stop, 35% of flights are one-stop, and the rest have 2 or more stops. Suppose that 20% of flights are non-stop with meals, 15% of flights are one-stop without meals, and 15% of flights are 2 or more stops with meals. What percentage of flights serve meals?

  6. §1.17, 12: Suppose that 60% of all cars brought into a repair shop are domestic and the rest are foreign. Suppose that 35% of all domestic cars have engine problems and 22% of all foreign cars have engine problems.

    (a) What fraction of all cars have engine problems?

    (b) Are engine problems independent of the type of car?

  7. §1.17, 13: Weekends account for 45% of the weeks business for a restaurant. On weekends, 25% of all customers order from the bar, while on the weeknights only 15% do.

    (a) Is ordering from the bar independent of weekend/weeknight?

    (b) What fraction of all customers order from the bar?

  8. §1.17, 24: Is a certain area of the ocean, a ships' sonar contacts are whales 60 % of the time and subs 40% of the time. If the contact is a whale then there is a 30% chance that the sonar registers a certain type of ``funny'' blip. A sub registers this ``funny'' blip 50% of the time.

    (a) Draw the Venn diagram. (Make a table.)

    (b) What percentage of all contacts show a ``funny'' blip?

    (c) If the contact has a ``funny'' blip, what is the probability it is a whale?

    (d) If the contact has a ``funny'' blip, what is the probability it is a sub?

    (e) Repeat all of the above questions, assuming that whales make up 75% of all contacts.

    (f) Find the probability for the contact being a whale which satisfies P(whale | funny blip)=P(sub | funny blip).

    (g) Assuming that whales make up 75% of all contacts, find values for P(funny blip | sub) and P(funny blip | whale) such that
    P(whale | funny blip)=P(sub | funny blip).

    (h) Find another set of probabilities for which this is true.

  9. §1.17, 29: Show that of events A and B are positively correlated then A and Bc are negatively correlated.

  10. §1.17, 31: Suppose that 5% of the students at a school get in disciplinary trouble. Suppose that 30% of the students are involved in the music program and that only 0.2% of the students are both in the music program and in disciplinary trouble. Let A be the event of being in trouble and B be the event of being in the music program.

    (a) Are A and B positively or negatively correlated?

    (b) How would you change the 0.2% figure to change the direction in the correlation between A and B.

  11. §1.17, 34: You've lost your car keys. They may be in your overcoat, your car, or in the parking lot on the ground. You are given: P(in coat)=0.1, P(in car)=0.05), P(in lot)=0.85. Let D be the event that they are found. You are given that: P(D | in coat)=0.75, P(D | in car)=1, P(D | in lot)=0.05.

    (a) If you look in all three places, what is the probability that you find your keys?

    (b) If you fail to find them after searching everywhere, what is the probability that they are in the parking lot?

    (c) If you instead decide to look in only one place, where should it be (to maximize your chances of finding them)?

    (d) If you look in that one place and don't find them, what is the probability it is in each of the other two places?

  12. §1.17, 35: An object is in one of 3 locations. If it is in location 1, there is a 30% chance of detection. If it is in location 2, there is a 40% chance of detection. If it is in location 3, there is a 50% chance of detection.

    (a) If there is a 50% chance that the object is in location 1, a 30% chance that the object is in location 2, a 20% chance that the object is in location 3, what is the chance we will find the object if we search all three locations?

    (b) If we could search only one location, where should we search? What is the probability of detection?

    (c) Repeat (a) and (b) if the probability of detection is reduced by a factor of 1/2.

    (d) If we searched all three locations, and did not find the object, what are the new probabilities it is in each location? Use the probabilities in (a). If you used the probabilities in (c), would your answer change?

    (f) In part (b), if we searched and did not find the object, what are the updated probabilities that the object is in each location?

