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Prob. problems

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Ankit Gupta

This document contains 34 quantitative probability problems involving concepts like binomial, normal, and Poisson distributions. The problems cover topics like finding the probability of certain outcomes occurring based on given probabilities, estimating values from sample data, and calculating confidence intervals.

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Quantitative Methods Probability Problems Prof. L. Shridharan 1. (a) What is the probability that a leap year selected at random will contain 53 Tuesdays? (b) What is the probability that a leap year selected at random will contain 53 Sundays or 53 Mondays? 2. A bag contains 8 black, 3 red and 9 white balls. If 3 balls are drawn at random , find the probability that (a) all are black, (b) 2 are black and 1 is white, (c) 1 of each colour, (d) the balls are drawn in the order black, red and white, (e) None is red. 3. From a pack of 52 cards, 4 are accidentally dropped. Find the chance that (a) they will consist of a joker, a queen, a king and an ace. (b) They are from the same suit. (c) They are one from each suit. (d) Two of them are red and two are black. 4. A six figure number is formed by the digits 4, 5, 6, 7, 8 & 9 ; no digit being repeated. Find the chance that the number formed is : (a) divisible by 5. (b) not divisible by 5. 5. The probability that a contractor will get a plumbing contract is 2/3, and the probability that he will get an electric contract is 5/9. If the probability of getting at least one contract is 4/5, what is the probability that he will get both the contracts? 6. The probability that a management trainee will remain with a company is 0.6. The probability that an employee earns more than Rs.10,000 per month is 0.5. The probability that an employee is management trainee who remained with the company or who earns more than Rs.10,000 per month is 0.7. What is the probability that an employee earns more than Rs.10,000 per month, given that he is a management trainee who stayed with the company? 7. A piece of electronic equipment has two essential parts A and B. In the past, part A failed 30% of the times, part B failed 20% of the times and both failed simultaneously 5% of the times. Assuming that both parts must operate to enable the equipment to function, what is the probability that the equipment will function? 8. A salesman has a 60% chance of making a sale to each customer. The behaviour of successive customers is independent. If two customers A and B enter the shop, what is the probability that the salesman will make a sale to A or B?
9. Suppose that a product is produced in 3 factories A, B & C. It is known that A produces twice as many items as B, and that B & C produce the same number of items. Assume that it is known that 2% of items produced by each of A & B are defective, while 4% of those of C are defective. All items produced in three factories are stocked, and an item is selected randomly. What is the probability of this item being defective? 10. In a population of workers, 40% are primary school graduates, 50% are high school graduates and 10% are college graduates. The unemployment rates among the 3 categories are respectively 10%, 5% & 2%. If a worker is chosen at random and found to be unemployed, what is the probability that he is a college graduate? 11. Air Corporation, having had just 2 aircrash during its first fifty years of existence, wants to make the next decade “air-crash free”. Assuming that the same trend will continue, what is the probability of the Corporation meeting the target? 12. A distributor of bean seeds determines from extensive tests that 5% of large batch of seeds will not germinate. He sells the seeds in packets of 200 and guarantees 90% germination. Determine the probability that a particular packet will violate the guarantee. 13. The life time in hours of certain electrical equipment has the normal distribution with mean at 80 hours and standard deviation at 16 hours. (a) What is the probability that the equipment lasts at least 100 hrs.? (b) If the equipment has already lasted 85 hours, what is the conditional probability that it will last another 15 hours? 14. A sample of 100 dry battery cells, tested to find the length of life produced, has mean 12 hrs. and standard deviation 3 hrs. Assuming the data to be normally distributed, what percentage of battery cells are expected to have life: (a) more than 15 hours, (b) less than 6 hours, (c) between 8 and 14 hours. 15. 2000 electric bulbs with an average life of 1000 hours and a standard deviation of 200 hours are installed in a town. Assuming the lives of the bulbs to be normally distributed, answer the following: (a) how many bulbs can be expected to fail in the first 700 burning hours? (b) What is the minimum burning life of the top one quarter of bulbs? 16. In a village of 21 inhabitants, a person tells a rumour to a second person, who in turn repeats it to a third person, etc. At each step the recipient of the rumour is chosen at random from the 20 people available. Find the probability that the rumour will be told 10 times without: (a) returning to the originator; (b) being repeated to any person. 17. Probability that a man will be alive 25 years hence is 0.3 and the probability that his wife will be alive 25 years hence is 0.4. Find the probability that 25 years hence (a) both will be alive, (b) only the man will be alive, (c) only the woman will be alive, (d) none will be alive, e) at least one of them will be alive.
