Lecture6 - lecture 6 Probability II 1/21 Conditional...

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Unformatted text preview: lecture 6, Probability II 1/21 Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox lecture 6, Probability II Outline 1 Conditional Probability 2 Multiplication Rule 3 Independence of Events 4 “Independent” versus “Mutually Exclusive” 5 The Birthday Problem 6 Simpson’s Paradox lecture 6, Probability II 2/21 Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox lecture 6, Probability II Conditional Probability 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 = . 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 lecture 6, Probability II 3/21 Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox lecture 6, Probability II Conditional Probability 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 ) > lecture 6, Probability II 4/21 Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox lecture 6, Probability II Conditional Probability 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 ) lecture 6, Probability II 5/21 Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox lecture 6, Probability II Conditional Probability 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 lecture 6, Probability II 6/21 Conditional Probability Multiplication Rule Independence of Events “Independent” versus “Mutually Exclusive” The Birthday Problem Simpson’s Paradox lecture 6, Probability II Conditional Probability 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?...
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This note was uploaded on 12/28/2010 for the course BUSINESS A ISOM 111 taught by Professor Yingyingli during the Fall '10 term at HKUST.

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Lecture6 - lecture 6 Probability II 1/21 Conditional...

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