  13. §1.17, 39: Saddam Hussein has hidden weapons of mass destruction in either a palace, a military base, or a children's orphanage. The probability of being in the palace is 0.5, the military base is 0.15, and the orphanage is 0.35. The probability of detection in the palace is 0.1, in the base is 0.25, and in the orphanage is 0.2.

    (a) If we only search one location at a time, where should we begin?

    (b) If we don't find the weapons in the location in (a), where should we search next?

    (c) If we still haven't found them, where should we search next?

    (d) What would the probability of detection in the palace have to be to start there?

  14. §1.17, 40: A cowboy has lost a dogie (a motherless calf). It is either in the ravine, the barn, or the next ranch. He figures that the probabilities of the dogie being in each location are 0.5 (ravine), 0.35 (barn), 0.15 (next ranch). The probabilities of being detected if the location is searched are 0.2 (ravine), 0.8 (barn), and 0.3 (next ranch).

    (b) If he searches all three locations, what is the probability that the dogie will be found?

    (c) If the dogie is not found, what is the probability it is in the barn?

    (d) If he searches all three locations a second time (not having found it the first time) and fails to find it, what is the probability it is in the ravine?

    (e) What are the chances of finding the dogie in the first or second search?

  15. §2.13, 1: Suppose that a fair 6-sided die is rolled 24 times.

    (a) What is the probability that a 1 or 2 occurs on 10 or more rolls?

    (b) If you roll the die until you get a 1 or 2 ten times, what is the probability that you roll the die not more than 20 times?

  16. §2.13, 2: A bag contains 5 red and 95 blue balls. If a ball is selected at random, replaced, and another ball is selected at random, what is the probability that exactly 1 of them is red?

  17. §2.13, 3: Suppose that there are 12 bags, each containing 75 red and 25 blue balls. If one ball is selected from each of the 12 bags, what is the probability that 5 or fewer are red?

  18. §2.13, 4: I'm going to give a quiz having 10 problems. There will be no partial credit.

    (a) If a student has a 60% chance of getting any one question correct, what is the probability that he will get 5 or fewer correct?

    (b) What is the probability that he will get 6 or more correct?

    (c) What is the probability that he will get exactly 6 correct?

    (d) A ``C'' student has a 70% chance of getting any one question correct. If a score of 5 or lower is an F, what proportion of such ``C'' students will pass?

    (e) A ``C'' student has a 70% chance of getting any one question correct. If I want 95% of the ``C'' students to pass, what should an F score on the quiz be?

    (f) I'm going to give a quiz having 20 problems. If a score of 15 or lower is an F, what proportion of such ``C'' students will pass?

    (g) On a quiz having 20 problems, a score of 15 or lower is an F. What should the probability be of answering any one question correctly in order to be 75% sure of passing?

    (h) The quiz has 20 problems. If I want 65% of the ``C'' students to pass, what should an F score on the quiz be?

    (i) The quiz has 20 problems. What is the probability that a ``C'' student will get none correct?

    (j) On a quiz having 20 problems, what is the probability that a ``C'' student will get all correct?

    (k) On a quiz having 20 problems, there is a 95% chance that a ``C'' student will get at least how many correct?

    (l) On a quiz having 20 problems, there is a 95% chance that a ``C'' student will get at most how many correct?

  19. §2.13, 6: In a lottery, the chances of winning are 1 in 20.

    (a) If you purchase 20 tickets, what is the probability that no tickets win?

    (b) If you purchase 20 tickets, what is the probability that 1 or more tickets win?

    (c) If you purchase 20 tickets, what is the probability that 2 or more tickets win?

  20. §2.13, 8: The Washington Post reported that in the 1996 U.S. Senate race, there were 10 races which were ``tossups''. To regain control of the Senate, the Republicans must win only 3 or these seats.

    (a) Assume that to be a ``tossup'' there must be a 50% chance that either party will win the race, find the probability that the Democrats will gain control of the Senate.

    (b) To have a 50% chance of winning the Senate, what must the Democrats' chances be in the ``tossup'' races?