18. A market research firm is interested in surveying certain attitudes in a small community. There are 125 households broken down according to income, ownership of a telephone or ownership of a TV. Households with Monthly Income < Rs.8000 Households with Monthly Income > Rs.8000 Telephone No Telephone Telephone No Telephone Owns TV 27 20 18 10 No TV 18 10 12 10 (a) What is the probability of obtaining of a TV owner in drawing at random? (b) If a household has a monthly income of over Rs.8000 and is a telephone subscriber, what is the probability that he has a TV? (c) What is the conditional probability of drawing a household that owns a TV, given that the household is a telephone subscriber? (d) Are the events ‘ownership of a TV’ and ‘telephone subscriber’ statistically independent? 19. In a railway reservation office, two clerks are engaged in checking reservation forms. On an average, the first clerk checks 55% of the form, while the second does the remaining. The first clerk has an error rate of 0.03 and the second has an error rate of 0.02. A reservation form is selected at random from the total number of forms checked during a day, and is found to have an error. Find the probability that it was checked (a) by the first clerk, (b) by the second clerk. 20. A manufacturing firm purchases a certain component for its manufacturing process from 3 sub-contractors A, B & C. They supply respectively 60%, 30% & 10% of the requirements. It is known that 2%, 5% & 8% of the items supplied by the respective suppliers are defective. On a particular day, a normal shipment arrives from each of the three suppliers and the contents get mixed. If a component is chosen at random from the day’s shipment, what is the probability that it is defective? 21. Due to turnover and absenteeism at an assembly plant, 20% of the items are assembled by inexperienced employees. Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. What is the probability that an item was assembled by an inexperienced employee, given that the item was returned? 22. An insurance salesman sells policies to 5 men, all identical age and good health. According to the actuarial tables, the probability that a man of this particular age will be alive 30 years hence is 2/3. Find the probability that in 30 years (a) all 5 will be alive? (b) At least one will be alive? (c) At most 3 will be alive?
23. A machine produces an average of 20%defective bolts. A batch is accepted if a sample of 5 bolts taken from that batch contains no defective and rejected if the sample contains 3 or more defectives. In other cases, a second sample is taken. What is the probability that the second sample is required? 24. In a certain factory turning out optical lenses, there is a small chance of 1/500 for any one lens to be defective. The lenses are supplied in packets of 10. Calculate the number of packets containing no defective, one defective, two defective, and three defective lenses in a consignment of 20,000 packets. 25. A cigarette company wants to promote the sales of Brand A with special advertising campaign. Fifty out of every 1000 cigarettes are rolled up in gold foil and randomly mixed up with the regular ones. The company offers to trade a new packet of cigarettes for each gold foiled cigarette a smoker finds in a packet of Brand A. What is the probability that buyers of Brand A will find 0, 1, 2, ….. gold cigarettes in a single pack of 10? 26. The daily wages of 1000 workmen are normally distributed with mean as Rs.70 and standard deviation as Rs.5. Estimate the number of workers with daily wages : (a) between Rs.70 and 74, (b) between Rs.67 and 73, (c) more than Rs.76 (d) less than Rs.60. Also estimate the lowest daily wages of the 100 highest paid workmen. 27. A factory turns out an article by mass production methods. From past experience it appears that 20 articles on an average are rejected out of every batch of 100. Find the variance of number of rejects in a batch. What is the probability that the number of rejects in a batch exceeds 30? 28. The arithmetic mean of purchases per day per customer in a large store is Rs.2500 with a standard deviation of Rs.1000. If on a particular day, 100 customers purchased for Rs.3780 or more, estimate the total number of customers who purchased from the store that day. 29. A factory turns out an article by mass production methods. From past experience it appears that 10 articles on the average are rejected out of every batch of 100. (a) Find the standard deviation of the number of rejects in a batch. (b) Write down the probability function for number of rejects assuming normal approximation. 30. Comment on the following: (a) The mean of a symmetrical binomial distribution is 5 and the number of trials is 12. (b) The mean of a Poisson distribution is 5 and the standard deviation is 3. (c) A binomial distribution with mean= 4 and variance = 3. (d) A Poisson distribution with mean= 10 and variance = 25. (e) A normal distribution with mean= 50 and median = 52.
