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Unformatted text preview: PROBABILITY THEORY Dr. V.R. Bencivenga Economic Statistics Economics 329 PROBABILITY THEORY PART 2 Outline Probability rules (1) Addition rule, rule of complements (2) Conditional probability (3) Multiplication rule, statistical independence (4) Bayes rule Bivariate probabilities Counting rules Objectives: Derive the rules of probability Understand how to identify which rule(s) apply when computing probabilities of events in a variety of situations involving randomness. Practice lots of problems! PART 3 2 PROBABILITY THEORY PROBABILITY RULES 3 PROBABILITY THEORY Example #5 Causes of recession A newspaper polls economists about the causes of the 20089 recession. The economists were asked about two possible causes, and they could name either or both. 80% identified the credit crunch that developed in the aftermath of the financial crisis. 60% identified the loss of housing and financial wealth due to the financial crisis. 50% identified both. What is the probability that a randomlychosen economist will not identify either of these causes? 4 PROBABILITY THEORY C = credit crunch W = loss of housing and financial wealth DeMorgans law: ) W C ( P ) W C ( P Rule of complements: ) W C ( P 1 ) W C ( P Addition rule: P(C W) = P(C) + P(W) P(C W) W C W C DeMorgans law: ) W C ( P ) W C ( P Rule of complements: ) W C ( P 1 ) W C ( P 5 PROBABILITY THEORY C = credit crunch W = loss of housing and financial wealth DeMorgans law: ) W C ( P ) W C ( P Rule of complements: ) W C ( P 1 ) W C ( P Addition rule: P(C W) = P(C) + P(W) P(C W) C W C W DeMorgans law: ) W C ( P ) W C ( P Rule of complements: ) W C ( P 1 ) W C ( P Addition rule: P(C W) = P(C) + P(W) P(C W) 6 PROBABILITY THEORY (1) Addition rule Addition rule for two events A and B: ) B A ( P ) B ( P ) A ( P ) B A ( P Addition rule for mutually exclusive events : ) B ( P ) A ( P ) B A ( P The probability of the intersection is subtracted to correct for double counting to make sure basic outcomes in the intersection are counted exactly once. The intersection of A and B is empty, and so its probability is zero! 7 PROBABILITY THEORY A and A are mutually exclusive and collectively exhaustive. ) A ( P ) A ( P ) A A ( P apply the addition rule for mutually exclusive events 1 ) A A ( P use the definition of collectively exhaustive Therefore Rule of complements: ) A ( P 1 ) A ( P 8 PROBABILITY THEORY Example #5 Causes of recession A newspaper polls economists about the causes of the 20089 recession. The economists were asked about three possible causes, and they could name none, any, or all....
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 Spring '12
 BENCIVENGA
 Economics

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