Questions
STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 1 1. For each of the following parts. Determine if the number is random or not. On a separate piece of paper, please explain each answer. The amount of snow on the ground at noon on February 15. A. Yes, it is random. B. No, it is not random. 2. The first digit in your student identification number. A. Yes, it is random. B. No, it is not random. 3. You draw a queen from a well-shuffled deck of cards. A. Yes, it is random. B. No, it is not random. 4. The first number of courses at Purdue. A. Yes, it is random. B. No, it is not random. Problem 2 1. For the following scenarios, indicate which type of probability this represents. The chances that you will get the job of your dreams after graduation. A. subjective B. empirical C. theoretical (equally likely) 1 STAT 350 (Findsen) Homework 4 Problems Spring 2015 2. The probability that a person in one of my two sections of STAT 350 is female. A. subjective B. empirical C. theoretical (equally likely) 3. The probability that a high school senior in Indiana will go to Purdue. A. subjective B. empirical C. theoretical (equally likely) 4. The winning ticket at a raffle. A. subjective B. empirical C. theoretical (equally likely) Problem 3 1. Please state whether each of the following scenarios is true or false. If the situation is false, on a separate piece of paper, please describe what is wrong and correct the statement. If two events are disjoint, we can multiply their probabilities to determine the probability that they will both occur. 2. If the probability of A is 0.25, then the probability of the complement of A is -0.25. 3. If the sample space consists of two outcomes, then each outcome has probability of 0.5. 4. If the probability of A is 0.2 , the probability of B is 0.3, and the probability of A and B is 0.06, then A and B are independent. Problem 4: Multiple questions Problem 5: Multiple questions 2 STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 6 1. People with type O-negative blood are universal donors. That is, any patient can receive a transfusion of Onegative blood. Only 7% of the American population have O-negative blood. If 10 people appear at random to give blood, what is the probability that at least 1 of them is a universal donor? Give your answer to 3 decimal places. Fill in the blank: The probability that at least 1 of the 10 random Americans is a universal donor is: _______ . Problem 7 1. Mendelian inheritance. Some traits of plants and animals depend on inheritance of a single gene. This is called Mendelian inheritance, after Gregor Mendel (1822–1884). This exercise is based on the following information about Mendelian inheritance of blood type. Each of us has an ABO blood type, which describes whether two characteristics called A and B are present. Every human being has two blood type alleles (gene forms), one inherited from our mother and one from our father. Each of these alleles can be A, B, or O. Which two we inherit determines our blood type. The following table shows what our blood type is for each combination of two alleles: We inherit each of a parent's two alleles with probability 0.5. We inherit independently from our mother and father. Hannah and Jacob both have alleles A and B. What blood types can their children have? A. A and B B. A, B and AB C. A, B and O D. All of the types. 3 STAT 350 (Findsen) Homework 4 Problems Spring 2015 2. Mendelian inheritance. Some traits of plants and animals depend on inheritance of a single gene. This is called Mendelian inheritance, after Gregor Mendel (1822–1884). This exercise is based on the following information about Mendelian inheritance of blood type. Each of us has an ABO blood type, which describes whether two characteristics called A and B are present. Every human being has two blood type alleles (gene forms), one inherited from our mother and one from our father. Each of these alleles can be A, B, or O. Which two we inherit determines our blood type. The following table shows what our blood type is for each combination of two alleles: We inherit each of a parent's two alleles with probability 0.5. We inherit independently from our mother and father. Hannah and Jacob both have alleles A and B. What is the probability that their next child has each of these blood types? Give your answers to 2 decimal places. Fill in the blanks: The requested probabilities are: A) _______ . B) _______ . C) D) _______ . _______ . 4 STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 8 1. The game of Texas hold ’em starts with each player receiving two cards. Let represent the number of aces in a randomly selected deal of two cards in this game. Here is the probability distribution for the random variable : Find , the mean of the probability distribution of Give your answer to 4 decimal places. Fill in the blank: . _______ 2. The game of Texas hold ’em starts with each player receiving two cards. Let represent the number of aces in a randomly selected deal of two cards in this game. Here is the probability distribution for the random variable : Fill in the blank. Give your answer to three decimal places. The standard deviation of the number of aces is ___. 5 STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 9 1. In which of the following games of chance would you be willing to assume independence of and in making a probability model? (a) In blackjack, you are dealt two cards and examine the total points on the cards (face cards count 10 points). You can choose to be dealt another card and compete based on the total points on all three cards. (b) In craps, the betting is based on successive rolls of two dice. is the sum of the faces on the first roll, and the sum of the faces on the next roll. A. (a) only. B. (b) only. C. Both (a) and (b). D. None of them. Problem 10 1. On a separate piece of paper, answer the following: A) Show: Var(X) = E(X2) - (E(X))2 (Chapter 4, Slide 40) B) BONUS: Show Rule 3 for the discrete case on Chapter 4, Slide 36. E(g(X)) = g(x) is a linear equation, that is, g(x) = ax + b. . Hint: Assume A report of the National Center for Health Statistics says that the heights of 20-year-old men have mean 176.8 centimeters (cm) and standard deviation 7.2 cm. There