31.The manager of Cardinal Electric’s light bulb division must estimate the average number of hours that a bulb made by each bulb machine will last. A sample of 40 bulbs was selected from machine A and the average burning time was 1,416 hours. The standard deviation of burning time is known to be 30 hours. (a) Compute the standard error of the mean. (b) Construct a 90% confidence interval for the true population mean. 32.A drug manufacturer wants to market a new drug only if he could be quite sure that the mean temperature of a healthy person taking the drug could not rise above 98.6 F; otherwise he will withhold the drug. The drug is administered to a random sample of 17 healthy persons. The mean temperature was found to be 98.4 F, with a standard deviation of 0.6 F. Assuming that the distribution of the temperature is normal, and α= 0.01, what should the manufacturer do? 33.Bill, the top advertising executive for Grapevine Concepts, as just launched a publicity campaign for a new restaurant in town, he Black Angus. Bill has just installed four billboards on a highway outside of town and he knows from experience the probabilities that each will be noticed by a randomly chosen motorist. The probability of the first billboard’s being noticed by a motorist is 0.75.The probability of the second’s being noticed is 0.82,the third has a probability of 0.87 of being noticed, and the probability of the fourth sign’s being noticed is 0.9.Assuming that the event that a motorist notices any particular billboard is independent of whether or not he notices the others, what is the probability that- (a) All four billboards will be noticed by a randomly chosen motorist? (b) The first and fourth, but not the second and third billboards will be noticed? (c) Exactly one of the billboards will be noticed? (d) None of the billboards will be noticed? (e) The third and fourth billboards won’t be noticed? 34.Martin Coleman, credit manager for Beck’s, knows that the company uses three methods to encourage collection of delinquent accounts. From past collection records, he learns that 70 percent of the accounts are called on personally, 20 percent are phoned, and 10 percent are sent a letter. The probabilities of collecting an overdue amount from an account with the three methods are 0.75, 0.60, and 0.65 respectively. Mr. Coleman has just received payment from a past due account. What is the probability that this account- a) Was called on personally? b) Received a phone call? c) Received a letter?
31.The manager of Cardinal Electric’s light bulb division must estimate the average number of hours that a bulb made by each bulb machine will last. A sample of 40 bulbs was selected from machine A and the average burning time was 1,416 hours. The standard deviation of burning time is known to be 30 hours. (a) Compute the standard error of the mean. (b) Construct a 90% confidence interval for the true population mean. 32.A drug manufacturer wants to market a new drug only if he could be quite sure that the mean temperature of a healthy person taking the drug could not rise above 98.6 F; otherwise he will withhold the drug. The drug is administered to a random sample of 17 healthy persons. The mean temperature was found to be 98.4 F, with a standard deviation of 0.6 F. Assuming that the distribution of the temperature is normal, and α= 0.01, what should the manufacturer do? 33.Bill, the top advertising executive for Grapevine Concepts, as just launched a publicity campaign for a new restaurant in town, he Black Angus. Bill has just installed four billboards on a highway outside of town and he knows from experience the probabilities that each will be noticed by a randomly chosen motorist. The probability of the first billboard’s being noticed by a motorist is 0.75.The probability of the second’s being noticed is 0.82,the third has a probability of 0.87 of being noticed, and the probability of the fourth sign’s being noticed is 0.9.Assuming that the event that a motorist notices any particular billboard is independent of whether or not he notices the others, what is the probability that- (a) All four billboards will be noticed by a randomly chosen motorist? (b) The first and fourth, but not the second and third billboards will be noticed? (c) Exactly one of the billboards will be noticed? (d) None of the billboards will be noticed? (e) The third and fourth billboards won’t be noticed? 34.Martin Coleman, credit manager for Beck’s, knows that the company uses three methods to encourage collection of delinquent accounts. From past collection records, he learns that 70 percent of the accounts are called on personally, 20 percent are phoned, and 10 percent are sent a letter. The probabilities of collecting an overdue amount from an account with the three methods are 0.75, 0.60, and 0.65 respectively. Mr. Coleman has just received payment from a past due account. What is the probability that this account- a) Was called on personally? b) Received a phone call? c) Received a letter?

