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Stat 491 Chapter 3--Probability

# Stat 491 Chapter 3--Probability - Probability and Inference...

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Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes’ Rule and Screening Tests Stat 491: Biostatistics Chapter 3: Probability Solomon W. Harrar The University of Montana Fall 2012 Chapter 3: Probability Stat 491: Biostatistics

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Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes’ Rule and Screening Tests Example: Verifying a Claim A manufacturer of pregnancy test kit claims that the accuracy (true positive rate) of their kit is over 75%. We conducted a clinical trial and out of 100 pregnant women, 77 tested positive. Can we have faith on the sample evidence? There is enough evidence to back your claim if Probability (Evidence given that the Claim is FALSE) = Small , say less than 0 . 05. This is one instance where probability and probability models come in handy. Chapter 3: Probability Stat 491: Biostatistics
Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes’ Rule and Screening Tests Definitions Sample Space: the set of all possible outcomes of a trial. Event (A,B,C, . . . ): any set of outcomes of interest. The probability of an event is the relative frequency of occurrence of this set of outcomes over an indefinitely large (or infinite) number of trials. That is, P ( E ) = n E n where n E the number of outcomes in favor of E in n (large) repetitions of the trial. Chapter 3: Probability Stat 491: Biostatistics

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Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes’ Rule and Screening Tests Interpretation of Probability Suppose it is known that The probability of developing a breast cancer over 30 years in 40-year-old women who have never had cancer is 1/11. This probability means over a large sample of 40-year-old women who have never had breast cancer, approximately 1 in 11 will develop the disease by age 70. This proportion becomes increasingly close to 1 in 11 as the number of women sampled increases. Chapter 3: Probability Stat 491: Biostatistics
Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes’ Rule and Screening Tests Properties of Probability Prop. 1 For any event A 0 P ( A ) 1 . Prop. 2 If A and B are two events that can not happen at the same time then P ( A or B occurs) = P ( A ) + P ( B ) . Definition Two events A and B are said to be mutually exclusive if they can not happen at the same time. Chapter 3: Probability Stat 491: Biostatistics

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Probability and Inference Definitions and Properties Event Relations Laws of Probability Conditional Probability Bayes’ Rule and Screening Tests Example: Properties of Probability Let X be diastolic blood pressure of a person. Let A = { X < 90 } and B = { 90 X < 95 } . Suppose P ( A ) = 0 . 7 and P ( B ) = 0 . 1. Let C = X < 95. Then, P ( C ) = P ( A or B ) = P ( A ) + P ( B ) = 0 . 7 + 0 . 1 = 0 . 8 .
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