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Example 1. How many ways are there to distribute 8 identical objects in 4 boxes if each box must contain at least one object?. Example 2. How many ways are there for a mailman to deposit 12 letters in 2 mailboxes such that there is at least one letter per mailbox?. Example 3.
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Example 1 • How many ways are there to distribute 8 identical objects in 4 boxes if each box must contain at least one object?
Example 2 • How many ways are there for a mailman to deposit 12 letters in 2 mailboxes such that there is at least one letter per mailbox?
Example 3 • A game is considered fair if the expected value of the winnings is at least equal to the price one pays to participate in this game. a) Consider the game that consists in predicting the result of tossing a die where you gain 4 $ if you guess correctly. You are asked to pay 1 $ to participate, is this game fair ? Justify your answer. b) Assume now that you must still pay 1 $ to participate but that you gain k $ if you guess correctly. For what values of k is this game fair ? Justify your answer.
Example 4 • A company receives on average 12 orders per day for a product that it sells for 50 $. A single person working 8 hours a day is responsible for the processing of these orders. Let N denote the number of orders received in a day and X the time interval between two consecutive orders. • What is the starndard deviation of the number of orders received in 5 days? • What is the probability of receiving 68 orders or more in 5 days? • What is the probability of receiving no orders in a two-hour period? • What is the probability that the orders on a given day amount to 750 $ or more?
Example 5 • A small company employs 100 persons, 60 of whom are men. A total of 10 employees are considered executives. The president of this company believes in the equal opportunity principle and therefore "being a male employee" and "being an executive employee" are two independent events in this company. a) How many female executives are there in this company? b) The president chooses at random two executives to participate in a conference. What is the probability that at least one woman will participate in the conference? c) The president meets a male employee while leaving the office on Friday night. What is the probability that this employee is an executive? d) The president must fire one non-executive employee. He will choose this person at random. What is the probability that the soon to be unemployed individual is a man ?
Example 6 • The communauté urbaine de Québec (CUQ) is studying different scenarios to encourage the use of public transportation. It is considering the possibility of installing an electric tramway. The following three projects were retained by the municipal council: • PROJET A : • Purchase new buses; • Improve bus routes; • Increase number of bus routes. • PROJET B : • Buy 50 tramways; • Install a tramway line along the axis Cité–Sainte-Foy. • PROJET C : • Buy 100 tramways; • Install a tramway line along the axis Cité–Sainte-Foy; • Install a tramway line along the axis Cité–Beauport; • Install a tramway line along the axis Cité–Charlesbourg.
Example 6 - continued • A person at CUQ estimates at 80 % the chances that one of the tramway projects will be chosen. He also estimates that project C is three times less likely to be chosen than project B. Volvo is interested in the tramway projects. It is thinking of relocating its Canadian head office to Québec. However, Volvo's decision will depend on the project that will be chosen by CUQ. If project B is chosen, the chances that Volvo moves its Canadian head office to Québec are 90 %. If the projet C is chosen, this probability drops to 70 %. Globally, the chances of Québec obtaining Volvo's Canadian head office are 75 %. • Find the individual probabilities of choosing the projects A, B or C. • What is the probability that there will be no tramway at Beauport? • What is the probability that Volvo will not move its head office to Québec if project A is chosen? • If project C is not chosen, what is the probability that Volvo will still move its head office to Québec? • What is the probability of the best combined scenario for Québec, that is, CUQ chooses project C and Volvo moves its head office to Québec?