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Prob. problems

  • 1. Quantitative Methods Probability Problems Prof. L. Shridharan 1. (a) What is the probability that a leap year selected at random will contain 53 Tuesdays? (b) What is the probability that a leap year selected at random will contain 53 Sundays or 53 Mondays? 2. A bag contains 8 black, 3 red and 9 white balls. If 3 balls are drawn at random , find the probability that (a) all are black, (b) 2 are black and 1 is white, (c) 1 of each colour, (d) the balls are drawn in the order black, red and white, (e) None is red. 3. From a pack of 52 cards, 4 are accidentally dropped. Find the chance that (a) they will consist of a joker, a queen, a king and an ace. (b) They are from the same suit. (c) They are one from each suit. (d) Two of them are red and two are black. 4. A six figure number is formed by the digits 4, 5, 6, 7, 8 & 9 ; no digit being repeated. Find the chance that the number formed is : (a) divisible by 5. (b) not divisible by 5. 5. The probability that a contractor will get a plumbing contract is 2/3, and the probability that he will get an electric contract is 5/9. If the probability of getting at least one contract is 4/5, what is the probability that he will get both the contracts? 6. The probability that a management trainee will remain with a company is 0.6. The probability that an employee earns more than Rs.10,000 per month is 0.5. The probability that an employee is management trainee who remained with the company or who earns more than Rs.10,000 per month is 0.7. What is the probability that an employee earns more than Rs.10,000 per month, given that he is a management trainee who stayed with the company? 7. A piece of electronic equipment has two essential parts A and B. In the past, part A failed 30% of the times, part B failed 20% of the times and both failed simultaneously 5% of the times. Assuming that both parts must operate to enable the equipment to function, what is the probability that the equipment will function? 8. A salesman has a 60% chance of making a sale to each customer. The behaviour of successive customers is independent. If two customers A and B enter the shop, what is the probability that the salesman will make a sale to A or B?
  • 2. 9. Suppose that a product is produced in 3 factories A, B & C. It is known that A produces twice as many items as B, and that B & C produce the same number of items. Assume that it is known that 2% of items produced by each of A & B are defective, while 4% of those of C are defective. All items produced in three factories are stocked, and an item is selected randomly. What is the probability of this item being defective? 10. In a population of workers, 40% are primary school graduates, 50% are high school graduates and 10% are college graduates. The unemployment rates among the 3 categories are respectively 10%, 5% & 2%. If a worker is chosen at random and found to be unemployed, what is the probability that he is a college graduate? 11. Air Corporation, having had just 2 aircrash during its first fifty years of existence, wants to make the next decade “air-crash free”. Assuming that the same trend will continue, what is the probability of the Corporation meeting the target? 12. A distributor of bean seeds determines from extensive tests that 5% of large batch of seeds will not germinate. He sells the seeds in packets of 200 and guarantees 90% germination. Determine the probability that a particular packet will violate the guarantee. 13. The life time in hours of certain electrical equipment has the normal distribution with mean at 80 hours and standard deviation at 16 hours. (a) What is the probability that the equipment lasts at least 100 hrs.? (b) If the equipment has already lasted 85 hours, what is the conditional probability that it will last another 15 hours? 14. A sample of 100 dry battery cells, tested to find the length of life produced, has mean 12 hrs. and standard deviation 3 hrs. Assuming the data to be normally distributed, what percentage of battery cells are expected to have life: (a) more than 15 hours, (b) less than 6 hours, (c) between 8 and 14 hours. 15. 2000 electric bulbs with an average life of 1000 hours and a standard deviation of 200 hours are installed in a town. Assuming the lives of the bulbs to be normally distributed, answer the following: (a) how many bulbs can be expected to fail in the first 700 burning hours? (b) What is the minimum burning life of the top one quarter of bulbs? 16. In a village of 21 inhabitants, a person tells a rumour to a second person, who in turn repeats it to a third person, etc. At each step the recipient of the rumour is chosen at random from the 20 people available. Find the probability that the rumour will be told 10 times without: (a) returning to the originator; (b) being repeated to any person. 17. Probability that a man will be alive 25 years hence is 0.3 and the probability that his wife will be alive 25 years hence is 0.4. Find the probability that 25 years hence (a) both will be alive, (b) only the man will be alive, (c) only the woman will be alive, (d) none will be alive, e) at least one of them will be alive.
