Chapter 6. QTM1310/ Sharpe. Randomness and Probability. 6.1 Random Phenomena and Probability. 6.1 Random Phenomena and Probability

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1 6.1 Random Phenomena and Probability Chapter 6 Randomness and Probability The Law of Large Numbers (LLN) states that if the events are independent, then as the number of trials increases, the longrun relative frequency of an event gets closer and closer to a single value. Empirical probability is based on repeatedly observing the event s outcome. Copyright 2015 Pearson Education. All rights reserved. 6-1 Copyright 2015 Pearson Education. All rights reserved Random Phenomena and Probability 6.2 The Nonexistent Law of Averages With random phenomena, we can t predict the individual outcomes, but we can hope to understand characteristics of their long-run behavior. For any random phenomenon, each attempt, or trial, generates an outcome. We use the more general term event to refer to outcomes or combinations of outcomes. Many people confuse the Law of Large numbers with the socalled Law of Averages. Many people believe that an outcome of a random event that hasn t occurred in many trials is due to occur. The Law of Averages doesn t exist. Copyright 2015 Pearson Education. All rights reserved. 6-2 Copyright 2015 Pearson Education. All rights reserved Random Phenomena and Probability 6.3 Different Types of Probability The sample space is the collection of all possible outcomes. We denote the sample space S or sometimes Ω. The probability of an event is its long-run relative frequency. Independence means that the outcome of one trial doesn t influence or change the outcome of another. Model-Based (Theoretical) Probability The (theoretical) probability of event A can be computed with the following equation: # outcomes in A P( A) total # of outcomes Copyright 2015 Pearson Education. All rights reserved. 6-3 Copyright 2015 Pearson Education. All rights reserved

2 6.3 Different Types of Probability Model-Based (Theoretical) Probability Example: Pew Research reports that of 10,190 randomly generated working phone numbers, the initial results of the calls were as follows: Rule 2: The Probability Assignment Rule The probability of the set of all possible outcomes must be 1. P( S) 1 where S represents the set of all possible outcomes and is called the sample space. Since the phone numbers were generated randomly, each was equally likely, and the probability of a contact is simply 7400 / 10,190 = Copyright 2015 Pearson Education. All rights reserved. 6-7 Copyright 2015 Pearson Education. All rights reserved Different Types of Probability Personal Probability A subjective, or personal probability expresses your uncertainty about the outcome. Although personal probabilities may be based on experience, they are not based either on long-run relative frequencies or on equally likely events. Rule 3: The Complement Rule The probability of an event occurring is 1 minus the probability that it doesn t occur. C P( A) 1 P( A ) where the set of outcomes that are not in event A is called the complement of A, and is denoted A C. Copyright 2015 Pearson Education. All rights reserved. 6-8 Copyright 2015 Pearson Education. All rights reserved Rule 1 If the probability of an event occurring is 0, the event can t occur. For Example: Lee s Lights sell lighting fixtures. Lee records the behavior of 1000 customers entering the store during one week. Of those, 300 make purchases. What is the probability that a customer doesn t make a purchase? If the probability is 1, the event always occurs. For any event A, 0 P( A). 1 If P(Purchase) = 0.30 then P(no purchase) = = 0.70 Copyright 2015 Pearson Education. All rights reserved. 6-9 Copyright 2015 Pearson Education. All rights reserved

