CondProbInd - Outline Conditional Probability and...

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Unformatted text preview: Outline Conditional Probability and Independence Definition and Calculus of Probability Michael Akritas Michael Akritas Definition and Calculus of Probability Outline Conditional Probability and Independence Conditional Probability and Independence Independence The Law of Total Probability Bayes Theorem Michael Akritas Definition and Calculus of Probability Outline Conditional Probability and Independence Independence The Law of Total Probability Bayes Theorem I In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. I For now, the word prediction will mean update the probabilities of events regarding the other variable. I The updated probabilities are called conditional probabilities . For example, 1. Knowing a mans height helps update the probability that he weighs over 170lb. 2. Knowing which assembly line a product came from, helps update the probability that it has a particular defect. 3. Knowing a persons education level helps update the probability of that person being in a certain income category. Michael Akritas Definition and Calculus of Probability Outline Conditional Probability and Independence Independence The Law of Total Probability Bayes Theorem Given partial information about the outcome of the experiment, results in a reduction of either the sample space or the number of eligible units. For example: Example I If the outcome of rolling a die is known to be even, what is the probability it is a 2? I If the selected card from a deck is known to be a figure card, what is the probability it is a king? I Given event A = { household subscribes to paper 1 } , what is the probability of B = { household subscribes to paper 2 } ? U U P(A B) U U U U P(A B ) P(A B) A B .1 .5 .3 Michael Akritas Definition and Calculus of Probability Outline Conditional Probability and Independence Independence The Law of Total Probability Bayes Theorem Definition The conditional probability of the event A given the information that event B has occurred is denoted by P ( A | B ) and equals P ( A | B ) = P ( A B ) P ( B ) , provided P ( B ) > Note that P ( B | B ) = 1, which highlights the fact that when we are given the information that B occurred, B becomes the new sample space. Proposition The set function P ( | B ) satisfies the three axioms of probability. Michael Akritas Definition and Calculus of Probability Outline Conditional Probability and Independence Independence The Law of Total Probability Bayes Theorem Example In the card game of bridge, the 52 cards are dealt out equally to 4 players called East, West, North and South. Given that North and South have a total of 8 spades among them, what is the probability that East has 3 of the remaining 5 spades? Use the reduced sample space defined by the given information....
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CondProbInd - Outline Conditional Probability and...

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