Copyright 2014, 2013, 2010 and 2007 Pearson Education, Inc. Chapter Probability Copyright 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section Probability Rules 5.1 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Apply the rules of probabilities 2.Compute and interpret probabilities using the empirical method 3.Compute and interpret probabilities using the classical method 4.Use simulation to obtain data based on probabilities 5.Recognize and interpret subjective probabilities 5-3 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. 5-4 Probability deals with experiments that yield random short-term results or outcomes, yet reveal long-term predictability. The long-term proportion in which a certain outcome is observed is the probability of that outcome. Copyright 2013, 2010 and 2007 Pearson Education, Inc. The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. 5-5 Copyright 2013, 2010 and 2007 Pearson Education, Inc. In probability, an experiment is any process that can be repeated in which the results are uncertain. The sample space, S, of a probability experiment is the collection of all possible outcomes. An event is any collection of outcomes from a probability experiment. An event may consist of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, e i. In general, events are denoted using capital letters such as E. 5-6 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Consider the probability experiment of having two children. (a)Identify the outcomes of the probability experiment. (b) Determine the sample space. (c) Define the event E = have one boy. EXAMPLEIdentifying Events and the Sample Space of a Probability Experiment e 1 = boy, boy, e 2 = boy, girl, e 3 = girl, boy, e 4 = girl, girl 5-7 {(boy, boy), (boy, girl), (girl, boy), (girl, girl)} {(boy, girl), (girl, boy)} Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Apply Rules of Probabilities 5-8 A probability model lists the possible outcomes of a probability experiment and each outcomes probability. A probability model must satisfy rules 1 and 2 of the rules of probabilities. 0 P(E) 1 P(e 1 ) + P(e 2 ) + + P(e n ) = 1 Copyright 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE A Probability Model In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model. ColorProbability Brown0.12 Yellow0.15 Red0.12 Blue0.23 Orange0.23 Green0.15 All probabilities are between 0 and 1, inclusive. Because = 1, (the sum of all probabilities must equal 1) is satisfied. 5-9 Copyright 2013, 2010 and 2007 Pearson Education, Inc. If an event is a certainty, the probability of the event is 1. If an event is impossible, the probability of the event is 0. An unusual event is an event that has a low probability of occurring. 5-10 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Compute and Interpret Probabilities Using the Empirical Method 5-11 The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment. P(E) relative frequency of E The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment. P(E) relative frequency of E Copyright 2013, 2010 and 2007 Pearson Education, Inc. Pass the Pigs TM is a Milton-Bradley game in which pigs are used as dice. Points are earned based on the way the pig lands. There are six possible outcomes when one pig is tossed. A class of 52 students rolled pigs 3,939 times. The number of times each outcome occurred is recorded in the table at right EXAMPLE Building a Probability Model OutcomeFreq Side with no dot1344 Side with dot1294 Razorback767 Trotter365 Snouter137 Leaning Jowler (a)Use the results of the experiment to build a probability model for the way the pig lands. (b)Estimate the probability that a thrown pig lands on the side with dot. (c)Would it be unusual to throw a Leaning Jowler? Prob yes Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 Compute and Interpret Probabilities Using the Classical Method 5-13 The classical method of computing probabilities requires equally likely outcomes. An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring. Copyright 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Computing Probabilities Using the Classical Method Suppose a fun size bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected. (a)What is the probability that it is yellow? (b) What is the probability that it is blue? 5-14 P(yellow) = 6/30 = 1/5 P(blue) = 2/30 = 1/15 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 4 Use Simulation to Obtain Data Based on Probabilities 5-15 Use the probability applet to simulate throwing a 6-sided die 100 times. Approximate the probability of rolling a 4. How does this compare to the classical probability? Repeat the exercise for 1000 throws of the die. EXAMPLEUsing Simulation Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 5 Recognize and Interpret Subjective Probabilities 5-16 The subjective probability of an outcome is a probability obtained on the basis of personal judgment. For example, an economist predicting there is a 20% chance of recession next year would be a subjective probability. Copyright 2013, 2010 and 2007 Pearson Education, Inc. In his fall 1998 article in Chance Magazine, (A Statistician Reads the Sports Pages, pp ,) Hal Stern investigated the probabilities that a particular horse will win a race. He reports that these probabilities are based on the amount of money bet on each horse. When a probability is given that a particular horse will win a race, is this empirical, classical, or subjective probability? EXAMPLEEmpirical, Classical, or Subjective Probability Subjective because it is based upon peoples feelings about which horse will win the race. The probability is not based on a probability experiment or counting equally likely outcomes. 