To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-1 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Chapter 2 Probability.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-2 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Learning Objectives Students will be able to: Understand the basic foundations of probability analysis Understand the difference between mutually exclusive and collectively exhaustive events Describe statistically dependent and independent events

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-3 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Learning Objectives - continued Describe and provide examples of both discrete and continuous random variables Explain the difference between discrete and continuous probability distributions Calculate expected values and variances

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-4 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Introduction Life is uncertain! We must deal with risk! What is the chance that Hurricane Frances will hit New Orleans? Who will win the election? What is the chance that the oil price will hit $50? Is it likely that SLU Faculty will get a pay raise since the tuition went up 7%? A probability is a numerical value that measure the likelihood that an event will occur

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-5 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 How many Christmas trees should be stored? How long will it take to complete an activity? What should be the inventory level?

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-7 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 The word "statistics" is used in several different senses. In the broadest sense, "statistics" refers to a range of techniques and procedures for analyzing data, interpreting data, displaying data, and making decisions based on data. This is what courses in "statistics" generally cover. In a second usage, a "statistic" is defined as a numerical quantity (such as the mean) calculated in a sample. Such statistics are used to estimate parameters.mean) sample.parameters. The term "statistics" sometimes refers to calculated quantities regardless of whether or not

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-8 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 they are from a sample. For example, one might ask about a baseball player's statistics and be referring to his or her batting average, runs batted in, number of home runs, etc. Or, "government statistics" can refer to any numerical indexes calculated by a governmental agency. Although the different meanings of "statistics" has the potential for confusion, a careful consideration of the context in which the word is used should make its intended meaning clear.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-9 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Basic Statements About Probability 1.The probability, P, of any event or state of nature occurring is greater than or equal to 0 and less than or equal to 1. That is: 0  P(event)  1 2. The sum of the simple probabilities for all possible outcomes of an activity must equal 1.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-10 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Example 2.1 Demand for white latex paint at Diversey Paint and Supply has always been 0, 1, 2, 3, or 4 gallons per day. (There are no other possible outcomes; when one outcome occurs, no other can.) Over the past 200 days, the frequencies of demand are represented in the following table:

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-11 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Example 2.1 - continued Quantity Demanded (Gallons) 0 1 2 3 4 Number of Days 40 80 50 20 10 Total 200 Frequencies of Demand

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-12 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Example 2.1 - continued Quant. Freq. Demand (days) 0 40 1 80 2 50 3 20 4 10 Total days = 200 Probability (40/200) = 0.20 (80/200) = 0.40 (50/200) = 0.25 (20/200) = 0.10 (10/200) = 0.05 Total Prob = 1.00 Probabilities of Demand

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-13 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Types of Probability Objective probability: Determined by experiment or observation: Probability of heads on coin flip Probably of spades on drawing card from deck occurrencesor outcomes ofnumber Total occursevent timesofNumber )(  eventP

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-14 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Types of Probability Subjective probability: Based upon judgement Determined by: judgement of expert opinion polls etc.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-15 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Mutually Exclusive Events Events are said to be mutually exclusive if only one of the events can occur on any one trial

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-16 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Collectively Exhaustive Events Events are said to be collectively exhaustive if the list of outcomes includes every possible outcome: heads and tails as possible outcomes of coin flip

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-17 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Example 2 Outcome of Roll 1 2 3 4 5 6 Probability 1/6 Total = 1 Rolling a die has six possible outcomes

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-18 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Example 2a Outcome of Roll = 5 Die 1 Die 2 1 4 2 3 3 2 4 1 Probability 1/36 Rolling two dice results in a total of five spots showing. There are a total of 36 possible outcomes.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-19 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Example 3 Draw a space and a club Draw a face card and a number card Draw an ace and a 3 Draw a club and a nonclub Draw a 5 and a diamond Draw a red card and a diamond Yes No Yes Yes No Yes No Draw Mutually Collectively Exclusive Exhaustive

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-20 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Probability : Mutually Exclusive P(event A or event B) = P(event A) + P(event B) or: P(A or B) = P(A) + P(B) i.e., P(spade or club) = P(spade) + P(club) = 13/52 + 13/52 = 26/52 = 1/2 = 50%

