Complete Each Pdf assignment
MAT2572 Probability and Mathematical Statistics I Assignment 1 The assignment needs be submitted in a single pdf file. Please show steps/reasoning. 1. A graduating engineer has signed up for three job interviews. She intends to categorize each one as being either a “success” or a “failure” depending on whether it leads to a plant trip. (a) Write out the appropriate sample space. (b) What outcomes are in the event A: Second success occurs on third interview? 2. A coin is tossed four times and the resulting sequence of heads and/or tails is recorded. Define the events A, B, and C as follows: A: exactly two heads appear; B: heads and tails alternate; C: first two tosses are heads (a) Which events, if any, are mutually exclusive? Why? (b) Which events, if any, are subsets of other sets? Why? 3. (1.9) Suppose you are eating at a pizza parlor with two friends. You have agreed to the following rule to decide who will pay the bill: Each person will toss a coin and the person who gets a result different from the others will pay the bill. If all three tosses are the same, the bill will be shared by all. Find the probability that (a) You will pay for everyone; (b) the bill will be shared; (c) you get a free lunch. 4. A total of 1200 graduates of State Tech have gotten into medical school in the past several years. Of that number, 1000 earned scores of 27 or higher on the MCAT and 400 had GPAs that were 3.5 or higher. Moreover, 300 had MCATs that were 27 or higher and GPAs that were 3.5 or higher. What proportion of those 1200 graduates with an MCAT lower than 27 and a GPA below 3.5? 5. (1.13) The experiment is to select a person from the U.S. at random. Event A is “person resides in New York’” and Event B is “person is an immigrant.” Suppose P (A) = 0.066, P (B) = 0.072, and P (A ∩B) = 0.02. (a) Interpret the events A ∩B and A ∪B. (b) What is P (A ∪B)? 6. An urn contains twenty-four chips, numbered 1 through 24. One is drawn at random. Let A be the event that the number is divisible by 2 and let B be the event that the number is divisible by 3. Find P (A ∪B). 7. Let A and B be any two events defined on S. Suppose that P (A) = 0.4, P (B) = 0.5, and P (A ∩B) = 0.1. What is the probability that A or B but not both occur? 8. (1.15) Suppose that in a certain country 10% of the elderly people have diabetes. It is also known that 10% of the elderly people are living below poverty level, and 35% of the elderly population falls into at least one of these categories. (a) What proportion of elderly people in this country have both diabetes and are living below poverty level? (b) Suppose we choose an elderly person in this country “at random.” What is the probability that the person will neither have diabetes nor be living at the poverty level? 9. What is a sample space? What is the difference between an experiment, an outcome, a sample space, and an event? What is a probability? Be specific and feel free to reference any of the above problems as examples. Page 2 MAT2572 Probability and Mathematical Statistics I Assignment 2 The assignment needs be submitted in a single pdf file. For the R code, please include code and the output. 1. Suppose you roll a fair, six-sided die 2 times. (a) Compute the probability that the sum of two numbers is greater than 8. (b) Write R code to approximate the probability that the sum of two numbers is greater than 8. 2. (2.7) In China, 48% of the population have blood type O, 28% have blood type A, 19% have blood type B, and 5% have blood type AB. Suppose four people in China are chosen “at random”. Write R code to simulate thee probability that there is one of each blood type. For the following two problems, please choose one to do. If you do both, you will receive extra credit. 3. (2.11) A corporation has 6 executives in Chicago, 8 in Los Angeles, and 12 in Tampa. Three executives are chosen “at random” to represent the corporation at a retreat. Use simulations in R to approximate the probability that none is from Tampa. 4. (2.13) A drawer contains 12 pair of socks, where each pair is a different color. Sam draws four socks “at random” from the drawer (without replacement). Write R code to determine the probability that there is no pair among the four socks. 5. Explain in words the general scheme for approximating probabilities experimentally. What are the obstacles, or the difficulties that you may encounter? Why is it that while a probability of an event is a fixed number, the experiment gives a different result each time? What then is true about your approximations and are you sure it is true? why or why not? MAT2572 Probability and Mathematical Statistics I Assignment 3 The assignment needs be submitted in a single pdf file. Please show steps/reasoning. 