  • 3. 18. A market research firm is interested in surveying certain attitudes in a small community. There are 125 households broken down according to income, ownership of a telephone or ownership of a TV. Households with Monthly Income < Rs.8000 Households with Monthly Income > Rs.8000 Telephone No Telephone Telephone No Telephone Owns TV 27 20 18 10 No TV 18 10 12 10 (a) What is the probability of obtaining of a TV owner in drawing at random? (b) If a household has a monthly income of over Rs.8000 and is a telephone subscriber, what is the probability that he has a TV? (c) What is the conditional probability of drawing a household that owns a TV, given that the household is a telephone subscriber? (d) Are the events ‘ownership of a TV’ and ‘telephone subscriber’ statistically independent? 19. In a railway reservation office, two clerks are engaged in checking reservation forms. On an average, the first clerk checks 55% of the form, while the second does the remaining. The first clerk has an error rate of 0.03 and the second has an error rate of 0.02. A reservation form is selected at random from the total number of forms checked during a day, and is found to have an error. Find the probability that it was checked (a) by the first clerk, (b) by the second clerk. 20. A manufacturing firm purchases a certain component for its manufacturing process from 3 sub-contractors A, B & C. They supply respectively 60%, 30% & 10% of the requirements. It is known that 2%, 5% & 8% of the items supplied by the respective suppliers are defective. On a particular day, a normal shipment arrives from each of the three suppliers and the contents get mixed. If a component is chosen at random from the day’s shipment, what is the probability that it is defective? 21. Due to turnover and absenteeism at an assembly plant, 20% of the items are assembled by inexperienced employees. Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. What is the probability that an item was assembled by an inexperienced employee, given that the item was returned? 22. An insurance salesman sells policies to 5 men, all identical age and good health. According to the actuarial tables, the probability that a man of this particular age will be alive 30 years hence is 2/3. Find the probability that in 30 years (a) all 5 will be alive? (b) At least one will be alive? (c) At most 3 will be alive?
  • 4. 23. A machine produces an average of 20%defective bolts. A batch is accepted if a sample of 5 bolts taken from that batch contains no defective and rejected if the sample contains 3 or more defectives. In other cases, a second sample is taken. What is the probability that the second sample is required? 24. In a certain factory turning out optical lenses, there is a small chance of 1/500 for any one lens to be defective. The lenses are supplied in packets of 10. Calculate the number of packets containing no defective, one defective, two defective, and three defective lenses in a consignment of 20,000 packets. 25. A cigarette company wants to promote the sales of Brand A with special advertising campaign. Fifty out of every 1000 cigarettes are rolled up in gold foil and randomly mixed up with the regular ones. The company offers to trade a new packet of cigarettes for each gold foiled cigarette a smoker finds in a packet of Brand A. What is the probability that buyers of Brand A will find 0, 1, 2, ….. gold cigarettes in a single pack of 10? 26. The daily wages of 1000 workmen are normally distributed with mean as Rs.70 and standard deviation as Rs.5. Estimate the number of workers with daily wages : (a) between Rs.70 and 74, (b) between Rs.67 and 73, (c) more than Rs.76 (d) less than Rs.60. Also estimate the lowest daily wages of the 100 highest paid workmen. 27. A factory turns out an article by mass production methods. From past experience it appears that 20 articles on an average are rejected out of every batch of 100. Find the variance of number of rejects in a batch. What is the probability that the number of rejects in a batch exceeds 30? 28. The arithmetic mean of purchases per day per customer in a large store is Rs.2500 with a standard deviation of Rs.1000. If on a particular day, 100 customers purchased for Rs.3780 or more, estimate the total number of customers who purchased from the store that day. 29. A factory turns out an article by mass production methods. From past experience it appears that 10 articles on the average are rejected out of every batch of 100. (a) Find the standard deviation of the number of rejects in a batch. (b) Write down the probability function for number of rejects assuming normal approximation. 30. Comment on the following: (a) The mean of a symmetrical binomial distribution is 5 and the number of trials is 12. (b) The mean of a Poisson distribution is 5 and the standard deviation is 3. (c) A binomial distribution with mean= 4 and variance = 3. (d) A Poisson distribution with mean= 10 and variance = 25. (e) A normal distribution with mean= 50 and median = 52.
