Lect6 - Conditional Probability Multiplication Rule Independence of Events Independent versus Mutually Exclusive The Birthday Prob Outline

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Conditional Probability Multiplication Rule l e m Outline Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox 1 / 21 Xinghua Zheng Probability II
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Conditional Probability Multiplication Rule l e m Example, Stein’s class ± What is the probability for a randomly selected student from Professor Stein’s class this year to receive an A? In the last 5 years Professor Stein has awarded 190 A’s out of 1000 students. Can use long run relative frequency : the probability can be estimated as 190 / 1000 = 0 . 19 ± Amy studies 10 hours or more every week for professor Stein’s course. Amy is really interested in knowing “What’s the probability for a student who studies 10 hours or more to be awarded an A?" Conditional probability 2 / 21 Xinghua Zheng Probability II
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Conditional Probability Multiplication Rule l e m 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 by P ( A | B ) ± Mathematically, P ( A | B ) := P ( A B ) P ( B ) , Assuming P ( B ) > 0 3 / 21 Xinghua Zheng Probability II
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Conditional Probability Multiplication Rule l e m Interpretation P ( A | B ) = P ( A B ) / P ( B ) Geometrically, = the ratio of the area of the shaded part to that of the right disk P ( B | A ) = P ( A B ) / P ( A ) 4 / 21 Xinghua Zheng Probability II
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Conditional Probability Multiplication Rule l e m Interpretation, ctd The conditional probability P ( A | B ) : Restrict sample space to just event B P ( A | B ) measures the chance of event A occurring in this new sample space In other words, if B occurs, then what is the chance of A occurring 5 / 21 Xinghua Zheng Probability II
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Conditional Probability Multiplication Rule l e m Example, Stein’s class, ctd ± What’s the probability for a randomly selected student from Prof. Stein’s class who studies 10 hours or more to be awarded an A? Past data shows that 200 students worked 10 hours or more for Stein’s course, and 120 of them were awarded A. Define events A={to be awarded an A}, B = {studies 10 hours or more} To find the answer to our question, we need to find out 6 / 21 Xinghua Zheng Probability II
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Conditional Probability Multiplication Rule l e m Example, Stein’s class, ctd P ( A | B ) = P ( A B ) / P ( B ) A B ={studies 10 hours or more and is to be awarded A} By the long-run relative frequency, P ( A B ) = 120 / 1000 = 0 . 12, P ( B ) = 200 / 1000 = 0 . 2 Hence
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This note was uploaded on 12/28/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Fall '09 term at HKUST.

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Lect6 - Conditional Probability Multiplication Rule Independence of Events Independent versus Mutually Exclusive The Birthday Prob Outline

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