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lecture34

# lecture34 - Lecture 34 Bayes Theorem Probability Given...

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Lecture 34 Bayes’ Theorem

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Probability Given sample space S, we assign a number p(s) to each outcome s S such that p is a function from S to [0 1] and called a probability distribution. S s s p S s s p 1 ) ( each for 1 ) ( 0
Event probabilities Recall that an event E is a subset of S, the sample space. Probability of E is the sum of the probabilities of the outcomes in E E s s p E p ) ( ) (

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Combining probabilities ) ( ) ( ) ( ) ( ) ( 1 ) ( 2 1 2 1 2 1 E E p E p E p E E p E p E p
Conditional Probability E and F are two events from the same sample space p(E|F) = probability of E given F

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Conditional Probability Example We toss a coin thrice E1 = two heads E2 = first one is H P(E1) = ? P(E2) = ? P(E1|E2) = ? THH HHH TTH HTH THT HHT HTT TTT E1 E2
Conditional Probability ) ( ) ( ) | ( F p F E p F E p

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Independent Events E and F are independent if p(E|F) = p(E) i.e. knowing that F happened does not affect the probability of E’s happening Equivalently: p(E F) = p(E)p(F)
Section 5.3 Bayes’ Theorem and Random Variables

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lecture34 - Lecture 34 Bayes Theorem Probability Given...

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