5-17 Copyright 2013, 2010 and 2007 Pearson Education, Inc. 5.1 Summary 1.Apply the rules of probabilities 2.Compute and interpret probabilities using the empirical method (based on past data values) 3.Compute and interpret probabilities using the classical method 4.Use simulation to obtain data based on probabilities 5.Recognize and interpret subjective probabilities 5-18 Copyright 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section The Addition Rule and Complements 5.2 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Use the Addition Rule for Disjoint Events 2.Use the General Addition Rule 3.Compute the probability of an event using the Complement Rule 5-20 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Use the Addition Rule for Disjoint Events 5-21 Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. Addition Rule for Disjoint Events If E and F are disjoint (or mutually exclusive) events, then Addition Rule for Disjoint Events If E and F are disjoint (or mutually exclusive) events, then Copyright 2013, 2010 and 2007 Pearson Education, Inc. We often draw pictures of events using Venn diagrams. The rectangle represents the sample space, and each circle represents an event. For example, suppose we randomly select a chip from a bag where each chip in the bag is labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Let E represent the event choose a number less than or equal to 2. Let F represent the event choose a number greater than or equal to 8. These events are disjoint as shown in the figure P(E) = P(F) = P(E or F) = 3 / 10 2 / 10 = 1 / 5 P(E) + P(F) = 5 / 10 = 1 / 2 Copyright 2013, 2010 and 2007 Pearson Education, Inc. The Addition Rule for Disjoint Events can be extended to more than two disjoint events. In general, if E, F, G,... each have no outcomes in common (they are pairwise disjoint), then 5-23 Copyright 2013, 2010 and 2007 Pearson Education, Inc. The probability model to the right shows the distribution of the number of rooms in housing units in the United States. Number of Rooms in Housing Unit Probability One0.010 Two0.032 Three0.093 Four0.176 Five0.219 Six0.189 Seven0.122 Eight0.079 Nine or more0.080 EXAMPLE The Addition Rule for Disjoint Events (a) Verify that this is a probability model. All probabilities are between 0 and 1, inclusive = Copyright 2013, 2010 and 2007 Pearson Education, Inc. Number of Rooms in Housing Unit Probability One0.010 Two0.032 Three0.093 Four0.176 Five0.219 Six0.189 Seven0.122 Eight0.079 Nine or more0.080 EXAMPLE The Addition Rule for Disjoint Events 5-25 (b) What is the probability a randomly selected housing unit has two or three rooms? P(two or three) = P(two) + P(three) = = 0.125 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Number of Rooms in Housing Unit Probability One0.010 Two0.032 Three0.093 Four0.176 Five0.219 Six0.189 Seven0.122 Eight0.079 Nine or more0.080 EXAMPLE The Addition Rule for Disjoint Events 5-26 (c) What is the probability a randomly selected housing unit has one or two or three rooms? P(one or two or three) = P(one) + P(two) + P(three) = = 0.135 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Use the General Addition Rule 5-27 The General Addition Rule For any two events E and F, The General Addition Rule For any two events E and F, Copyright 2013, 2010 and 2007 Pearson Education, Inc. Suppose that a pair of dice are thrown. Let E = the first die is a two Let F = the sum 0, the probability of event A i for i = 1, 2, , n given the event E, is Bayess Rule Let A 1, A 2, A 3, , A n be a partition of a sample space S. Then, for any event E for which P(E) > 0, the probability of event A i for i = 1, 2, , n given the event E, is 5-113 Copyright 2013, 2010 and 2007 Pearson Education, Inc. The business majors example from before Business Not Business Business Not Business Female Male Total = EXAMPLE Bayess Rule 5-114 Copyright 2013, 2010 and 2007 Pearson Education, Inc. If we chose a random business major, what is the probability that this student is female? A 1 = Female student A 2 = Male student E = business major We want to find P(A 1 | E), the probability that the student is female (A 1 ) given that this is a business major (E) EXAMPLE Bayess Rule 5-115 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Do it in a straight way first We know that 8.25% of the students are female business majors We know that 9% of the students are male business majors Choosing a business major at random is choosing one of the 17.25% The probability that this student is female is 8.25% / 17.25% = 47.83% EXAMPLE Bayess Rule (continued) 5-116 Copyright 2013, 2010 and 2007 Pearson Education, Inc. Now do it using Bayess Rule its the same calculation Bayess Rule for a partition into two sets (n = 2) P(A 1 ) =.55, P(A 2 ) =.45 P(E | A 1 ) =.15, P(E | A 2 ) =.20 We know all of the numbers we need EXAMPLE Bayess Rule (continued) 5-117 Copyright 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Bayess Rule (continued) 5-118