are 2.54 centimeters in an inch. What is the mean height in inches? Give your answer to 2 decimal places. Fill in the blank: The mean height in inches is: _______ in. 2. A report of the National Center for Health Statistics says that the heights of 20-year-old men have mean 176.8 centimeters (cm) and standard deviation 7.2 cm. There are 2.54 centimeters in an inch. What is the standard deviation of the height in inches? Give your answer to 2 decimal places. Fill in the blank: The standard deviation in inches is: _______ in. 6 STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 11 1. Assume that a 25-year-old man has these probabilities of dying during the next five years: Age at death Probability 25 26 27 28 29 0.00039 0.00044 0.00051 0.00057 0.00060 What is the probability that the man does not die in the next five years? Give your answer to 5 decimal places. Fill in the blank: The probability to survive the next five years is: _______ . 2. Assume a 25-year-old man has these probabilities of dying during the next five years: Age at death Probability 25 26 27 28 29 0.00039 0.00044 0.00051 0.00057 0.00060 An online insurance site offers a term insurance policy that will pay $100,000 if a 25-year-old man dies within the next 5 years. The cost is $175 per year. So the insurance company will take in $875 from this policy if the man does not die within five years. If he does die, the company must pay $100,000. Its loss depends on how many premiums were paid, as follows: What is the insurance company's mean cash intake from such polices? Give your answer in dollars to 2 decimal places. Fill in the blank: The company's mean cash intake is: $ _______ . 7 STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 12 1. You insure the life of a 25-year-old friend under the following terms: You will pay $100,000 if your 25-year-old friend dies within the next five years. The cost for your friend is $175 per year. So you will take in $875 from this policy if your friend does not die within five years. If he does die, you must pay $100,000. Your monetary loss depends on how many premiums your friend paid. See the below figure detailing your monetary loss. State whether the following is True or False: "Selling insurance is less risky for an insurance company that insures many thousands of 25-year-old men than for you." Please explain your answer on a separate piece of paper. Problem 13 1. Julie is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let , , and be the events that Julie is offered a job by = the Connecticut Office of the Chief Medical Examiner = the New Jersey Division of Criminal Justice = the federal Disaster Mortuary Operations Response Team Julie writes down her personal probabilities for being offered these jobs: What is the probability that Julie is offered at least one of the three jobs? Give your answer to 1 decimal place. Fill in the blank: _______ . 8 STAT 350 (Findsen) Homework 4 Problems Spring 2015 2. Julie is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let , , and be the events that Julie is offered a job by = the Connecticut Office of the Chief Medical Examiner = the New Jersey Division of Criminal Justice = the federal Disaster Mortuary Operations Response Team Julie writes down her personal probabilities for being offered these jobs: What is the probability that Julie is offered both the Connecticut and New Jersey jobs, but not the federal job? Give your answer to 1 decimal place. Fill in the blank: _______ . 3. Julie is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let , , and be the events that Julie is offered a job by = the Connecticut Office of the Chief Medical Examiner = the New Jersey Division of Criminal Justice = the federal Disaster Mortuary Operations Response Team Julie writes down her personal probabilities for being offered these jobs: What is the probability that Julie is offered the federal job or the New Jersey job? Give your answer to 1 decimal place. Fill in the blank: P(B or C) = _______ . 9 STAT 350 (Findsen) Homework 4 Problems Spring 2015 4. Julie is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let , , and be the events that Julie is offered a job by = the Connecticut Office of the Chief Medical Examiner = the New Jersey Division of Criminal Justice = the federal Disaster Mortuary Operations Response Team Julie writes down her personal probabilities for being offered these jobs: If Julie is offered the federal job, what is the conditional probability that she is also offered the New Jersey job? Give your answer to 1 decimal place. Fill in the blank: The conditional probability is: _______ . 5. Julie is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let , , and be the events that Julie is offered a job by = the Connecticut Office of the Chief Medical Examiner = the New Jersey Division of Criminal Justice = the federal Disaster Mortuary Operations Response Team Julie writes down her personal probabilities for being offered these jobs: If Julie is offered the New Jersey job, what is the conditional probability that she is also offered the federal job? Give your answer to 1 decimal place. Fill in the blank: The conditional probability is: _______ . 10 STAT 350 (Findsen) Homework 4 Problems Spring 2015 Problem 14 1. Muscular dystrophy is an incurable muscle-wasting disease. The most common and serious type, called DMD, is caused by a sex-linked recessive mutation. Specifically: women can be carriers but do not get the disease; a son of a carrier has probability 0.50 of having DMD; a daughter has probability 0.50 of being a carrier. As many as 1/3 of DMD cases, however, are due to spontaneous mutations in sons of mothers who are not carriers. Toni has one son, who has DMD. In the absence of other information, the probability is 1/3 that the son is the victim of a spontaneous mutation and 2/3 that Toni is a carrier. There is a screening test called the CK test that is positive with probability 0.70 if a woman is a carrier and with probability 0.10 if she is not. Toni's CK test is positive. What is the probability that she is a carrier? Give your answer to 2 decimal places. Fill in the blank: The probability that Toni is a carrier is: _______ . 11
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