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included content:Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive Events 5. Conditional Probability 6. The Multiplicative Rule and Independent Events 7. Bayes’s Rule, chapter_3.pdf Unformatted Attachment Preview Chapter 3 Probability ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 1 Introduction to Probability Read An Informal Introduction to Probability. You can view a pre-recorded lecture on the above reading. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 2 Contents 1. 2. 3. 4. Events, Sample Spaces, and Probability Unions and Intersections Complementary Events The Additive Rule and Mutually Exclusive Events 5. Conditional Probability 6. The Multiplicative Rule and Independent Events 7. Bayes’s Rule ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 3 Learning Objectives 1. Develop probability as a measure of uncertainty 2. Introduce basic rules for finding probabilities 3. Use probability as a measure of reliability for an inference 4. Provide an advanced rule for finding probabilities ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 4 Thinking Challenge What’s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). So toss a coin twice. Do it! Did you get one head & one tail? What’s it all mean? ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 5 Many Repetitions!* Total Heads Number of Tosses 1.00 0.75 0.50 0.25 0.00 0 25 50 75 100 125 Number of Tosses ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 6 3.1 Events, Sample Spaces, and Probability ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 7 Experiments & Sample Spaces 1. Experiment • Process of observation that leads to a single outcome that cannot be predicted with certainty 2. Sample point • Most basic outcome of an experiment 3. Sample space (S) • Collection of all sample points ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 8 Visualizing Sample Space 1. Listing S = {Head, Tail} 2. Venn Diagram H T S ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 9 Sample Space Examples Experiment Sample Space Toss a Coin, Note Face Toss 2 Coins, Note Faces Select 1 Card, Note Kind Select 1 Card, Note Color Play a Football Game Inspect a Part, Note Quality Observe Gender ALWAYS LEARNING {Head, Tail} {HH, HT, TH, TT} {2♥, 2♠, ..., A♦} (52) {Red, Black} {Win, Lose, Tie} {Defective, Good} {Male, Female} Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 10 Events 1. Specific collection of sample points 2. Simple Event Contains only one sample point 3. Compound Event Contains two or more sample points ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 11 Venn Diagram Experiment: Toss 2 Coins. Note Faces. Sample Space S = {HH, HT, TH, TT} TH Outcome HH Compound Event: At least one Tail HT TT S ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 12 Event Examples Experiment: Toss 2 Coins. Note Faces. Sample Space: Event 1 Head & 1 Tail Head on 1st Coin At Least 1 Head Heads on Both ALWAYS LEARNING HH, HT, TH, TT Outcomes in Event HT, TH HH, HT HH, HT, TH HH Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 13 Probabilities ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 14 What is Probability? 1. Numerical measure of the likelihood that event will occur P(Event) P(A) 2. Lies between 0 & 1 1 Certain 0.5 3. Sum over sample points is 1 0 ALWAYS LEARNING Impossible Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 15 Steps for Calculating Probability 1. Define the experiment; describe the process used to make an observation and the type of observation that will be recorded. 2. List the sample points. 3. Assign probabilities to the sample points. 4. Determine the collection of sample points contained in the event of interest. 5. Sum the sample points probabilities to get the event probability. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 16 Combinations Rule A sample of n elements is to be drawn from a set of N elements. Then, the number of different samples possible is denoted by  N  and is equal to  n   N N!  n   n!N  n ! where the factorial symbol (!) means that n!  n  n  1 n  2  ... 3 2 1 For example, 5!  5  4  3 2 1 0! is defined to be 1. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 17 Example Suppose you plan to invest equal amounts of money in each of five business ventures. If you have 20 ventures from which to make the selection, how many different samples of five ventures can be selected from the 20? For this example, N = 20 and n = 5. Then the number of different samples of 5 that can be selected from the 20 ventures is  20  20! 20!  5   5!(20  5)!  5!15!   20  19  18...3  2  1 20  19  18  17  16    15,504 (5  4  3  2  1)(15  14  13...3  2  1) 5  4  3  2 1 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 18 More on Combinations For further discussion, review Properties of Combinations You can find a pre-recorded lecture on Properties of Combinations here. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 19 Suggested Exercises Work out the following exercises from the Textbook : 3.1, 3.2, 3.3, 3.4, 3.6, 3.7, 3.11, 3.19, 3.23, 3.26, 3.29 These exercises will not be collected or graded, but let me know as questions arise. Also, view this example. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 20 3.2 Unions and Intersections. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 21 Compound Events Compound events: Composition of two or more other events. Can be formed in two different ways. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 22 Unions & Intersections 1. Union Outcomes in either events A or B or both ‘OR’ statement Denoted by  symbol (i.e., A  B) 2. Intersection Outcomes in both events A and B ‘AND’ statement Denoted by  symbol (i.e., A  B) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 23 Event Union: Venn Diagram Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space: 2, 2, 2, ..., A Ace Event Ace: A, A, A, A ALWAYS LEARNING Black S Event Black: 2, 2, ..., A Event Ace  Black: A, ..., A, 2, ..., K Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 24 Event Union: Two–Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Simple Sample Space Type (S): Ace 2, 2, 2, ..., A Non-Ace Total Event Ace  Black: A,..., A, 2, ..., K ALWAYS LEARNING Total Ace & Ace & Ace Red Black Non & Non & NonRed Black Ace Red Black S Red Black Event Ace: A, A, A, A Simple Event Black: 2, ..., A Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 25 Event Intersection: Venn Diagram Experiment: Draw 1 Card. Note Kind, Color & Suit. Sample Space: 2, 2, 2, ..., A Ace Event Ace: A, A, A, A ALWAYS LEARNING Black Event Black: 2,...,A S Event Ace  Black: A, A Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 26 Event Intersection: Two–Way Table Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Sample Space Type (S): Ace 2, 2, 2, ..., A Non-Ace Event Ace  Black: A, A Total ALWAYS LEARNING Total Ace & Ace & Ace Red Black Non & Non & NonRed Black Ace Red Black S Red Black Simple Event Ace: A, A, A, A Simple Event Black: 2, ..., A Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 27 More on Union and Intersection Review this example to see how the union and intersection of two events is constructed in a real life situation. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 28 Compound Event Probability 1. Numerical measure of likelihood that compound event will occur 2. Can often use two–way table Two variables only ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 29 Event Probability Using Two–Way Table Event Event B1 B2 Total A1 P(A 1  B1) P(A 1  B2) P(A 1) A2 P(A 2  B1) P(A 2  B2) P(A 2) Total P(B 1) Joint Probability ALWAYS LEARNING P(B 2) 1 Marginal (Simple) Probability Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 30 Two–Way Table Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Ace 2/52 2/52 Total 4/52 Non-Ace 24/52 24/52 48/52 Total 26/52 26/52 52/52 P(Red) ALWAYS LEARNING P(Ace) P(Ace  Red) Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 31 Thinking Challenge What’s the Probability? 