UNCERTAINTY, DATA & JUDGMENT EXTRA EXERCISES SET 1

Transcription

1 UNCERTAINTY, DATA & JUDGMENT EXTRA EXERCISES SET 1 INSEAD MBA Programme September/October 2009

2 1. The service lives of Sultania super light bulbs are normally distributed with a mean of 1200 hours and a standard deviation of 36 hours. a) What percentage of the bulbs will last for longer than 1272 hours? b) What percentage of the bulbs will last for less than 1146 hours? c) The 10 per cent of the bulbs with the longest service life will last for longer than how many hours? d) The 20 per cent of the bulbs with the shortest service life will last for less than how many hours? 2. Dow-Jones stock index Monthly percentage changes in the Dow-Jones stock price index from 1949 to 1970 were normally distributed with a mean of and a standard deviation of 3.5. Answer each question if a month is selected at random. a) What is the probability that the index rose during the month? b) What is the probability of an increase exceeding 5%? c) What percentage change is exceeded with probability 0.05? 3. Messages An INSEAD professor receives on average 24 messages per working day. The rate of s per hour is constant. If his working day lasts 8 hours, what is the probability that a) He receives no s in half an hour? b) He receives more than 5 s in two hours? 4. Villas by the beach A recent census indicated that villas on the French Riviera sell at an average price of 6800 per square meter, with a standard deviation of 1970 per square meter. At an official dinner one night, a potential business partner eating at your table claims to have just bought a villa on the French Riviera that belonged to a high-ranked Saudi family for less that 2000 per square meter. What is the maximum probability that his story is true? 1

3 5. Golden M&Ms The M&M company is considering a new marketing concept to reward addictive customers: the golden M&Ms. These would be of a rare kind: the probability of finding a golden candy in a regular bag will be only 10%. For every ten golden M&Ms found, one will get a special prize (for example, a free CD). a) Suppose Rosalind buys 10 bags of M&Ms. What is the probability that she will find at least one golden candy? b) Suppose Rosalind convinces 9 of her friends to go after the golden candies. They (Rosalind + her nine friends) will each buy 10 bags and pool the candies together. What would be the expected value and the standard deviation of the golden candies in the pool? What is the probability that Rosalind and her friends will win the prize? 6. Holiday Inn in Paris often grants reservations in excess of capacity to minimize losses due to no-shows. Suppose that the records of the hotel show that, on average, 10% of their prospective guests will not claim their reservation. If the hotel accepts 215 reservations for a given day and there are only 200 rooms in the hotel, what is the probability that all the guests who arrive to claim a room will receive one? 7. Suppose that the proportion of engines which contain a defect in an assembly operation is 0.10, and a sample of 200 engines is included in a particular shipment. What is the probability that at least 30 of the 200 engines contain a defect? 8. Suppose that the proportion of engines which contain a defect in an assembly operation is 0.01, and a sample of 200 engines is included in a particular shipment. What is the probability that three or fewer engines contain a defect? 9. American Express claims that its card is accepted in 60% of the hotels in Europe. If we consider the group of 10 hotels in Fontainebleau, what is the probability that only 3 will accept American Express cards? 2

4 10. Easy Stats EasyJet advertises that by 2007, 40% of their customers will fly for free. Revenues foregone from such promotion will be compensated by increased market size, and improved profitability from existing customers, based on more accurate forecasting of their willingness to pay. To collect this information, customers will find out if their ticket is free only after they have completed the internet booking (including credit card number), at the posted price. In order to be equitable across destinations and customers, EasyJet guarantees that each ticket issued has a 40% chance of being free. With this new promotion in place, EasyJet estimate that on average 5 bookings will be made on their website every 10 minutes, that is two free tickets will be offered on average every 10 minutes. a) If you would fly Easyjet four times in 2007: (i) How many flights do you expect to get for free during that year? (ii) How likely is it that you get at least one free flight? (iii) How likely is it that you get all flights for free? b) If you would fly EasyJet once a week in 2007 (there are 52 weeks in a year): (i) How many flights do you expect to get for free that year? (ii) What is the probability that you will get at least one free flight during the year? (iii) What is the probability that you will get on average one free flight per month, or more, that is at least 12 free flights per year? c) Suppose that each ticket that is not free costs 100 Euros, and you have an airfare budget of 800 Euros. How many flights can you expect to make within that budget? d) It is 4pm October 25, 2007 and you want to book a ticket on EasyJet in the next 15 minutes. (i) What is the probability that your ticket will be free? (ii) What is the probability that no free tickets will be issued during that time? 3

