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QMS Chapter5+6(Fall2012)-1.doc

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Department
Quantitative Methods
Course
QMS 102
Professor
Jason Chin- Tiong Chan
Semester
Fall

Description
Business Statistics I-QMS102 Chapter5and 6 Probability The chance that an event will occur 1. Discuss the meaning of “10% chance of snowing this late afternoon”. 2. For a true or false question, what is the chance of getting the correct answer if I completely guess the answer? Why? 3. What is the purpose of buying insurance? 4. For a multiple-choice question with four choices, which are A, B, C and D, what is the chance of getting the correct answer if I completely guess the answer? Why? 5. Give 2-3 examples of situations where you use probability concept in your daily life. 6.1 PROBABILITY DISTRIBUTION Outcome: Understand the concepts of discrete and continuous random variables and their associated distributions Definitions 1. random variable – a variable whose value is determined by the outcomes of a random experiment 2. discrete random variable – a random variable that can take on any value from a list of distinct possible values 3. continuous random variable – a random variable that can take on any value from a continuous range. 4. probability distribution for a discrete random variable – a list of all the possible values of the random variable and their associated probabilities Example1 Which of the following random variables are discrete, and which are continuous? a. The number of passengers on a flight from Toronto to Hong Kong. b. The number of students in Business Statistics class. c. The time required driving from home to Ryerson University in the morning. d. The number of traffic fatalities per year in Toronto. Fall2012 Page # 1 Business Statistics I-QMS102 Chapter5and 6 Probability distributions 1.can be represented by a.table b. graphical chart 2.can be summarized with a. expected values(or means), which are measures of centre tendency (location) of the probability distribution b. standard deviations, which are measures of variability of the probability distribution Expected value = mean = E(X) = xP(x) ∑ Standard deviation = ∑ x P(x) − μ2 Example2 The probability distribution for a random variable X is as shown in the following table. x 1 2 3 4 5 P(X=x) 0.20 0.25 0.40 0.10 0.05 Determine a. the expected value b. the variance c. the standard deviation Example3 The probability distribution for a random variable X is as shown in the following table. x 1 2 3 4 5 P(X=x) 0.1 0.3 y 0.2 0.1 Determine a. the value of y b. the expected value c. the standard deviation Fall2012 Page # 2 Business Statistics I-QMS102 Chapter5and 6 Example 4 Amanda, the marketing manager for Third Cups Coffee Store is implementing a new marketing strategy that offers free refills on all coffee order. To review her strategy, she gathered the following information on coffee refills. Refills 0 1 2 3 4 Percent 21 37 32 10 0 Compute a. the probability that a customer refill his/her coffee one or more times b. the expected value, variance and standard deviation for the distribution of number of refills. Example 5 A fast food company plans to install a new ice-cream dispensing unit in one of the two store locations. The company figures that the probability of a unit being successful in a location is 5/8 and the annual profit in this case is $185,000. If it is not successful there will be losses of $36,800. At the location B the probability of succeeding is ½ but the potential profit and loss are $250,000 and $58,000 respectively. a. Where should the company locate to maximize expected profit? b. Which location is less risky, i.e has the lowest relative variability? Fall2012 Page # 3 Business Statistics I-QMS102 Chapter5and 6 Example6 The normal weekly demand of a certain perishable product sold by Ryerson Inc. is given by the following distribution Demand 21 22 23 24 25 Probability 0.4 0.2 0.2 0.1 0.1 The product costs Ryerson $9 each. The product sells for $16 each. If not sold by the end of the week, the leftover units must be scrapped. The supplier only has 23, 24 or 25 units for Ryerson to purchase. How many would you recommend Ryerson to purchase based on expected profit? 23 units Demand Profit P(X) 21 22 23 24 25 24 units Demand Profit P(X) 21 22 23 24 25 25 units Demand Profit P(X) 21 22 23 24 25 Fall2012 Page # 4 Business Statistics I-QMS102 Chapter5and 6 Example 7 You can insure an $80,000 24K gold watch for its total value by paying a premium of P dollars. If the probability of theft in a given year is estimated to be 0.02,what premium should the insurance company charge if it wants the expected gain to be $3000? Fall2012 Page # 5 Business Statistics I-QMS102 Chapter5and 6 Example 8. Christopher owns an IT business in downtown Toronto. The probability model below describes the number of employees that may call in sick on any given day. Number of Employee 0 1 2 3 4 5 P(X=x) a 0.4 0 1.5b 1.5a 0.05 Note: b=2a What is the expected value of the number of employees calling in sick each day? Fall2012 Page # 6 Business Statistics I-QMS102 Chapter5and 6 Example 9 A tire manufacturer has four tire designs. The manufacturing cost of each type of tire is shown below. Tire Design Fixed Cost/year Variable Cost A $41,000 $28 B $65,000 $21 C $50,000 $25 D $128,000 $18 There are four possible levels of annual demand: 4,000 tires, 6,000tires, 8,000 tires and 10,000 tires. The respective probabilities are 0.30, 0.25, 0.4, and 0.05. The selling price will be $73 for A, $71 for B, $ 68 for C and $74 for D. a. Calculate the expected annual demand for tires b. Calculate the expected profit for tire Design C Fall2012 Page # 7 Business Statistics I-QMS102 Chapter5and 6 6.2 THE BINOMIAL PROBABILITY DISTRIBUTION Outcome: Recognize situations when the binomial probability distribution applies, and use formula to calculate binomial probabilities Binomial probability distribution 1. Often applies when we are interested in the number of times a particular characteristic turns up. Example: we might want to know how many of the next 50 customers who come into the store will purchase some merchandise. 2. The count (how many out of a particular number), which can also be expressed as a percentage or proportion. 3. This count is a random variable, because its outcome is determined by chance. 4. A binomial random variable counts the number of times one of only two possible outcomes takes place (thus the bi-part of the binomial’s name). 5. A binomial experiment must satisfy the following four conditions. a. there are n identical trials b. each tria
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