  • 5. 31.The manager of Cardinal Electric’s light bulb division must estimate the average number of hours that a bulb made by each bulb machine will last. A sample of 40 bulbs was selected from machine A and the average burning time was 1,416 hours. The standard deviation of burning time is known to be 30 hours. (a) Compute the standard error of the mean. (b) Construct a 90% confidence interval for the true population mean. 32.A drug manufacturer wants to market a new drug only if he could be quite sure that the mean temperature of a healthy person taking the drug could not rise above 98.6 F; otherwise he will withhold the drug. The drug is administered to a random sample of 17 healthy persons. The mean temperature was found to be 98.4 F, with a standard deviation of 0.6 F. Assuming that the distribution of the temperature is normal, and α= 0.01, what should the manufacturer do? 33.Bill, the top advertising executive for Grapevine Concepts, as just launched a publicity campaign for a new restaurant in town, he Black Angus. Bill has just installed four billboards on a highway outside of town and he knows from experience the probabilities that each will be noticed by a randomly chosen motorist. The probability of the first billboard’s being noticed by a motorist is 0.75.The probability of the second’s being noticed is 0.82,the third has a probability of 0.87 of being noticed, and the probability of the fourth sign’s being noticed is 0.9.Assuming that the event that a motorist notices any particular billboard is independent of whether or not he notices the others, what is the probability that- (a) All four billboards will be noticed by a randomly chosen motorist? (b) The first and fourth, but not the second and third billboards will be noticed? (c) Exactly one of the billboards will be noticed? (d) None of the billboards will be noticed? (e) The third and fourth billboards won’t be noticed? 34.Martin Coleman, credit manager for Beck’s, knows that the company uses three methods to encourage collection of delinquent accounts. From past collection records, he learns that 70 percent of the accounts are called on personally, 20 percent are phoned, and 10 percent are sent a letter. The probabilities of collecting an overdue amount from an account with the three methods are 0.75, 0.60, and 0.65 respectively. Mr. Coleman has just received payment from a past due account. What is the probability that this account- a) Was called on personally? b) Received a phone call? c) Received a letter?
  • 6. 31.The manager of Cardinal Electric’s light bulb division must estimate the average number of hours that a bulb made by each bulb machine will last. A sample of 40 bulbs was selected from machine A and the average burning time was 1,416 hours. The standard deviation of burning time is known to be 30 hours. (a) Compute the standard error of the mean. (b) Construct a 90% confidence interval for the true population mean. 32.A drug manufacturer wants to market a new drug only if he could be quite sure that the mean temperature of a healthy person taking the drug could not rise above 98.6 F; otherwise he will withhold the drug. The drug is administered to a random sample of 17 healthy persons. The mean temperature was found to be 98.4 F, with a standard deviation of 0.6 F. Assuming that the distribution of the temperature is normal, and α= 0.01, what should the manufacturer do? 33.Bill, the top advertising executive for Grapevine Concepts, as just launched a publicity campaign for a new restaurant in town, he Black Angus. Bill has just installed four billboards on a highway outside of town and he knows from experience the probabilities that each will be noticed by a randomly chosen motorist. The probability of the first billboard’s being noticed by a motorist is 0.75.The probability of the second’s being noticed is 0.82,the third has a probability of 0.87 of being noticed, and the probability of the fourth sign’s being noticed is 0.9.Assuming that the event that a motorist notices any particular billboard is independent of whether or not he notices the others, what is the probability that- (a) All four billboards will be noticed by a randomly chosen motorist? (b) The first and fourth, but not the second and third billboards will be noticed? (c) Exactly one of the billboards will be noticed? (d) None of the billboards will be noticed? (e) The third and fourth billboards won’t be noticed? 34.Martin Coleman, credit manager for Beck’s, knows that the company uses three methods to encourage collection of delinquent accounts. From past collection records, he learns that 70 percent of the accounts are called on personally, 20 percent are phoned, and 10 percent are sent a letter. The probabilities of collecting an overdue amount from an account with the three methods are 0.75, 0.60, and 0.65 respectively. Mr. Coleman has just received payment from a past due account. What is the probability that this account- a) Was called on personally? b) Received a phone call? c) Received a letter?