5 11. Executive Airlines operates small luxury jets on commuter flights from Orly airport to Heathrow (London). Although each of its jets holds 10 passengers, Executive Airlines takes up to 12 passenger reservations per flight, a practice known as overbooking. This is because they have found that the probability is 20% that a person with a reservation fails to show up for the flight. Experience has shown that most passengers are traveling alone, so you may assume that any person(s) showing up doesn t change the probability of others showing up. a) If Executive Airlines has 12 reservations for its next flight, what is the probability that there will not be enough seats on the jet for all the passengers with reservations who show up? b) Mechanical breakdowns cause significant scheduling problems for small airlines if one plane has problems, there may not be another plane available to make a scheduled flight. Executive Airlines planes are among the most reliable in service: they have found that, on average, one plane experiences mechanical problems every 10 weeks. The breakdowns appear to occur independently over time. Suppose one plane has broken down and will take one week to repair. What is the probability that at least one more plane will break down while this one is being fixed? c) Executive Airlines has found that its flight time from Orly to Heathrow is normally distributed with a mean of one hour and a standard deviation of 6 minutes. To promote its service, the airline Management is considering offering customers a money-back guarantee: if any flight takes more than 70 minutes to get to London, each passenger riding that plane will receive a full refund of the 380 (one-way) fare. Thus, sometimes the airline will pay out the refunds, but usually it will not. What is the expected loss of revenue per passenger because of this policy? 4

6 12. Getting the royal treatment Getting timely medical attention for serious health problems in the UK is becoming more and more difficult. Suppose a patient, called Harry, who is diagnosed with a certain health problem must get surgery done within 15 days in order to have a reasonable chance of survival. The two hospitals in the region have a waiting list for this surgery, and the patient can only choose one of the hospitals because of the preliminary procedures for the surgery. Hospital A handles an average of eight surgeries in 60 days and Hospital B handles an average of six surgeries in 60 days. There are three people on the waiting list for the surgery in Hospital A and there are two people on the waiting list for the surgery in Hospital B. The rates for the two hospitals are independent. a) If Harry wants to maximize the probability of getting the surgery done within 15 days, which hospital should he choose? b) Suppose there is only one (combined) waiting list for the two hospitals (basically, the two hospitals work as one unit with two operating theatres and all the preliminary procedures are performed at one place) and Harry is now sixth on the waiting list. The patient on top of the waiting list undergoes surgery at whichever hospital is ready first. Then, what is the probability that Harry will undergo surgery within 15 days? 5

7 13. Speculate.com A sizable part of internet-based trading is due to day-trading (extremely short-term speculations, usually in highly volatile stocks). a) Bill, a veteran day-trader, believes that there is a 60% chance that he makes a profit at the end of any given trading day. Since he closes his positions at the end of every trading day, Bill believes that making a profit or losing money on any given trading day is independent of what happened or will happen on any other trading day. Based on Bill s estimates, what is the probability that he will make a profit on at most 5 trading days in the next 3 weeks of trading? (Note that markets are open Mon.-Fri., so each week of trading consists of 5 trading days; assume no holidays during the next three weeks.) b) An Internet based trading service, Speculate.com, estimates that at the end of the day the return on 1 invested in day-trading is normally distributed. Speculate.com s trading records suggest that, on average, 1 invested in day-trading is worth only 95c at the end of the day (i.e., day-traders lose money as a group even though there might be some individual day-traders who make a fortune). The records also suggest a 45% chance that a 1 day-trading investment will turn out profitable at the end of the day. Using Speculate.com s records and estimates, find the standard deviation of the return at the end of the day on 1 invested in day-trading. c) Some analysts do not completely agree with the estimates provided by Speculate.com. While they do agree that at the end of the day the return on 1 invested in day-trading is normally distributed, they believe that, on average, it will be worth only 90c with a standard deviation of 20c. Using the analysts estimates, find the probability that 1 invested in day-trading will be worth more than 1.25 at the end of the day. d) Nancy is concerned about her Internet trading software crashing in critical moments. A technical support expert told her that these crashes occur independently and at a steady rate, but are not very likely. In fact, the probability of five or more crashes during the last trading hour (when many day-traders try to close their positions) is one hundredth of one percent (i.e., ). How many crashes are expected during the last trading hour? What is the probability of having no crashes in the last trading hour? 6