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lecture2.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 2
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Recall: Chevalier de M´ er´ e Problem Figure: A die (or dice) A game of dice-throwing: Two dice. 24 throws. Both show Ace. de M´ er´ e lost money - he kept loosing money.
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The de M´ er´ e argument There is 1 / 6 chance that an Ace will turn up. By throwing a die 4 times there is more than 50% chance that there will be an Ace For the second die an Ace will turn up on average in 6 throws So 4 × 6 = 24 throws must give more than 50% chance of winning But he still lost more than he won
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Model of Probability A Sample Space Ω Probability Law: A Ω; P ( A ) Events Sets Modeling: More likely event to get more probability
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Axioms of Probability Nonnegativity: P ( A ) 0 Additivity: For any two disjoint sets A , B , P ( A B ) = P ( A ) + P ( B ) Holds for infinitely many disjoints events A 1 , A 2 , A 3 , . . . P ( i A i ) = X i P ( A i ) . Normalization: P (Ω) = 1
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That’s all we need Question: What is P ( )?
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