Statistics Data Analysis Course Number Course Section Meeting

Statistics & Data Analysis Course Number Course Section Meeting Time B 01. 1305 31 Wednesday 6: 00 -8: 50 pm CLASS #2 Professor S. D. Balkin -- February 5, 2003

Class #2 Outline § § Brief review of last class Class introduction with Birthday Problem Questions on homework Chapter 3: A First Look at Probability Professor S. D. Balkin -- February 5, 2003 2

Class Introduction and The Birthday Problem § Everyone introduce yourselves, giving your name, job/industry, and birthday § Question: How likely is it that two people in your class have the same birthday? § Let’s make a bet: I bet that at least two people in this class share the same birthday. • What should we bet? • Should I be so certain? Professor S. D. Balkin -- February 5, 2003 3

Review of Last Class § Distinguish between quantitative and qualitative variables § Graphical representations of single variables § Numeric measures of center and variation Professor S. D. Balkin -- February 5, 2003 4

Chapter 3 A First Look At Probability Professor S. D. Balkin -- February 5, 2003

Chapter Goals § Be able to interpret probabilities § Understand the differences between statistics and probability § Understand basic principles of probability • Addition, Complements, Multiplication § Understand statistical independence and conditional probability § Be able to construct probability trees § Understand managerial implications of probability Professor S. D. Balkin -- February 5, 2003 6

Probability in Everyday Life § § § There is a 90% chance the Yankees will win the game tomorrow There is a sixty percent chance of thunderstorm this afternoon That bill has a 35% chance of being passed There is a 20% chance of rain today There is a 37% chance my hand will beat the dealer’s Professor S. D. Balkin -- February 5, 2003 7

Probability in Everyday Life (cont) § Your company is deciding on launching a new product in the consumer market. Success based on reaction from competition, ability of suppliers to meet demand, unknown adverse events or issues, economic and regulatory conditions, etc. § An airplane has multiple engines and can make a journey safely as long as at least one is operating. Despite designers’ best efforts, what is the chance of a disaster occurring? Which parts of the plane should receive the most attention? § You’re still waiting for Ed Mc. Mahan to knock on your door? Professor S. D. Balkin -- February 5, 2003 8

What is Probability? § Quantification of uncertainty and variability § Basis for statistical inference and business decision making § Probability theory is a branch of mathematics and it beyond the scope of this class Professor S. D. Balkin -- February 5, 2003 9

Illustrative Questions… If you toss a coin, what is the probability of getting a head? § If you toss a coin twice, what is the probability of getting exactly one Head? • § § How can you verify your answer? If you toss a coin 10 times and count the total number of Heads, do you think probability of 0 heads equals the probability of 5 heads? Do you think probability of 4 heads equals the probability of 6 heads? Professor S. D. Balkin -- February 5, 2003 10

History of Probability § Originated from the study of games of chance • • Tossing a dice Spinning a roulette wheel § Probability theory as a quantitative discipline arose in the seventeenth century when French gamblers prominent mathematicians for help in their gambling § In the eighteenth and nineteenth centuries, careful measurements in astronomy and surveying led to further advances in probability. § In the twentieth century probability is used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; figure out the energy states of subatomic particles; Estimate the spread of rumors; and predict the rate of return in risky investments. Adapted from Probability Central Professor S. D. Balkin -- February 5, 2003 11

Example: New York Times Online Cellphones Not Killing Real Ones (May 26, 2002) Despite their growing affection for cellphones, most Americans are not ready to pull the plug on traditional phones, according to a survey by Maritz Research. The results were released this month. When asked about the probability that they would use only cellphones for their calls in the next year, only 8 percent said that they were very likely or certain to do so; 79 percent answered "very unlikely" or "absolutely not. " Maritz surveyed 803 adults nationwide this spring. Each respondent, or someone in the household, subscribed to a wireless phone service, Forty-two percent, however, said their wireless phones had led them to use their existing long-distance companies less than they did previously. "Just five years ago, cellphones were viewed as a luxury; now they've become ingrained in everyday life for all members of a family, " said Paul Pacholski, a vice president at Maritz. Professor S. D. Balkin -- February 5, 2003 12

