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Unformatted text preview: lecture 5, Probability II lecture 5, Probability II Outline • The Concept of Probability • Sample Spaces and Events • Some Elementary Probability Rules Conditional Probability and Independence lecture 5, Probability II lecture 5, Probability II Example, Conditional Probability • Professor Stein has awarded 156 A’s among 1000 students that he has taught. What’s the chance for a randomly selected student from Professor Stein’s class to obtain an A? • Amy works 10 hours or more every week for professor Stein’s course. Amy’s really interested in knowing “What’s the chance for a student who work 10 hours or more to be awarded an A?" • Conditional Probability lecture 5, Probability II lecture 5, Probability II Conditional Probability • The probability of an event A, given that the event B has occurred, is called the conditional probability of A given B • Denoted as P ( A  B ) • Further, P ( A  B ) = P ( A ∩ B ) / P ( B ) • Assuming P ( B ) 6 = lecture 5, Probability II lecture 5, Probability II Interpretation • Restrict sample space to just event B • The conditional probability P(AB) measures the chance of event A occurring in this new sample space • In other words, if B occurred, then what is the chance of A occurring lecture 5, Probability II lecture 5, Probability II Interpretation, ctd • P ( A  B ) = P ( A ∩ B ) / P ( B ) • P ( B  A ) = P ( A ∩ B ) / P ( A ) lecture 5, Probability II lecture 5, Probability II General Multiplication Rule • We have P ( A  B ) = P ( A ∩ B ) / P ( B ) • Multiplying both sides by P ( B ) , we have P ( A ∩ B ) = P ( B ) P ( A  B ) • Similarly, P ( B  A ) = P ( A ∩ B ) / P ( A ) ⇒ P ( A ∩ B ) = P ( A ) P ( B  A ) • General multiplication rule: P ( A ∩ B ) = P ( B ) P ( A  B ) = P ( A ) P ( B  A ) lecture 5, Probability II lecture 5, Probability II Independence of Events • Two events A and B are said to be independent if and only if P ( A  B ) = P ( A ) • This is equivalent to P ( B  A ) = P ( B ) . • We assume that P ( A ) > 0 and P ( B ) > 0. lecture 5, Probability II lecture 5, Probability II Multiplication rule for Independent events • If A and B are independent events, then P ( A ∩ B ) = P ( A ) P ( B ) • If A 1 , A 2 , · · · , A N are independent events, then P ( A 1 ∩ A 2 ∩ · · · A N ) = P ( A 1 ) P ( A 2 ) · · · P ( A N ) lecture 5, Probability II lecture 5, Probability II Example, Newspaper Subscribers • Define events: • A = event that a randomly selected household subscribes to the Atlantic Journal • B = event that a randomly selected household subscribes to the Beacon News • Given: • total number in city, N = 1,000,000 • number subscribing to A, N ( A ) = 650,000 • number subscribing to B, N ( B ) = 500,000 • number subscribing to both, N ( A ∩ B ) = 250,000 lecture 5, Probability II lecture 5, Probability II Contingency Tables Events B ¯ B Total A 250,000 ? 650,000 ¯ A ? ? ?...
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 Fall '09
 AnthonyChan

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