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# lec.04 - ENGG2430C Lecture 4 Total Probability Theorem...

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ENGG2430C Lecture 4: Total Probability Theorem, Bayes’ Law, and Counting Minghua Chen ([email protected]) Department of Information Engineering The Chinese University of Hong Kong Reading: Ch. 1.4 and 1.5 of the textbook M. Chen ENGG2430C lecture 4 1 / 19

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Review: Conditional Probability Given the outcome is in B with P ( B ) > 0, what is the chance it is in A ? P ( A | B ) = P ( A B ) P ( B ) . It defines a probability law on the new “universe" B I P ( B | B ) = 1 , P ( A c | B ) = 1 - P ( A | B ) I P ( A 1 A 2 | B ) = P ( A 1 | B )+ P ( A 2 | B ) , if A 1 and A 2 are disjoint in B Multiplication rule: P ( A B ) = P ( A ) P ( B | A ) P ( A B C ) = P ( A ) P ( B | A ) P ( C | A B ) M. Chen ENGG2430C lecture 4 2 / 19
Review: Independence of Two Events Definition: Events A and B are independent if P ( A B ) = P ( A ) · P ( B ) . If A and B are independent, then I P ( A | B ) = P ( A ) ( meaning: knowing B happens tells nothing about A ) I By symmetry, P ( B | A ) = P ( B ) . Definition: n Events A 1 , A 2 , ··· , A n are mutually independent if for any subset S ⊆ { 1 , 2 , ··· , n } P ( i S A i ) = Π i S P ( A i ) Remarks: I Disjoint events are NOT independent I Conditioning can change independence M. Chen ENGG2430C lecture 4 3 / 19

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Motivating Example: Divide and Conquer (1) Certain survey provides the following statistics of IE graduates I 10% of them join the bank industry, with an average monthly salary 40K HKD I 65% of them join the IT industry, with an average monthly salary 13K HKD I 25% of them continue graduate-school study, with an average monthly salary 15K HKD Question : what is average salary of IE graduates? 40 × 0 . 1 + 13 × 0 . 65 + 15 × 0 . 25 M. Chen ENGG2430C lecture 4 4 / 19
Motivating Example: Divide and Conquer (2) Certain survey provides the following statistics of IE graduates I 10% of them join the bank industry, with 50% chance to get a monthly salary at least 20K HKD I 65% of them join the IT industry, with 10% chance to get a monthly

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