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-21 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Probability: Not Mutually Exclusive P(event A or event B) = P(event A) + P(event B) - P(event A and event B both occurring) or P(A or B) = P(A)+P(B) - P(A and B)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-24 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Statistical Dependence Events are either statistically independent (the occurrence of one event has no effect on the probability of occurrence of the other) or statistically dependent (the occurrence of one event gives information about the occurrence of the other)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-25 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Which Are Independent? (a) Your education (b) Your income level (a) Draw a Jack of Hearts from a full 52 card deck (b) Draw a Jack of Clubs from a full 52 card deck (a) Chicago Cubs win the National League pennant (b) Chicago Cubs win the World Series

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-26 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Probabilities - Independent Events Marginal probability: the probability of an event occurring: [P(A)] Joint probability: the probability of multiple, independent events, occurring at the same time P(AB) = P(A)*P(B) Conditional probability (for independent events): the probability of event B given that event A has occurred P(B|A) = P(B) or, the probability of event A given that event B has occurred P(A|B) = P(A)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-28 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Statistically Independent Events 1. P(black ball drawn on first draw) P(B) = 0.30 (marginal probability) 2. P(two green balls drawn) P(GG) = P(G)*P(G) = 0.70*0.70 = 0.49 (joint probability for two independent events) A bucket contains 3 black balls, and 7 green balls. We draw a ball from the bucket, replace it, and draw a second ball

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-29 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Statistically Independent Events - continued 1. P(black ball drawn on second draw, first draw was green) P(B|G) = P(B) = 0.30 (conditional probability) 2. P(green ball drawn on second draw, first draw was green) P(G|G) = 0.70 (conditional probability)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-30 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Probabilities - Dependent Events Marginal probability: probability of an event occurring P(A) Conditional probability (for dependent events): the probability of event B given that event A has occurred P(B|A) = P(AB)/P(A) the probability of event A given that event B has occurred P(A|B) = P(AB)/P(B)

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-33 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Statistically Dependent Events Assume that we have an urn containing 10 balls of the following descriptions: 4 are white (W) and lettered (L) 2 are white (W) and numbered N 3 are yellow (Y) and lettered (L) 1 is yellow (Y) and numbered (N) Then: P(WL) = 4/10 = 0.40 P(WN) = 2/10 = 0.20 P(W) = 6/10 = 0.60 P(YL) = 3/10 = 0.3 P(YN) = 1/10 = 0.1 P(Y) = 4/10 = 0.4

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-34 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Statistically Dependent Events - Continued Statistically Dependent Events - Continued Then: P(L|Y) = P(YL)/P(Y) = 0.3/0.4 = 0.75 P(Y|L) = P(YL)/P(L) = 0.3/0.7 = 0.43 P(W|L) = P(WL)/P(L) = 0.4/0.7 = 0.57

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-35 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Joint Probabilities, Dependent Events Your stockbroker informs you that if the stock market reaches the 10,500 point level by January, there is a 70% probability the Tubeless Electronics will go up in value. Your own feeling is that there is only a 40% chance of the market reaching 10,500 by January. What is the probability that both the stock market will reach 10,500 points, and the price of Tubeless will go up in value?

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-36 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Joint Probabilities, Dependent Events - continued Then: P(MT) =P(T|M)P(M) = (0.70)(0.40) = 0.28 Let M represent the event of the stock market reaching the 10,500 point level, and T represent the event that Tubeless goes up.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-37 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Revising Probabilities: Bayes’ Theorem Bayes’ theorem can be used to calculate revised or posterior probabilities Prior Probabilities Bayes’ Process Posterior Probabilities New Information

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-39 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Posterior Probabilities A cup contains two dice identical in appearance. One, however, is fair (unbiased), the other is loaded (biased). The probability of rolling a 3 on the fair die is 1/6 or 0.166. The probability of tossing the same number on the loaded die is 0.60. We have no idea which die is which, but we select one by chance, and toss it. The result is a 3. What is the probability that the die rolled was fair?