1. (3.3) Given an area code, how many different (seven digit) phone numbers are possible if none of the numbers can start with zero? 2. (3.6) Suppose there are 24 kids in a sports competition. (a) How many different sets of winners of gold, silver, and bronze medals are there? (b) If three of them are to get (unranked) medals, how many different sets of winners are there? 3. (3.7) A shop makes deluxe ice cream sundaes with three scoops of ice cream. If there are 12 ice cream flavors, how many different sundaes can be made if each of the scoops is a different flavor? (You can decide whether or not order is important, and explain why.) 4. (3.5) A corporation has 6 executives in Chicago, 8 in Los Angeles, and 12 in Tampa. Three executives are chosen “at random” to represent the corporation at a retreat. What is the probability that none is from Tampa? 5. (3.12) A deck of 40 cards contains ten of each of the following colors: red, blue, green, and yellow. If the deck is well shuffled, and a hand of four cards is randomly chosen (without replacement), what is the probability that all four colors are in the hand? (You will receive extra credit if you use simulations in R to verify your answer). 6. An urn contains twenty chips, numbered 1 through 20. Two are drawn simultaneously. What is the probability that the numbers on the two chips will differ by more than 2? 7. Suppose that 5 fair six-sided dice are rolled. What is the probability that all 5 faces will be the same? 8. Explain the difference between ”with replacement” and ”without replacement” and when order matters vs when it does. Give examples with your explanation. MAT2572 Probability and Mathematical Statistics I Assignment 4 The assignment needs be submitted in a single pdf file. Please show steps/reasoning. 1. Suppose that two fair dice are tossed. What is the probability that the sum equals 10 given that it exceeds 8? 2. A fair coin is tossed three times. What is the probability that at least two heads will occur given that at most two heads have occurred? 3. (4.3) In Smalltown, 40% of households have at least one dog and 60% of households have at least one cat, while 20% of households have neither dogs nor cats. If a household is chosen at random and found to have at least one dog, what is the probability that it also has at least one cat? 4. Suppose that P (A ∩B) = 0.2, P (A) = 0.6, and P (B) = 0.5. (a) Are A and B mutually exclusive? (b) Are A and B independent? (c) Find P (AC ∪BC). 5. Suppose that P (A) = 1/4 and P (B) = 1/8. (a) What does P (A ∪B) equal if i. A and B are mutually exclusive? ii. A and B are independent? (b) What does P (A|B) equal if i. A and B are mutually exclusive? ii. A and B are independent? 6. It is known that Blood A type is 40%, Blood B type is 10%, Blood AB type is 5%, and Blood O type is 45%. If we randomly selected two people, what is the probability that they have different blood types? 7. (4.6) Suppose that in a certain country 10% of the elderly people have diabetes. It is also known that 30% of the elderly people are living below the poverty level, and 35% of the elderly population falls into at least one of these categories. (a) Given that a randomly selected elderly person is living below the poverty level, what is the probability that this person has diabetes? (b) Are the events “has diabetes” and ”living below the poverty level” disjoint (mu- tually exclusive) in this elderly population? Explain. (c) Are the events “has diabetes” and ”living below the poverty level” independent in this elderly population? Explain. 8. Explain what conditional probability is in your own words. MAT2572 Probability and Mathematical Statistics I Assignment 5 The assignment needs be submitted in a single pdf file. Please show steps/reasoning. 1. An urn contains five balls numbered 1 to 5. Two balls are drawn simultaneously. (a) what is the sample space? (b) Let X be the larger of the two numbers drawn. Find the probability mass function of X. (c) What is E(X) ? (d) Calculate V (X) and sd(X). 2. A fair coin is tossed three times. Let X be the number of heads in the tosses minus the number of tails. Find the probability mass function of X. 3. (5.2) A quiz consists of two multiple choice questions with choices (a), (b), (c) for each. If an unprepared student marks answers at random, what are the probabilities of the following events: (a) Both answers are correct. (b) Exactly one correct answer. (c) Both answers wrong. (d) Let Y be the random variable representing the number of correct answers on the quiz. What is E(Y )? 4. (5.16) Harry will play a lottery game repeatedly until he wins or until he runs out of the money. It costs $20 to play the game, and he has only $60. The probability of winning for each game is p = 0.15, and each game is independent. Let Y be