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Lect5_L6_L7_handout1

Lect5_L6_L7_handout1 - lecture 5 Probability II lecture 5...

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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
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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
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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 ) = 0
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lecture 5, Probability II lecture 5, Probability II Interpretation Restrict sample space to just event B The conditional probability P(A|B) 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
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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 )
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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 )
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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.
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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 )
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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
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lecture 5, Probability II lecture 5, Probability II Contingency Tables Events B ¯ B Total A 250,000 ? 650,000 ¯ A ? ? ?
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