Example: Wall Street Journal Online European Markets Close Little Changed (May 21, 2002) …Retail-price data published Tuesday showed that inflation in the United Kingdom was steady in April at an annual rate of 2. 3%, lower than the expected 2. 4%. However, Lehman Brothers economist Michael Hume said the numbers are no obstacle to an interest-rate hike. "We continue to look for a rate hike in June, but would put the probability of a move at no more than 60%, " he said…. Professor S. D. Balkin -- February 5, 2003 13

Interesting Probability Quotes § Aristotle: The probable is what usually happens § Sir Arther Conan Doyle, The Sign of Four : When you have eliminated the impossible, what ever remains, however improbable, must be the truth. § Blaise Pascal: The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it. Professor S. D. Balkin -- February 5, 2003 14

Types of Occurrences § Predictable Occurrence: Occurrence whose value can be accurately determined using science: • Position of a meteor in 25 years § Unpredictable Occurrence: Occurrence whose value is based on a random process: • Toss of a coin • Gender of a baby § Random Process: An event or phenomenon is called random if individual outcomes are uncertain but there is, however, a regular distribution of relative frequencies in a large number of repetitions. Professor S. D. Balkin -- February 5, 2003 15

Probability and Statistics § Statistics: Observed data to generalizations about how the world works § Probability: Start from an assumption about how the world works, and then figure out what kinds of data you are likely to see Probability is the only scientific basis for decision making in the face of uncertainty Professor S. D. Balkin -- February 5, 2003 16

Terminology § Random Experiment: A process or course of action that results in one of a number of possible outcomes • The outcome that occurs cannot be predicted with certainty § Outcome: Single possible results of a random experiment § Sample Space: The set of all possible outcomes of the experiment § Event: Any subset of the sample space § Simple Event: Event consisting of just one outcome Professor S. D. Balkin -- February 5, 2003 17

Example § If we toss a nickel and a dime: • What are the possible outcomes? • Which outcome is the event “no heads”? • Which outcomes are in the event “one head and one tail”? • Which outcomes are in the event “one or more heads”? Professor S. D. Balkin -- February 5, 2003 18

Defining Probabilities § Probability has no precise definition!! § All attempts to define probability must ultimately rely on circular reasoning § Roughly speaking, the probability of an event is the chance or likelihood that the event will occur § To each event A, we want to attach a number P(A), called the probability of A, which represents the likelihood that A will occur Professor S. D. Balkin -- February 5, 2003 19

Defining Probabilities (cont. ) § There are various ways to define P(A), but in order to make sense, any definition must satisfy • P(A) is between zero and 1 • P(E 1) + P(E 2) + ··· = 1, where E 1, E 2, ··· are the simple events in the sample space § The three most useful approaches to obtaining a definition of probability are: • classical • relative frequency • subjective Professor S. D. Balkin -- February 5, 2003 20

Classical Approach Assume that all simple events are equally likely. Define the classical probability that an event A will occur as: So P(A) is the number of ways in which A can occur, divided by the number of possible individual outcomes, assuming all are equally likely. Professor S. D. Balkin -- February 5, 2003 21

Example: Classical Approach § In tossing a coin twice, if we take: S = {HH, HT, TH, TT}, then the classical approach assigns probability 1/ 4 to each simple event. § If A = {Exactly One Head} = {HT, TH}, then P(A) = 2/ 4 = 1/ 2. Question : Does this tell you how often A would occur if we repeated the experiment (“toss a coin twice”) many times? Professor S. D. Balkin -- February 5, 2003 22

Relative Frequency Approach § The probability of an event is the long run frequency of occurrence. § To estimate P(A) using the frequency approach, repeat the experiment n times (with n large) and compute x/n, where x = # Times A occurred in the n trials. § The larger we make n, the closer x/ n gets to P(A). Coin Flipping Example Professor S. D. Balkin -- February 5, 2003 23