    (c) To use the binomial distribution, we must assume that all the trials are independent. Is this assumption reasonable or not in this case? Explain.

  21. §2.13, 9: A ship owner plans to send 12 ships to the Caribbean and wished to insure each for $ 10,000. Assume that there is a 75% chance that the ship will return safely. How much should the insurance company charge to be 90% sure of making a profit?

  22. §2.13, 12: On a typical day, 20% of a cabbie's riders want to go to the airport. Today, he will pick up 19 riders.

    (a) He is 90% certain he will go to the airport at least how many times?

    (b) He is 90% certain he will go to the airport at most how many times?

    (c) Suppose a novice cabbie arrives (who knows a lot of probability theory) and that only two out of his 19 rides wanted to go to the airport today. Let p be the probability that a rider will want to go to the airport. Based on this, he is 90% confident p is at most what?

  23. §2.13, 16: For a university honor board, we need to find 3 people to serve who don't know the defendant. The defendant is rather popular and only 30% of the student body doesn't know him.

    (a) If we only have time to interview 8 people, what is the probability that we will find the 3 we need?

    (b) Suppose that we call ``a bunch'' of people in to interview. How many is ``a bunch'' if we want to be 90% certain of finding our 3 people?

  24. §2.13, 17: It's you first solo flight in an airplane. You have an 80% chance of making a successful landing.

    (a) What is the chance of making a successful landing within the first 5 attempts?

    (b) 99% of all students make a successful landing within the first how many attempts?

  25. §2.13, 19: It is known that 10% of squirrels in a large popoulation is rabid. We need 6 for an experiemnt.

    (a) If we capture 30 squirrels, what is the probability that 6 of them will be rabid? Is 30 enough or too many?

    (b) If we capture 60 squirrels, what is the probability that 6 of them will be rabid? Is 60 enough or too many?

    (c) How many squirrels need to be captured to be 80% sure we will have 6 rabid ones?

  26. §2.13, 20: Godzilla is attacking and you have a laser cannon. It requires 3 hits to kill Godzilla. The probability that your shot hits him is 80%. You only have 5 shots.

    (a) What is the probability you will stop Godzilla?

    (b) What is the number of shots you need to be 99% sure of stopping him?

  27. §2.13, 32: The mathematics department has 40 civilian and 20 military faculty. Suppose that we form a committee by randomly selecting 15 faculty.

    (a) We are 90% sure we will have at least how many military faculty?

    (b) We are 90% sure we will have at most how many military faculty?

  28. §3.13, 2: Suppose the shuttle bus leaves every 10 minutes and can hold 8 people, and people arrive at the bus stop in a Poisson process.

    (a) If, on average, 6 people arrive every 10 minutes, how many buses will leave with less than 8 people on board?

    (b) How many buses will have to leave some people behind when they depart?

    (c) What should the capacity of the bus be so that no more than 5% of the buses have to leave people behind?

    (d) Instead of larger buses leaving every 10 minutes, we could use the 8 passenger buses and have them leave more often. How often should they be scheduled to leave so that there is only a 5% chance they will have to leave people behind?

    (e) After surveying the passengers, we determine that several people will stop using the shuttle bus. How low would the passenger rate have to be so that the 8 passenger bus can leave on a 10 minute schedule and have only a 5% chance of leaving people behind?

    (f) Suppose that the rate is 6 passengers every 10 minutes. We send the 8 passenger buses in pairs, every 20 minutes. How often are passengers left behind?

  29. §3.13, 4: Suppose that a nearby volcano erupts every 25 years, and its a Poisson process.

    (a) What is the probability it will erupt in the next 10 years?

    (b) By what time (to the nearest 0.1 year) would there be at least a 25% chance of an eruption?

    (c) There is a 90% chance it will erupt within how many years?

    (d) Suppose we feel protected from the next eruption, but figure that the one after that will destroy us. What is the chance of being destroyed in the next 10 years?

