Understanding Discrete Probability Distributions in Assignments

1 ASSIGNMENT 5, Chapter 5. Discrete probability distributions. Ch.5.1 Discrete random variables and their probability distributions 1. Ch.5.1, 12 -15, p.174 OPTIONAL 2. Ch.5.1, 17 - 21, p.174

2 3. Ch.5.1, 22, p.174 4. Ch.5.1, 23, p.174 5. Ch.5.1, 31, p.175 6. A service club sells 4000 tickets, $ 10 each, for a cash prize of $800. If you buy one ticket, what is expected gain? 7. In a lottery conducted to benefit the local fire station, 8000 tickets are sold at $5 each. The prize is a $12, 000 car. If you purchase 2 tickets, what is your expected gain?

3 8. A charitable organization raises funds by selling 2,000 raffle tickets, $ 10 each, for a first prize of $ 500 and a second price of $100. If you buy one ticket, what is expected gain? 9. Example 5.4, p. 173 An insurance company needs to know how much to charge for a $100,000 policy insuring an event against cancellation due to inclement weather. The probability of inclement weather during the time of event is assessed as 2 in 100. a. Calculate the value of the yearly premium such that the expected gain for insurance company will be zero. b. Calculate the value of the yearly premium such that the expected gain for insurance company will equal to $500. 10. Ch.5.1, 33, p.175 11. Mr. X wants to insure his car priced at $20, 000 for its total. The probability of the accident in the coming year is estimated to be 0.08. a. What premium should Mr. X pay to the insurance company, if the company wants the expected gain to be $ 4000? b. What premium should Mr. X pay to the insurance company, if the company wants just to break even?

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4 12.Ch.5.1, 36, p.176 13. Ch.5.1, 37, p.176 14. A manufacturing representative is considering taking out an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, the representative feels that a loss of $ 80, 000 would be incurred: if it is only moderately successful, a loss of $ 25, 000 would be incurred. Insurance actuaries have determined from marked surveys and other available information that the probabilities that the product will be a failure or only moderately successful are 0.01 and 0.05 respectively. Assuming that the manufacturing representative is willing to ignore all other losses, what premium should the insurance company charge for a policy in order to break even?

5 Ch.5.2 The binomial probability distribution 1. Ch.5.2, 39, p.187 2. Ch.5.2, 40, p.187 3. Ch.5.2, 40, p.187

6 4. Ch.5.2, 45, p.187 5. Ch.5.2, 50, p.188 6. Ch. 5.2, 51, p.188) modified. Car color preferences change over the years and according to the particular model that the customer selects. In a recent year, 10% of all luxury cars sold were black. If 25 cars of that year and type are randomly selected, find the following probabilities. a. At least five cars are black. b. At most six cars are black. c. More than four cars are black. d. Less than seven cars are black. e. Exactly four cars are black. f. Between three and five cars (inclusive) are black. g. More than 20 cars are not black.

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7 7. Ch. 5.2, 53, p.188 8. Ch. 5.2, 53, p.188 9. A home security system is designed to have a 99% reliability rate. Suppose that nine homes, equipped with this system, experience an attempted burglary. Find the probabilities that a. Only one alarm will be triggered. b. All alarms will be triggered. c. At least one of the alarms will be triggered. d. More than seven alarms will be triggered. e. At least seven alarms will be triggered (seven or more alarms will be triggered). f. Less than eight alarms will be triggered. g. At most eight alarms will be triggered (eight or fewer alarms will be triggered). h. Find the mean number of alarms triggered and standard deviation of number of alarms triggered.

8 10. A student had no time to prepare for the test and randomly selects the answers for 25 multiple choice questions. Each of 25 multiple choice questions has 5 possible answers and only one answer for each question is correct. a. What is probability that the student gets exactly 10 correct answers? b. What is probability that the student gets at least one correct answer? c. What is probability that the student gets at least two correct answers? d. What is the mean number (expected number) of correct answers and what is the variance? The test can be considered as binomial experiment with n=25 independent trials, and probability of success p = 1/5 (success is identified with the choice of correct answer). Ch.5.4 The hypergeometric probability distribution 1. Ch. 5.4, 32, p.199 2. Ch. 5.4, 31, p.199

9 3. Ch. 5.4, 30, p.199 4. Ch. 5.4, 35, p.200 5. Example 5.17, p.197 A case of wine has 12 bottles, 3 of which contain spoiled wine. A sample of 4 bottles is randomly selected from the case. Find probability distribution for the number of bottles of spoiled wine in the sample and the mean and variance of spoiled bottles in the sample. 6.Example 5.18, p.198 A particular industrial product is shipped in lots of 20. Testing whether the item is defective is costly; hence, the manufacturer samples production rather than using a 100% inspective plan. A sample of five items is selected from each lot, and a lot is rejected if more than one defective item is observed. If a lot contains four defective items, what is probability that it will be accepted?

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10 Ch.5.3 The Poisson probability distribution 1. Example 5.13, p.190 The average number of traffic accidents on a certain section of highway is two per week. Assume that the number of accidents follows Poisson distribution with the mean equal to 2. a. What is probability of no accidents on his section of highway during a 1 – week period? b. What is probability of at most three accidents on this section of highway during a 2 – week period? 2. Example 5.16, p.193 A manufacturer of power mowers buys engines in lots of 1000 from the supplier and equips each of the mowers produced by her plant with one of the engines. History shows that the probability of any one engine from that supplier being defective is 0.001. a. In a shipment of 1000 engines, what is probability that none of engines is defective? Three are defective? b. In a shipment of 1000 engines, what is probability that at most three engines are defective? 3. Ch. 5.3, 15, p.195 4. Ch. 5.3, 19, p.195

11 5. Ch. 5, Review, 25, p.206 6. Ch. 5.3, 13, p.195 7. Ch. 5.3, 17, p.195

ASSIGNMENT 5

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