1. P(A) = 2. P(D) = 3. P(C  B) = Event C D 4 2 4. P(A  D) = Event A 5. P(B  D) = B 1 3 4 Total 5 5 10 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Total 6 Slide - 32 Solution* The Probabilities Are: 1. P(A) = 6/10 2. P(D) = 5/10 3. P(C  B) = 1/10 Event C D 4 2 4. P(A  D) = 9/10 Event A 5. P(B  D) = 3/10 B 1 3 4 Total 5 5 10 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Total 6 Slide - 33 3.3 Complementary Events ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 34 Complementary Events Complement of Event A The event that A does not occur All sample points not in A Denote complement of A by AC AC A S ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 35 Rule of Complements The sum of the probabilities of complementary events equals 1: P(A) + P(AC) = 1 AC A S ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 36 Complement of Event Example Experiment: Draw 1 Card. Note Color. Black Sample Space: 2, 2, 2, ..., A Event Black: 2, 2, ..., A ALWAYS LEARNING S Complement of Event Black, BlackC: 2, 2, ..., A, A Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 37 De Morgan’s Rule Given two events A and B, the intersection of the complement of A and the complement of B equals the complement of the union of A and B. Here is an example illustrating this rule. This note explains why the rule works; here is a pre-recorded lecture on the note. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 38 3.4 The Additive Rule and Mutually Exclusive Events ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 39 Mutually Exclusive Events Mutually Exclusive Events Events do not occur simultaneously A  B does not contain any sample points ALWAYS LEARNING  Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 40 Mutually Exclusive Events Example Experiment: Draw 1 Card. Note Kind & Suit. Sample Space: 2, 2, 2, ..., A   Event Spade: 2, 3, 4, ..., A ALWAYS LEARNING S Outcomes in Event Heart: 2, 3, 4 , ..., A Events  and are Mutually Exclusive Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 41 Additive Rule 1. Used to get compound probabilities for union of events 2. P(A OR B) = P(A  B) = P(A) + P(B) – P(A  B) 3. For mutually exclusive events: P(A OR B) = P(A  B) = P(A) + P(B) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 42 Additive Rule Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Ace Red Black 2 2 Total 4 Non-Ace 24 24 48 Total 26 26 52 P(Ace  Black) = P(Ace) + P(Black) – P(Ace  Black) 4 26 2 28 = + – = 52 52 52 52 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 43 Thinking Challenge Using the additive rule, what is the probability? 1. P(A  D) = 2. P(B  C) = ALWAYS LEARNING Event A Event C D 4 2 Total 6 B 1 3 4 Total 5 5 10 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 44 Solution* Using the additive rule, the probabilities are: 1. P(A  D) = P(A) + P(D) – P(A  D) 6 5 2 9 = + – = 10 10 10 10 2. P(B  C) = P(B) + P(C) – P(B  C) 4 5 1 8 = + – = 10 10 10 10 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 45 Approximation P(A  B) = P(A) + P(B) – P(A  B) This is an equation connecting 4 quantities. As long as you know any three of them, you can use this equation to find the fourth. However, in practice, you often know only P(A) and P(B). Obviously, in such a situation you cannot calculate either P(A  B) or P(A  B), but you can approximate both of them. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 46 Suggested Exercises Work out the following exercises from the Textbook : 3.30, 3.32, 3.33, 3.39, 3.43, 3.46, 3.49 These exercises will not be collected or graded, but let me know as questions arise. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 47 Solved Examples Review The Country Club Problem. Review The Sprinkler Problem. Review The Election Problem. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 48 3.5 Conditional Probability ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 49 Conditional Probability 1. Event probability given that another event occurred 2. Revise original sample space to account for new information Eliminates certain outcomes 3. P(A | B) = P(A and B) = P(A  B P(B) P(B) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 50 Conditional Probability Using Venn Diagram Ace Black S Black ‘Happens’: Eliminates All Other Outcomes Black (S) Event (Ace  Black) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 51 Conditional Probability Using Two–Way Table Experiment: Draw 1 Card. Note Kind & Color. Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Revised Sample Space P(Ace  Black) 2 / 52 2 P(Ace | Black) =   P(Black) 26 / 52 26 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 52 Thinking Challenge Using the table then the formula, what’s the probability? 