Stat 491 Chapter 3--Probability

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Unformatted text preview: 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Multiplication and Addition Laws Multiplication Law: For mutually independent events A1 , A2 , . . . , Am , P(A1 A2 Am ) = P(A1 ) P(A2 ) P(Am ). Addition Law: For any two events A and B, P(A B) = P(A) + P(B) - P(A B). Use venn diagram. For the STD example, suppose a patient will be referred for further lab test if at least one of the doctors makes a positive diagnosis. Then P(A+ B + ) = P(A+ )+P(B + )-P(A+ B + ) = 0.1+0.17-0.08 = .19 Thus 19% of all patients will be referred for further lab tests. Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Addition Law for Independent Events If A and B are independent events then P(A B) = P(A) + P(B)[1 - P(A)]. Example: Let A = {Wife's DBP > 95} and B = {Husband's DBP > 95} where DBP stands for diastolic blood pressure. Assume that P(A) = 0.1 and P(B) = 0.2. Then the probability of a hypertensive household is P(AB) = P(A)+P(B)[1-P(A)] = 0.1+0.2[1-0.1] = 0.28. Therefore, 28% of all households will be hypertensive. Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Addition Law for More Than Two Events For any events A, B and C , P(A B C ) = P(A) + P(B) + P(C ) - P(A B) - P(A C ) - P(B C ) + P(A B C ) The addition law also generalizes to an arbitrarily number of events, although that is beyond the scope of this course. Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Conditional Probability Let A and B be two events with P(B) > 0. Then the conditional probability of event A given event B, written as...
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This note was uploaded on 10/30/2013 for the course STAT 491 taught by Professor Solomonharrar during the Fall '12 term at Montana.

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