    (e) At what point is there a 20% chance wi will be destroyed?

  30. §3.13, 8: Fast food restaurants occur about every 10 miles along the highway. Assume it is a Poisson distribution.

    (a) What is the chance that I will pass a restaurant in the next 5 miles?

    (b) It's been at least 20 miles since I last saw a place to eat. What is the chance of finding some food in the next 5 miles?

    (c) My kids are hungry. I want to give them a reasonable estimate of when we will eat. To be 80% sure of being right, I whould tell them we will eat within how many miles?

    (d) My wife never wants to stop at the first place we see. She will consent to stopping at the second restaurant. What is the chance we will eat within the next 15 miles?

    (e) I am 75% sure that my wife will let us stop within how many miles?

  31. §3.13, 9: Shipbuilders used to require tall, straight trees to make the masts of sailing ships. An acre of forest would contain only 1 or 2 such trees. (Assume that there are, on average, 1.5 trees/acre , and that it is a Poisson distribution.)

    (a) What is the probability that an acre will not contain any such trees?

    (b) If I have 10 acres of land, I am 90% sure I will have at most how many trees?

    (c) A buyer wants to find 5 trees suitable for masts. What is the least acreage he should be prepared to search if he wants to be 90% sure he'll get 5?

    (d) There is a 90% chance he will have to search more than how many acres?

  32. §3.13, 12: A tow truck operator estimates that on a typical week, there are between 3 and 4 abandoned cars that he has to haul. (Assume that it is a Poisson process with an average of 3.5 cars/week.)

    (a) What is the probability that he will haul no cars in a given week?

    (b) What is the proability that he will haul 3 or fewer cars in a week?

    (c) Suppose he accumulates cars all week long and takes them to the dump on Saturday. He can transport 6 cars at a time. If he has to make a second trip, he has to get up earlier on Saturday, which he hates because it cuts into his fishing time. How often will he have to make 2 trips on a Saturday?

    (d) 3 trips on a Saturday?

    (e) He is considering a new transporter. What capacity should he buy so that 75% of the time he needs to make at most 1 trip on a Saturday?

    (f) What capacity is needed to be 99% sure of at most 1 Saturday trip?

    (g) Given the 99% sure transporter from (f), what is the probability he will not need a second Saturday trip?

    (h) With his 6 car transporter, what is the chance of going a month without needing a second Saturday trip? (Assume 4 Saturdays in a month.)

    (i) At most 1 Saturday with an extra trip?

  33. §3.13, 20: In the Happy Raptor Wildlife Preserve, bald eagles are sighted, on average, every 1.5 hours. (Assume it's a Poisson process.)

    (a) If we take a 3 hour tour, we are 95% sure we will see at least how many bald eagles?

    (b) At most how many?

    (c) We are 95% sure we will see a bald eagle before how long?

    (d) We spent 6 hours and saw 1 bald eagle. Based on this data alone, find a 90% confidence interval for the average bald eagle sighting rate.

  34. §3.13, 21: When network cards go bad, we replace them. (Assume they go bad in a Poisson fashion.)

    (a) If the manufacturer claims the failure rate is 1 per month, we are 90% sure we will have at most how many failures in 3 months?

    (b) We are 90% sure we will see at least how many failures in 6 months?

    (c) We collect the bad cards, and ship them back to the manufacturer. We ship them in lots of 4. We are 90% sure we won't have to ship a lot back for at least how many months?

    (d) We had 2 failures in the first 3 weeks. Based on this data alone, what is a 90% confidence interval for the failure rate?

  35. §3.13, 22: When a star explodes, it produces a "nova". These happen, on average, at rate of 2 per year. (Assume it is a Poisson process.)

    (a) We are 95% sure we will see at most how many in the next 10 years?

    (b) We are 90% sure we will see at most how many in 6 months?

    (c) We are 90% sure there will be a nova before how long?