1. P(A|D) = 2. P(C|B) = ALWAYS LEARNING Event A Event C D 4 2 Total 6 B 1 3 4 Total 5 5 10 Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 53 Solution* Using the formula, the probabilities are: 𝑃(𝐴 ∩ 𝐷) 2/10 2 𝑃 𝐴𝐷 = = = 𝑃(𝐷) 5/10 5 P C  B  110 1 P C B     4 P B  4 10 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 54 3.6 The Multiplicative Rule and Independent Events ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 55 Multiplicative Rule 1. Used to get compound probabilities for intersection of events 2. P(A and B) = P(A  B) = P(A)  P(B|A) = P(B)  P(A|B) 3. For Independent Events: P(A and B) = P(A  B) = P(A)  P(B) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 56 Multiplicative Rule Example Experiment: Draw 1 Card. Note Kind & Color. Color Type Ace Red Black 2 2 Total 4 Non-Ace 24 24 48 Total 26 26 52 P(Ace  Black) = P(Ace)∙P(Black | Ace)  4  2  2       52  4  52 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 57 Statistical Independence 1. Event occurrence does not affect probability of another event Toss 1 coin twice 2. Causality not implied 3. Tests for independence P(A | B) = P(A) P(B | A) = P(B) P(A  B) = P(A)  P(B) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 58 Thinking Challenge Using the multiplicative rule, what’s the probability? 1. P(C  B) = Event C D 4 2 2. P(B  D) = Event A 3. P(A  B) = B 1 3 4 Total 5 5 10 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Total 6 Slide - 59 Solution* Using the multiplicative rule, the probabilities are: 5 1 1 P C  B   P C  P B C     10 5 10 4 3 6 P B  D   P B  P D B     10 5 25 P A  B   P A  P B A  0 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 60 More on Independence Read this note on Independence; here is a pre-recorded lecture on the note. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 61 Tree Diagram Experiment: Select 2 pens from 20 pens: 14 blue & 6 red. Don’t replace. Dependent! 5/19 6/20 P(R  R)=(6/20)(5/19) =3/38 B R P(R  B)=(6/20)(14/19) =21/95 B P(B  B)=(14/20)(13/19) =91/190 R 14/19 14/20 R 6/19 P(B  R)=(14/20)(6/19) =21/95 B 13/19 ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 62 Suggested Exercises Work out the following exercises from the Textbook: 3.53, 3.55, 3.57, 3.60, 3.69, 3.46, 3.49, 3.80 These exercises will not be collected or graded, but let me know as questions arise. Review The Class Composition Problem Review The Avon Sale Problem ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 63 3.7 Bayes’s Rule ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 64 Bayes’s Rule Given k mutually exclusive and exhaustive events B1, B1, . . . Bk , such that P(B1) + P(B2) + … + P(Bk) = 1, and an observed event A, then P(Bi  A) P(Bi | A)  P( A) P(Bi )P( A | Bi )  P(B1 )P( A | B1 )  P(B2 )P( A | B2 )  ...  P(Bk )P( A | Bk ) ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 65 Bayes’s Rule Example Bayes’s Rule is hard to implement as stated, but it is an extremely useful tool, as illustrated by the The Rare Disease Problem; here is a pre-recorded lecture on this problem. The best way to learn how to use this rule is to understand how it works using examples. To that end: Review The MP3 Player Problem Review The LAN Shutdown Problem Review The Mayoral Election Problem ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 66 Suggested Exercises Work out the following exercises from the Textbook: 3.81, 3.88, 3.93 These exercises will not be collected or graded, but let me know as questions arise. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 67 Key Ideas Probability Rules for k Sample Points, S1, S2, S3, . . . , Sk 1. 0 ≤ P(Si) ≤ 1 2.  P S   1 i ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 68 Key Ideas Random Sample All possible such samples have equal probability of being selected. ALWAYS LEARNING Copyright © 2018, 2014, and 2011 Pearson Education, Inc. Slide - 69 Key Ideas Combinations Rule Counting number of samples of n elements selected from N elements N  N  1 N  2  N N!  n   n! N  n !  n  n  1 n  2      ALWAYS LEARNING  N  n  1  2 1 Copyright © 2018, 2014, and 2011 Pearso ... Purchase answer to see full attachment