lecture3.pdf - COMPSCI 240 Reasoning Under Uncertainty Arya...

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COMPSCI 240: Reasoning Under Uncertainty Arya Mazumdar University of Massachusetts at Amherst Fall 2016
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Lecture 3
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Think really hard: who you are? You study all night and wake up 15 minutes before your exam. You could take a 10 minute bus ride but you have to catch a bus first: each minute, a bus to school passes your house with probability 1/5. Your other choice is to bike to the University from home, but it will take you 20 minutes. Should you wait for the bus or hop on your bike?
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Sequential Model for Conditional Probabilities Three coin tosses. The first toss is unbiased. Whatever the first toss is, the second toss is biased towards that 60-40. Whatever the second toss is the third toss is biased towards that 70-30.
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Sequential Model for Conditional Probabilities First Toss Third Toss Second Toss Root H T HH HT TH TT HHH HHT HTH HTT THH THT TTH TTT 0.5 0.5 0.6 0.4 0.6 0.4 0.7 0.3 0.3 0.3 0.3 0.7 0.7 0.7
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Recall from conditional probability P ( A | B ) = P ( A B ) P ( B ) Therefore, P ( A B ) = P ( B ) P ( A | B ) P ( A B C ) = P ( C ) P ( A B | C ) = P ( C ) P ( B | C ) P ( A | B C )
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Multiplication Rule Shorthand: P ( n i =1 A i ) P ( A 1 A 2 . . . A n ) The Rule: P ( n i =1 A i ) = P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 2 ) . . . P ( A n | ∩ n - 1 i =1 A i )
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Sequential Model Probability of a leaf of this tree? First Toss Third Toss Second Toss Root H T HH HT TH TT HHH HHT HTH HTT THH THT TTH TTT 0.5 0.5 0.6 0.4 0.6 0.4 0.7 0.3 0.3 0.3 0.3 0.7 0.7 0.7
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Small exercise Example: Throw of two dice. Each of the 36 outcomes are equally likely A = max of two dice is at most 4 B = min of the two dice is at least 2 What is P ( A | B )?
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