    (d) If we observe no novas in 3 months, what is a 90% confidence interval for the nova rate, based on this data alone?

    (e) If we observe 4 in 1 year, get a 95% confidence interval for the nova rate, based on this data alone.

  36. §4.17, 10: The time required for directory assistance operators to service a query has a normal distribution with a mean time of 12 seconds, and a standard deviation of 1.8 seconds.

    (a) What is the probability a call will require over 15 seconds?

    (b) Less than 10 seconds?

    (c) To detect if an operator is taking personal calls, a device is triggered when a call exceeds a prescribed length. How should the length be set so that less than 1% of legitimate calls will trigger it?

    (d) 90% of all calls fall within how many seconds of the mean?

    (e) Same for 80% ?

    (f) If a call has been in progress for 11 seconds, what is the probability that it will require at least 15 seconds total?

    (g) What is the probability that 3 inquiries will take a total of more than 1 minute to service?

    (h) An operator has agreed to take the next 10 calls before leaving for the day. She calls security to escort her to the parking lot. How long should the security guard wait before leaving to be 90% sure she will be done when he arrives?

    (i) The record for number if inquiries in 1 minute was set in 1989 at 7 calls finished within 60 seconds. What is the probability of bettering this record?

  37. §4.17, 16: Suppose the length from tip to tip of adult male blue crabs in the Chesapeake Bay is normally distributed with a mean of 6 inches and a standard deviation of 1.1 inches. The legal limit for keeping crabs is 5 inches, i.e. crabs smaller than 5 inches must be thrown back in the Bay.

    (a) What fraction of crabs is legal?

    (b) Crabs are called "large" if they exceed 6.5 inches. What fraction of crabs is large?

    (c) What raction of legal crabs is large?

    (d) If a crabber caught 500 crabs and will risk keeping all that exceed 4.75 inches, approximately how many illegal crabs has he kept?

    (e) 95% of all crabs are how long? Answer this question 3 ways: 95% are smaller than what? 95% are larger than what? 95% are within how many inches of the mean?

    (f) Same using 90%.

  38. §4.17, 23: The weights of packages handled by a delivery service have probabilities (respectively) as follows: 
    0.40    ,   0.25   ,   0.10   ,   0.10   ,  0.10    ,   0.05    for weights
0-1 lbs,   1-2 lbs ,  2-3 lbs ,   3-4 lbs , 4-5 lbs ,   5-6 lbs.
    They do not handle packages over 6 lbs.

    (a) What is the mean weight (expectation value) of the packages?

    (b) What is the standard deviation of the weights?

    (c) If we have 30 packages, what is the (approximate) probability that the total weight exceeds 75 lbs? (Hint: Central Limit Theorem.)

    (d) We are 90% sure a cart with 50 packages will at least how much?

    (e) To assess fairness to drivers, we calculate each driver's average weight. One driver had 25 packages and the average weight was 2.2 lbs. What is the probability of this, if packages are truly assigned randomly?

    (f) If we randomly weigh and average 25 packages for every driver, we are 90% sure the averages will be at most how many lbs?

  39. §4.17, 24: A crab pot is lowered into the bay and retrieved 2 hours later. Suppose that a crab wanders into the pot, on average, once every 20 minutes, and this process is Poisson. Now the crab pot can hold at most 4 crabs - if a 5th comes by, it cannot get into the pot.

    (a) What is the probability distribution function for the number of crabs in a pot?

    (b) What are the expectation value and standard deviation of this distribution?

    (c) If we put out 50 pots, how mny crabs might we expect to catch? Give a range that has a 95% probability of occuring.

    (d) If the Dept. of Natural Resources said that they had averaged the number of crabs observed in 30 pots, you are 95% sure this average will be at least how much?

  40. §4.17, 29: In surveying, we measure from point A to point B by measuring a sequence of "waypoints" and summing the results. The errors would tend to accumulate. Suppose that each waypoint measurement has mean error of 0 and standard deviation of 0.6 ft.