3 Rule 4: The Multiplication Rule For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. P( A and B) P( A) P( B) provided that A and B are independent. For Example: Some customers prefer to see the merchandise but then make their purchase online. Lee determines that there s an 8% chance of a customer making a purchase in this way. We know that about 30% of customers make purchases when they enter the store. What is the probability that a customer who enters the store makes no purchase at all? P(purchase in the store or online) = P (purchase in store) + P(purchase online) = = 0.39 P(no purchase) = = 0.61 Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved For Example: If we can assume that customers behave independently, what is the probability that the next two customers entering Lee s Lights both make purchases? We can use the multiplication rule because the events are independent: P( A and B) P( A) P( B) P(first customer makes a purchase and second customer makes a purchase) = P(purchase) x P(purchase) = 0.30 x 0.30 = 0.09 Rule 6: The General Addition Rule The General Addition Rule calculates the probability that either of two events occurs. It does not require that the events be disjoint. P( A or B) P( A) P( B) P( A and B) Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Rule 5: The Addition Rule Two events are disjoint (or mutually exclusive) if they have no outcomes in common. The Addition Rule allows us to add the probabilities of disjoint events to get the probability that either event occurs. where A and B are disjoint. P( A or B) P( A) P( B) For Example: Lee notices that when two customers enter the store together, their purchases are not disjoint. In fact, there s a 20% they ll both make a purchase. When two customers enter the store together, what is the probability that at least one of them will make a purchase? P(at least one purchases) = P(A purchases or B purchases) = P(A purchases) + P(B purchases) P(A and B both purchase) = = 0.40 Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved

4 Example: Car Inspections You and a friend get your cars inspected. The event of your car s passing inspection is independent of your friend s car. If 75% of cars pass inspection what is the probability that Your car passes inspection? Your car doesn t pass inspection? Both cars pass inspection? At least one of two cars passes? Neither car passes inspection? 6.5 Joint Probability and Contingency Tables Events may be placed in a contingency table such as the one in the example below. As part of a Pick Your Prize Promotion, a store invited customers to choose which of three prizes they d like to win. The responses could be placed in the following contingency table: Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Example (continued): Car Inspections You and a friend get your cars inspected. The event of your car s passing inspection is independent of your friend s car. If 75% of cars pass inspection what is the probability that 6.5 Joint Probability and Contingency Tables Marginal probability depends only on totals found in the margins of the table. Your car passes inspection? P(Pass) = 0.75 Your car doesn t pass inspection? P(Pass C ) = = 0.25 Both cars pass inspection? P(Pass)P(Pass) = (0.75)(0.75) = Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Example (continued): Car Inspections You and a friend get your cars inspected. The event of your car s passing inspection is independent of your friend s car. If 75% of cars pass inspection what is the probability that 6.5 Joint Probability and Contingency Tables In the table below, the probability that a respondent chosen at random is a woman is a marginal probability. At least one of two cars passes? 1 (0.25) 2 = OR = Neither car passes inspection? = P(woman) = 251/478 = Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved

5 6.5 Joint Probability and Contingency Tables Joint probabilities give the probability of two events occurring together. Rule 7: The General Multiplication Rule The General Multiplication Rule calculates the probability that both of two events occurs. It does not require that the events be independent. P( A and B) P( A) P( B A) P(woman and camera) = 91/478 = Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Each row or column shows a conditional distribution given one event. What is the probability that a randomly selected customer wants a bike if the customer selected is a woman? In the table above, the probability that a selected customer wants a bike given that we have selected a woman is: P(bike woman) = 30/251 = P( A and B) P( B A) P(bike woman) P( A) P(bike and woman) P(woman) Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved In general, when we want the probability of an event from a conditional distribution, we write P(B A) and pronounce it the probability of B given A. A probability that takes into account a given condition is called a conditional probability. Events A and B are independent whenever P(B A) = P(B). This means knowing event A has occurred has no impact on the probability of event B occurring. P( A and B) P( B A) P( A) Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved

6 Are Prize preference and Sex independent? If so, P(bike woman) will be the same as P(bike). P(bike woman)= 30/251 = 0.12 P(bike) = 90/478 = If you re given probabilities without a contingency table, you can often construct a simple table to correspond to the probabilities and use this table to find other probabilities. Since the two probabilities are not equal, Prize preference and Sex and not independent. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Independent vs. Disjoint For all practical purposes, disjoint events cannot be independent. Don t make the mistake of treating disjoint events as if they were independent and applying the Multiplication Rule for independent events. A survey classified homes into two price categories (Low and High). It also noted whether the houses had at least 2 bathrooms or not (True or False). 56% of the houses had at least 2 bathrooms, 62% of the houses were Low priced, and 22% of the houses were both. Translating the percentages to probabilities, we have: Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved The 0.56 and 0.62 are marginal probabilities, so they go in the margins. The 22% of houses that were both Low priced and had at least 2 bathrooms is a joint probability, so it belongs in the interior of the table. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved

7 Because the cells of the table show disjoint events, the probabilities always add to the marginal totals going across rows or down columns. Example (continued): Online Banking A national survey indicated that 30% of adults conduct their banking online. It also found that 40% are under the age of 50, and that 25% are under the age of 50 and conduct their banking online. Construct a contingency table showing joint and marginal probabilities. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Example: Online Banking A national survey indicated that 30% of adults conduct their banking online. It also found that 40% are under the age of 50, and that 25% are under the age of 50 and conduct their banking online. What percentage of adults do not conduct their banking online? What type of probability is the 25% mentioned above? Construct a contingency table showing joint and marginal probabilities. What is the probability that an individual who is under the age of 50 conducts banking online? Are Banking online and Age independent? Example (continued): Online Banking A national survey indicated that 30% of adults conduct their banking online. It also found that 40% are under the age of 50, and that 25% are under the age of 50 and conduct their banking online. What is the probability that an individual who is under the age of 50 conducts banking online? 0.25 / 0.40 = Are Banking online and Age independent? No. P(banking online under 50) = 0.625, which is not equal to P(banking online) = Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Example (continued): Online Banking A national survey indicated that 30% of adults conduct their banking online. It also found that 40% are under the age of 50, and that 25% are under the age of 50 and conduct their banking online. What percentage of adults do not conduct their banking online? 100% 30% = 70% What type of probability is the 25% mentioned above? Marginal Some business decisions involve more subtle evaluation of probabilities. Given the probabilities of various states of nature, we can use a picture called a probability tree or tree diagram to help think through the decision-making process. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved

8 Example: Personal electronic devices, such as smart phones and tablets, are getting more capable all the time. Microscopic and even submicroscopic flaws that can cause intermittent performance failures can develop during their manufacture. Defects will always occur, so the quality engineer in charge of the production process must monitor the number of defects and take action if the process seems out of control. Example: At the end of each branch, joint probabilities can be calculated by applying the multiplication rule for independent events. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved Example: The probability tree on the next slide shows that a manufacturing problem can occur with the motherboard, the memory, or the case alignment for an electronic device. The next set of branches denote the probability that a minor adjustment will fix each type of problem. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved *6.9 Reversing the Conditioning: Bayes Rule When you need to find reverse conditional probabilities, we recommend drawing a tree and finding the appropriate probabilities by using the definition of conditional probability (see text). Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved

9 *6.9 Reversing the Conditioning: Bayes Rule In general, we can write for n events A i that are mutually exclusive (each pair is disjoint) and exhaustive (their union is the whole sample space): What Have We Learned? Know the rules of probability and how to apply them. The Complement Rule says that P(A) = 1 P(A c ). The Multiplication Rule for independent events say that P(A and B) = P(A)*P(B), provided A and B are independent. The General Multiplication Rule says that P(A and B) = P(A)*P(B A). The Addition Rule for disjoint events says that P(A or B) = P(A) + P(B), provided events A and B are disjoint. The General Addition Rule says that P(A or B) = P(A) + P(B) P(A and B). Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved What Have We Learned? Beware of probabilities that don t add up to 1. Don t add probabilities of events if they re not disjoint. Don t multiply probabilities of events if they re not independent. Don t confuse disjoint and independent. Know how to construct and read a contingency table. Know how to define and use independence. Events A and B are independent if P(B A) = P(B). Know how to construct tree diagrams and use them to calculate probabilities. Know how to use Bayes Rule to compute conditional probabilities. Copyright 2015 Pearson Education. All rights reserved Copyright 2015 Pearson Education. All rights reserved What Have We Learned? Apply the facts about probability to determine whether an assignment of probabilities is legitimate. Probability is long-run relative frequency. Individual probabilities must be between 0 and 1. The sum of probabilities assigned to all outcomes must be 1. Understand the Law of Large Numbers and that the common understanding of the Law of Averages is false. Copyright 2015 Pearson Education. All rights reserved