Question

MATH220 Lab04 Due Date: See ACE Excel File: Lab2204-182.xlsx 1 Question 1: (by hand) A retail establishment accepts either the American Express card or the VISA credit card. A total of 30% of its customers carry an American Express card, 70% carry a VISA card, and 20% carry both. a) Find the probability that a randomly selected customer carries at least one of the two credit cards. i) Draw a Venn diagram and shade the region corresponding to at least one. b) Find the probability that a randomly selected customer carries one, and only one, of the two credit cards. i) Draw a Venn diagram and shade the region corresponding to one and only one. c) Given that a randomly selected customer is found to carry a VISA card, what is the probability he/she also carries an American Express card. d) Given that a randomly selected customer is found to carry and American Express card, probability he/she also carries a VISA? e) Are carrying an American Express card and carrying a VISA card independent? Question 2: (by hand) According to Erin Newton of Medscape, mammography remains the most cost-effective approach for breast cancer screening, the sensitivity (67.8%) and specificity (75%) are not ideal (http://emedicine.medscape.com/article/1945498-overview). Sensitivity is the probability of a positive test given a person has the disease. Specificity is the probability of a negative test given no disease. The breast cancer incidence rate for women over 50 in Indiana is 327.8 per 100,000. a) What is the probability that a randomly selected woman over 50 in Indiana has breast cancer? b) A randomly selected woman over 50 has a positive mammogram. What is the probability that she actually has breast cancer? Hint: Use a probability tree. c) A randomly selected woman over 50 has a negative mammogram. What is the probability that she actually has breast cancer? Hint: Use a probability tree. d) Based on these numbers, are mammograms useful? Question 3: (by hand) A gambler plays a sequence of games that she either wins or loses. The outcomes of the games are independent, and the probability that the gambler wins is 70%. The gambler stops playing as soon as she either has won a total of 2 games or has lost a total of 3 games. Let T be the number of games played by the gambler. a) Draw a tree diagram to illustrate this situation. b) What values can T take? c) Make a table of the probabilities associated with each value of T. d) Find the mean of T. e) Find the standard deviation of T. MATH220 Lab04 Due Date: See ACE Excel File: Lab2204-182.xlsx 2 Question 4: (by hand) Heights of basketball players in a certain league are normally distributed with a mean of 74 inches and a standard deviation of 3 inches. a) What is the probability that a randomly selected player is taller than 72 inches? b) Given that a randomly selected player is taller than 72 inches, what is the probability that the player is taller than 75 inches? Question 5: (by hand) A random number generator generates uniformly distributed numbers between 0 and 1. a. What is the probability that a number generated by the generator is less than 0.70? b. Given that a number is less than 0.80, what is the probability that it is greater than 0.20? Question 6: (by hand & Excel) According to the current Commissioners' Standard Ordinary mortality table, adopted by state insurance regulators in December 2002, a 25-year-old man has these probabilities of dying during the next five years: Age at death 25 26 27 28 29 Probability 0.00039 0.00044 0.00051 0.00057 0.00060 a) What is the probability that the man does not die in the next five years? b) An online insurance site offers a term insurance policy that will pay $100,000 if a 25-yearold 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: c) d) What is the insurance company's mean cash intake from such polices? Hint: These losses should be considered as negative numbers whereas the payments are positive. c) The associated lab Excel file contains a simulation of insuring 1000 25-year-old men with the probabilities in this problem. i. To run the simulation, type a 0 in one of the blank cells in column H and notice that the number alive in row 3 changes. Copy and paste the first four rows, columns A-F, into your lab report. ii. Repeat this 4 more times. iii. Answer the questions below: iv. In these 5 trials, what was the maximum number of men who died? v. If the insurance company had to pay out $100,000 for a death, how much money did the insurance company pay out for the trial with the most deaths? vi. How many deaths will cause the company to take a loss?

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