Stat 491 Chapter 3--Probability

Then a b x 75 chapter 3 probability stat 491

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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 The Intersection of Events The intersection of events A and B written as A B is defined as A B = the event that both A and B occur = the collection of all outcomes in the sample space that are both in A and B Using a venn diagram For the hypertension example,let A = {X 90} and B = {75 X 100} then P(A B) = {90 X 100} If A and B are mutually exclusive, then A B = and P(A B) = 0. Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests The Intersection of Events Cont'd ... Two events A and B are said to be independent if P(A B) = P(A) P(B). Otherwise, A and B are said to be dependent. Example: Let A = {Wife's DBP > 95}, B = {Husband's DBP > 95} and C = {first-born child's DBP > 80} where DBP stands for diastolic blood pressure. Assume that P(A) = 0.1 and P(B) = 0.2. What can we say about P(A B)? If we are willing to assume that the wife's hypertensive status does not depend on the husband's hypertensive status then P(A B) = 0.1 0.2 = 0.02. If P(C ) = .2 and P(B C ) = .05, are B and C independent? Chapter 3: Probability Stat Is the result unexpected? Explain.491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests The Intersection of Events: Example Suppose two doctors, A and B, test all patients coming into a clinic for syphilis. A+ = {Dr. A makes a positive diagnosis} B + = {Dr. B makes a positive diagnosis} Suppose P(A+ ) = 0.1 , P(B + ) = 0.17 and P(A+ B + ) = 0.08. Are the events A+ and B + independent? Now P(A+ B + ) = 0.08 > P(A+ ) P(B + ) = 0.017. Thus the two events are dependent. This result is NOT unexpected. Why? Chapter...
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