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Unit 1: Probability Wenyaw Chan Division of Biostatistics School of Public Health University of Texas - Health Science Center at Houston

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Definition of Probability Three ways of defining Probability: 1.Objective Probability 2.Deductive Logic Definition of Probability 3.Subjective Definition of Probability

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Definition of Probability Objective Probability –If E is an event in an experiment, the experiment is repeated a very large number of times, say N, and the event E is observed in n of these N trials then Prob(E) =n/N. Deductive Logic Definition of Probability –The probability of an event is determined logically from symmetry or geometric considerations associated with the experiment. Subjective Definition of Probability –The probability of an event is determined subjectively, reflecting a person’s “ degree of belief” that an event will occur!

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Examples 1. Toss a coin times. Pr(Head)= Throw a dart on the circular Target Pr(region A)=area(A)/total area 3. What is the probability that John will pass this course? Who is answering this question? John, Professor or his friend.

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Properties of probability 1.0 <= P(E) <= 1 2.P(A or B occurs)=P(A) +P(B), if A and B can not happen at the same time Mutually Exclusive –Two events are mutually exclusive, if A B =0 –A 1 = {1,2}, A 2 = {3,4} and A 3 = {5,6} are mutually exclusive

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Properties of probability II 3. P(A B)=P(A) + P(B) - P(A B) –Two events A and B are independent, then P(A B) = P(A) P(B) 4. If events A and B are independent, then P(A B)=P(A) + P(B) - P(A) P(B) =P(A) + P(B) [1- P(A)]

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Conditional Probability Conditional probability of B given A: P(B A)= P(A B)/P(A). –If A and B are independent, then P(B A)= P(B)= P(B A). –If A and B are not independent, then P(B A) P(B) P(B A). Toss two dices: what is the probability that sum=6, given that sum=even? (5/36)/(1/2)

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Exhaustive Events A set of events A 1, A 2, A 3 A k is exhaustive if at least one of the events must occur. i.e. A 1 A 2 A 3 A k = sample space Toss a die A 1 = {1,2,3}, A 2 = {1,3,4} and A 3 = {2,5,6} are exhaustive

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Law of Total Probability P(B)= P(B Ai) P(Ai), if A1 A2 A3 Ak are mutually exclusive and exhaustive Toss a blue die and a red die: Pr(red= even)= Pr(red= even| sum=2) +Pr(red= even| sum=3) + Pr(red= even| sum=4) Pr(red= even| sum=12)

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Bayes’ Rule Let A and B be two events, then P(B|A) =[P(A|B)P(B)]/[ P(A|B) P(B) + P(A| B) P( B)] Toss a blue die and a red die: Pr(sum= even| red=2) = Pr(red=2|sum= even)Pr(sum= even ) {Pr(red=2|sum= even)Pr(sum= even )+ Pr(red=2|sum= odd)Pr(sum= odd)}

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Population and Samples Random Sample –is a selection of some members of the population such that each member is independently chosen and has a known nonzero probability of being selected. Simple Random Sample –is a random sample in which each group member has the same probability of being selected. Cluster Sampling –involves selecting a random sample of clusters and then looking at all study units within the chosen clusters. –(one-stage) –In two-stage sampling, a random sample of clusters is selected and then, within each cluster, a random sample of study units is selected.

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