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-40 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Posterior Probabilities Continued We know that: P(fair) = 0.50 P(loaded) = 0.50 And: P(3|fair) = 0.166 P(3|loaded) = 0.60 Then: P(3 and fair) = P(3|fair)P(fair) = (0.166)(0.50) = 0.083 P(3 and loaded) = P(3|loaded)P(loaded) = (0.60)(0.50) = 0.300

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-41 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Posterior Probabilities Continued A 3 can occur in combination with the state “fair die” or in combination with the state ”loaded die.” The sum of their probabilities gives the unconditional or marginal probability of a 3 on a toss: P(3) = 0.083 + 0.0300 = 0.383. Then, the probability that the die rolled was the fair one is given by:

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-42 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Further Probability Revisions To obtain further information as to whether the die just rolled is fair or loaded, let’s roll it again. Again we get a 3. Given that we have now rolled two 3s, what is the probability that the die rolled is fair?

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-43 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Further Probability Revisions - continued P(fair) = 0.50, P(loaded) = 0.50 as before P(3,3|fair) = (0.166)(0.166) = 0.027 P(3,3|loaded) = (0.60)(0.60) = 0.36 P(3,3 and fair) = P(3,3|fair)P(fair) = (0.027)(0.05) = 0.013 P(3,3 and loaded) = P(3,3|loaded)P(loaded) = (0.36)(0.5) = 0.18 P(3,3) = 0.013 + 0.18 = 0.193

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-45 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 To give the final comparison: P(fair|3) = 0.22 P(loaded|3) = 0.78 P(fair|3,3) = 0.067 P(loaded|3,3) = 0.933 Further Probability Revisions - continued

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-46 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Random Variables Discrete random variable - can assume only a finite or limited set of values- i.e., the number of automobiles sold in a year Continuous random variable - can assume any one of an infinite set of values - i.e., temperature, product lifetime

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-47 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Random Variables (Numeric) Experiment Outcome Random Variable Range of Random Variable Stock 50 Xmas trees Number of trees sold X = number of trees sold 0,1,2,, 50 Inspect 600 items Number acceptable Y = number acceptable 0,1,2,…, 600 Send out 5,000 sales letters Number of peoplee responding Z = number of people responding 0,1,2,…, 5,000 Build an apartment building %completed after 4 months R = %completed after 4 months 0  R  100 Test the lifetime of a light bulb (minutes) Time bulb lasts- up to 80,000 minutes S = time bulb burns 0  S  80,000

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-48 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Random Variables (Non-numeric) Experiment Outcome Random Variable Range of Random Variable Students respond to a questionnaire Strongly agree (SA) Agree (A) Neutral (N) Disagree (D) Strongly Disagree (SD) X = 5 if SA 4 if A 3 if N 2 if D 1 if SD 1,2,3,4,5 One machine is inspected Defective Not defective Y = 0 if defective 1 if not defective 0,1 Consumers respond to how they like a product Good Average Poor Z = 3 if good 2 if average 1 if poor 1,2,3

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-50 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Expected Value of a Discrete Probability Distribution    n i ii )X(PX)X(E 

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-52 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Binomial Distribution Assumptions: 1. Trials follow Bernoulli process – two possible outcomes 2. Probabilities stay the same from one trial to the next 3. Trials are statistically independent 4. Number of trials is a positive integer

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-53 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Binomial Distribution Probability of r successes in n trials n = number of trials r = number of successes p = probability of success q = probability of failure

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-56 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458Example Seventy percent of married couples paid for their honeymoon themselves. You randomly select 10 married couples and ask each if they paid for their honeymoon themselves. Find the probability that the number of couples who say they paid for their honeymoon themselves is a.Exactly seven b. at least 7 c. Less than seven

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-57 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Probability Distribution Continuous Random Variable Probability density function - f(X) Normal Distribution

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-58 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Normal Distribution for Different Values of   =50  =60  =40

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-59 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Normal Distribution for Different Values of   =0.1  =0.2  =0.3  = 1

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-60 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Three Common Areas Under the Curve Three Normal distributions with different areas

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-61 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Three Common Areas Under the Curve Three Normal distributions with different areas

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-66 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 The Negative Exponential Distribution  =5 Expected value = 1/  Variance = 1/  2

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-68 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Hmwwork Assignment September 7 2-30, 2-32, 2-34, 2-38, 2-41 0n Page 70

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-1 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Chapter 2 Probability.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-1 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Chapter 2 Probability.

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Presentation on theme: "To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2-1 © 2003 by Prentice Hall, Inc., Upper Saddle River, NJ 07458 Chapter 2 Probability."— Presentation transcript:

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