    (a) I took 10 waypoints to get from point A to point B. What is the probability that my total error (in absolute value) is less than 1 ft?

    (b) Less than 2 ft?

    (c) 90% of the time I am within how much of the correct value?

    (d) Repeat for 20 waypoints.

  41. §4.17, 30: We are going to load seabags on a cart. The cart can hold 650 lbs. The weight of bags has a normal distribution with mean 50 lbs and a standard deviation of 10 lbs.

    (a) What is the probability that the cart can hod 10 bags?

    (b) 12 bags?

    (c) 15 bags?

    (d) If the standard deviation was 15 lbs instead of 10 lbs, what is the probability it can hold 15 bags?

    (e) For safety reasons, I want to have only a 1% chance of overloading it. How many bags can I load? Do it for standard deviation of 10 lbs., then for standard deviation 15 lbs.

  42. §4.17, 33: A car transport carries 8 cars. The weight of each car has normal distribution of with mean 1500 lbs and standard deviation 100 lbs. We designed the transport to be able to carry 14000 lbs.

    (a) What is the probability that the transport will be overloaded?

    (b) What weight should the transport be able to carry to be 99% confident that it will not be overloaded?

  43. §4.17, 34: Gallon milk bottles are filled to be accurate within 1%. Assume that 0.01 gal is equal to 2 standard deviations. Suppose 20 milk bottles are emptied into a vat.

    (a) I can be 90% certain of having at least how much milk?

    (b) I can be 90% certain of having at most how much milk?

  44. §5.8, 1: When we fill the tank with gas, the point at which the pump shuts off has a uniform distribution on [0, 0.2] gallons. (This is the amount by which we fail to completely fill the tank.)

    (a) What is the probability that the shutoff will occur within 0.05 gallons of the top of the tank?

    (b) What is the probability that the shutoff will leave more than 0.18 gallons empty?

    (c) If we fill up 10 times, what is the probability that all 10 of them will leave more than 0.15 gallons empty?

    (d) If we fill up 8 times, what is the probability all of them will leave less than 0.1 gallons empty?

    (e) We fill up a fleet of 50 cars. What is the (approximate) probability that the total of empty tank space is more than 5.5 gallons?

    (f) If I fill up 50 cars, what is a reasonable upper bound on the total amount of empty tank space? You should be 90% sure, given your (approximate) probability distribution for the total empty space.

  45. §5.8, 2: We use one digital watch to set another, but can only see the minutes. When we set the second watch, it assumes the seconds are zero. (a) What is a reasonable choice for the distribution of the number of seconds difference between the two watches? (If you use (time of reference watch) - (time of watch that is set), you will have a positive quantity.)

    (b) What is the probability that the difference between the 2 watches is more than 20 seconds?

    (c) Same for less than 10 seconds?

    (d) If we own a watch shop and set 10 watches (from the same master) what is the probability that all the watches are within 15 seconds of the master watch?

    (e) What is the probability that the master watch beeps the hour, but no other watch (of the 10 we set using it) beeps for the next 5 seconds?

    (f) I am 90% sure that after the master watch beeps there will be silence for how long (until the first set watch beeps)? (g) After the master beeps, I am 90% sure all 10 set watches will have beeped within how many seconds?

    (h) Suppose I set one watch from my master, then use the second to set a third, and the third to set a fourth, and so on, until I have set 10 watches. I am 90% sure the 10th will be at most how far off from the 1st? (This is asking about the total error in setting 10 watches.)

    (i) Same for 30 watches.


Bibliography

[T] J. Turner, Elementary Probability: A Calculating Approach, 2-nd edition, 1998.

[WMM] Walpole, Myers, and Myers, Probability and Statistics for Engineers and Scientists, 6-th edition, Prentice Hall, 1998.

created 1-4-2002 by wdj@usna.edu , last updated 1-4-2002