Classical and Frequency Approaches § If we can find a sample space in which the simple events really are equally likely, then the Law of Large Numbers asserts that the classical and frequency approaches will produce the same results. § For the experiment “Toss a coin once”, the sample space is S = {H, T} and the classical probability of Heads is 1/2. § According to the Law of Large Numbers (LLN), if we toss a fair coin repeatedly, then the proportion of Heads will get closer and closer to the Classical probability of 1/2. Professor S. D. Balkin -- February 5, 2003 24

Subjective Approach § This approach is useful in betting situations and scenarios where one- time decision- making is necessary. In cases such as these, we wouldn’t be able to assume all outcomes are equally likely and we may not have any prior data to use in our choice. § The subjective probability of an event reflects our personal opinion about the likelihood of occurrence. Subjective probability may be based on a variety of factors including intuition, educated guesswork, and empirical data. § Eg: In my opinion, there is an 85% probability that Stern will move up in the rankings in the next Business Week survey of the top business schools. Professor S. D. Balkin -- February 5, 2003 25

Example: Not Equally Likely Events § A market research survey asks the planned number of children for newly married couples giving the following data. What are the probabilities of a couple planning: • 1 or 2 children? • 3 or 4 children? • 4 or more children? Professor S. D. Balkin -- February 5, 2003 26

Complement Rule § The probability of the complement of an event is equal to 1 minus the probability of the event itself Professor S. D. Balkin -- February 5, 2003 27

Example: Complement Rule § A market research survey asks the planned number of children for newly married couples giving the following data. • Use the complement rule to find the probability of a couple planning to have any children at all Professor S. D. Balkin -- February 5, 2003 28

Odds § Odds are often used to describe the payoff for a bet. § If the odds against a horse are a: b, then the bettor must risk b dollars to make a profit of a dollars. § If the true probability of the horse winning is b/(a+b), then this is a fair bet. § In the 1999 Belmont Stakes, the odds against Lemon Drop Kid were 29. 75 to 1, so a $2 ticket paid $61. 50. § The ticket returns two times the odds, plus the $2 ticket price. Professor S. D. Balkin -- February 5, 2003 29

Example: Odds § If a fair coin is tossed once, the odds on Heads are 1 to 1 § If a fair die is tossed once, the odds on a six are 5 to 1. § In the game of Craps, the odds on getting a 6 before a 7 are 6 to 5. (We will show this later). Professor S. D. Balkin -- February 5, 2003 30

Combining Events § The union A B is the event consisting of all outcomes in A or in B or in both. § The intersection A B is the event consisting of all outcomes in both A and B. § If A B contains no outcomes then A, B are said to be mutually exclusive. § The Complement of the event A consists of all outcomes in the sample space S which are not in A. Professor S. D. Balkin -- February 5, 2003 31

Combining Events (cont. ) Professor S. D. Balkin -- February 5, 2003 32

Rules for Combining Events Professor S. D. Balkin -- February 5, 2003 33

Example 1: Combining Events § Based on the past experience in your copier repair shop suppose… • Probability of a blown fuse is 6% • Probability of a broken wire is 4% • 1% of copiers to be repaired come in with both a blown fuse AND a broken wire § What is the probability of a copier coming in with a blown fuse OR a broken wire? Professor S. D. Balkin -- February 5, 2003 34

Example 2: Combining Events § Market research firm tests a potential new product § 200 male respondents, selected at random, gave their opinions for the product and their marital status giving the following data: Professor S. D. Balkin -- February 5, 2003 35

Conditional Probability § Calculating probabilities given some restrictive condition § Example: Absenteeism Last Year for 400 Employees. § Compute the probability that a randomly selected employee is a smoker. § If we are told that the employee was absent less than 10 days, does this partial knowledge change the probability that the employee is a smoker? Professor S. D. Balkin -- February 5, 2003 36

Conditional Probability (cont. ) Professor S. D. Balkin -- February 5, 2003 37

Multiplication Law Professor S. D. Balkin -- February 5, 2003 38

Statistical Independence Events A and B are statistically independent if and only if P(B|A) = P(B). Otherwise, they are dependent. If events A and B are independent, then P(A B) = P(A)P(B) Professor S. D. Balkin -- February 5, 2003 39

