Probability Lecture 3 - Purdue University
ConditionalProbabilityNotesLecture 3Conditional ProbabilityText: A Course in Probability by Weiss 4.1STAT 225 Introduction to Probability ModelsJanuary 20, 2014ConditionalProbabilityGeneralMultiplication RuleWhitney HuangPurdue itionalProbability1Conditional Probability2General Multiplication RuleGeneralMultiplication Rule3.2
Motivating ExampleConditionalProbabilityNotesIn a certain population, the probability a person lives tobe 80 is 80% while the probability a person lives to be 90is 68%. Given that a person lives to be 80, what is theprobability that she/he will live to be 90?ConditionalProbabilityGeneralMultiplication Rule3.3Conditional ProbabilityConditionalProbabilityNotesLet A and B be events. The probability that event Aoccurs given (knowing) that event B occurs is called aconditional probability. It is denoted as P(A B). Theformula of conditional probability isP(A B)P(A B) P(B)ConditionalProbabilityGeneralMultiplication RuleThe above formula works so long as P(B) 0. Underthe equally likely framework the formula above can bewritten as#(A B)P(A B) #(B)3.4
ConditionalProbabilityMotivating ExampleNotesIn a certain population, the probability a person lives tobe 80 is 80% while the probability a person lives to be 90is 68%. Given that a person lives to be 80, what is theprobability that she/he will live to be ation RuleEvent A: a person lives to be 90Event B: a person lives to be 80P(A B)P(B) P(a person lives to be 80 AND a person lives to be 90)P(a person lives to be 80)P(a person lives to be 90)0.68P(a person lives to be 80) 0.80 0.85P(A B) 3.5Venn Diagram Illustration of Conditional robabilityGeneralMultiplication Rule3.6
General Multiplication RuleConditionalProbabilityNotesSuppose we know the conditional probability P(A B) andthe marginal probability i.e. the probability of the givenevent P(B). Then the formula of conditional probabilityprovides a way to compute the joint probability P(A B)ConditionalProbability2 events:P(A B) P(B) P(A B)GeneralMultiplication RuleMore than 2 events:P( ni 1 Ai ) P(A1 ) P(A2 A1 ) P(A3 A1 A2 ) · · · P(An An 1 · · · A1 )3.7Example 11A Morgan Stanley Consumer Research Survey sampledmen and women and asked each whether they preferredto drink plain bottled water or a sports drink such asGatorade or Propel Fitness water (The AtlantaJournal-Constitution, December 28, 2005). Suppose 200men and 200 women participated in the study, and 280reported they preferred plain bottled water. Of the grouppreferring a sports drink, 80 were men and 40 werewomen. tyGeneralMultiplication RuleM : the event the consumer is a manW : the event the consumer is a womanB : the event the consumer preferred plain bottledwaterS : the event the consumer preferred a sports drink3.8
ConditionalProbabilityExample 11 (cont’d)NotesAnswer the following:1What is the probability a person in the studypreferred plain bottled water?2What is the probability a person in the studypreferred a sports drink?ConditionalProbability3What are the conditional probabilities P(M S) andP(W S)?4What are the joint probabilities P(M S) andP(W S)?5Given a consumer is a man, what is the probabilityhe will prefer a sports drink?GeneralMultiplication Rule3.9ConditionalProbabilityExample 11Solution.2804001204001P(B) 2P(S) 3P(M S) 4040012040045 0.7ConditionalProbability 0.3P(M S)P(S) 80400120400 23 , P(W S) P(W S)P(S) GeneralMultiplication Rule13P(M S) P(S) P(M S) 0.3 23 0.2, P(W S) P(S) P(W S) 0.3 P(S M) P(S M)P(M) 8040020040013 0.1 0.43.10Notes
ConditionalProbabilityExample 12 (Example 10 revisit)NotesUsing the Venn Diagram summarizing the distribution ofoperating systems previously described, calculate thefollowing:1The probability that a randomly chosen student usesall three operating systems, given the student usesWindows2The probability that a randomly chosen student usesall three operating systems, given the student doesnot use Windows3The probability that a randomly chosen student usesWindows, given the student uses Mac OS4The probability that a randomly chosen student doesnot use any of the operating systems, given thestudent does not use WindowsConditionalProbabilityGeneralMultiplication Rule3.11ConditionalProbabilityExample 12 Venn DiagramLinuxConditionalProbabilityMac OSGeneralMultiplication Rule94392217Windows43.12Notes
ConditionalProbabilityExample 12NotesSolution.1P(W M L W ) 9503050P((W M L) W )P(W ) P(W M L)P(W ) ConditionalProbability 0.3P((W M L) W c )2P(W M L W c ) 3P(W M) 4P((W M L)c W c ) P((W M L)c )P(W c )P(W M)P(M) 1 46502050P(W c ) 11501850 P( )P(W c ) 0502050GeneralMultiplication Rule 01118P((W M L)c W c )P(W c ) 0.23.13SummaryConditionalProbabilityIn this lecture, we learnedConditional probability: definition, formula, venndiagram representationGeneral multiplication ruleConditionalProbabilityGeneralMultiplication Rule3.14Notes
Jan 20, 2014 · 1 46 50 20 50 0:2 Conditional Probability Conditional Probability General Multiplication Rule 3.14 Summary In this lecture, we learned Conditional probability:definition, formula, venn diagram representation General multiplication rule Notes Notes. Title: Conditional Probability - Text: A Course in
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
Joint Probability P(A\B) or P(A;B) { Probability of Aand B. Marginal (Unconditional) Probability P( A) { Probability of . Conditional Probability P (Aj B) A;B) P ) { Probability of A, given that Boccurred. Conditional Probability is Probability P(AjB) is a probability function for any xed B. Any
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Pros and cons Option A: - 80% probability of cure - 2% probability of serious adverse event . Option B: - 90% probability of cure - 5% probability of serious adverse event . Option C: - 98% probability of cure - 1% probability of treatment-related death - 1% probability of minor adverse event . 5
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Introduction In this unit we shall try to know about Aristotle and his life and works and also understand about the relationship between Criticism and Creativity. We shall see how criticism is valued like creative writings. We shall know the role and place given to 'the critic' in the field of literary criticism.
