Stat 491 Chapter 3--Probability

Then the conditional probability of event a given

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Unformatted text preview: P(A|B), is defined as P(A|B) = P(A B) P(B) and the conditional probability of event B given event A is defined in similar way assuming that P(A) > 0. Using a venn diagram. If A and B are independent then P(B|A) = P(B) = P(B|A). Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Relative Risk The relative risk (RR) of B given A is RR = Obviously, 0 RR < . Clearly, if A and B are independent then RR = 1. The more the dependence between events increase, the further the RR will be from 1. P(B|A) . P(B|A) Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Relative Risk: Example Suppose that 1500 smokers in 10,000 develop lung cancer in 20 years and 50 non-smokers in 5000 developed lung cancer in 20 years. What is the relative risk of lung cancer in 20 years given a person smokes? Let A = {Smoker} and B = {Develop lung cancer}. Then RR = P(B|A) 0.15 = 0.01 = 15. P(B|A) Therefore, smokers are 15 times more likely to develop lung cancer in 20 years than nonsmokers. Chapter 3: Probability Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Total Probability Rule Let A and B be any two events. Clearly, P(B) = P(B A) + P(B A). The above relation implies P(B) = P(B|A) P(A) + P(B|A) P(A) which is known as the Total Probability Rule. Generalization of the total probability rule: Let A1 , A2 , , Am be a set of mutually exclusive and exhaustive events. Then, m P(B) = i=1 Chapter 3: Probability P(B|Ai ) P(Ai ). Stat 491: Biostatistics Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes' Rule and Screening Tests Total Probability Rule: Example Suppose the rate of type II diabetes mellitus (DM) in 40- to 59-year old is 7% among Caucasians, 10% among African Americans, 12% among Hispanics and 5% among Asian Americans. Suppose the ethnic distribution in Houston, TX is 30% Caucasian, 25% African American, 40% Hispanic...
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