Example: Independence § Seattle corporations with 500 or more employees • 468 executives; 30 whom are women • Conditional probability of a person being a woman given that the person is an executive is 30/468 = 0. 064 § In the population, 51. 2% are women § Since the probability of randomly choosing a women changes when conditioning on “being an executive”, being a women and being an executive are dependent events Professor S. D. Balkin -- February 5, 2003 40

Another Independence Example § You are responsible for scheduling a construction project • In order to avoid trouble, it will be necessary for the foundation to be completed by July 27 th and for the electricity to be installed before August 6 th • Based on your experiences, you fix probabilities of 0. 83 and 0. 91 for these events to occur • Assume you have a 96% chance of meeting one deadline or the other (or both) § What is the probability of missing both deadlines? § Are these events mutually exclusive? How? § Are these events independent? How? Professor S. D. Balkin -- February 5, 2003 41

Revisiting the Birthday Problem § What is the probability that at least two people in this class share the same birthday? § Can be formulated as: What is the probability no one in this class shares the same birthday, and take the complement Professor S. D. Balkin -- February 5, 2003 42

Probability Tables and Trees § Human resources found that 46% of its junior executives have two-career marriages, 37% have single-career marriages, and 17% are unmarried. § HR estimates that 40% of the two-career marriage executives would refuse to transfer, as would 15% of the single-careermarriage executives, and 10% of the unmarried executives. § If a transfer offer is made to randomly selected executives, what is the probability it will be refused? Professor S. D. Balkin -- February 5, 2003 43

Probability Tables § Fill in this probability table: Professor S. D. Balkin -- February 5, 2003 44

Constructing Probability Trees 1. Events forming the first set of branches must have known marginal probabilities, must be mutually exclusive, and should exhaust all possibilities 2. Events forming the second set of branches must be entered at the tip of each of the sets of first branches. Conditional probabilities, given the relevant first branch, must be entered, unless assumed independence allows the use of unconditional probabilities 3. Branches must always be mutually exclusive and exhaustive Professor S. D. Balkin -- February 5, 2003 45

Probability Tree § Construct a probability tree Professor S. D. Balkin -- February 5, 2003 46

Let’s Make a Deal § In the show Let’s Make a Deal, a prize is hidden behind on of three doors. The contestant picks one of the doors. § Before opening it, one of the other two doors is opened and it is shown that the prize isn’t behind that door. § The contestant is offered the chance to switch to the remaining door. § Should the contestant switch? § Solve by making a tree… Professor S. D. Balkin -- February 5, 2003 47

Employee Drug Testing § A firm has a mandatory, random drug testing policy § The testing procedure is not perfect. • If an employee uses drugs, the test will be positive with probability 0. 90. • If an employee does not use drugs, the test will be negative 95% of the time. • Confidential sources say that 8% of the employees are drug users § 8% is an unconditional probability; 90 and 95% are conditional probabilities Professor S. D. Balkin -- February 5, 2003 48

Employee Drug Testing (cont. ) § Create a probability tree and verify the following probabilities: • • Probability of randomly selecting a drug user who tests positive = 0. 072 Probability of randomly selecting a non-user who tests positive = 0. 046 Probability of randomly selecting someone who tests positive = 0. 118 Conditional probability of testing positive given a non-drug user = 0. 05 Professor S. D. Balkin -- February 5, 2003 49

Next Time… § Random variables and probability distributions Professor S. D. Balkin -- February 5, 2003 50

Homework #2 § Hildebrand/Ott • • • 3. 3 3. 4 3. 5 3. 8 3. 10, 3. 11, and 3. 12 on pages 76 -77. These all draw on the same data, so it’s easy to deal with them together. Note that those who recalled the commercial correctly are in the “favorable” and “unfavorable” columns. 3. 14 3. 24, pages 90 -91. Observe that the rows of the given table sum to 1. These are thus conditional probabilities for the retest, given the results of the first test. For example, P(Retest = minor | First = major) = 0. 5. Part (c) asks you to supply two numbers. 3. 28 3. 29 Professor S. D. Balkin -- February 5, 2003 § Verzani • NONE 51