Probability Conditional

# Probability Conditional - Conditional Probability P A B P B...

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1 Conditional Probability [] [|] PA B PAB PB = A B S

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2 Example. Toss 2 dice and take the sum. : odd toss, {3, 5, 7, 9, 11} : {2,3} [ ] [{3}] 2 / 36 2 [|] [ ] [{2,3}] 1/ 36 2 / 36 3 A B PA B P PAB PB P = = = = + Homework. A couple has two kids and at least one is a boy. What is the probability that both are boys? Ans. 1/3.
3 Bayes Rule [|] [] PA B PB A PA PB = Notice that [ ] [|][] [|][] PA B PAB PB PB A PA = =

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4 Partitioning of the Sample Space The events 12 , ,, K AA A partition the sample space S iff ij A A for all i and j i j φ = and 1 K j j AS = = . Notice that in this case, 1 1 [ ][ ] 1 K K jj j j P APAP S = = = = = 1 A 2 A 3 A
5 Conditional Probability after Partitioning We want to find [ ]. PB Suppose the event B depends on some other events . j A Find the events that partition the sample space S . 123 1 2 3 , , are disjoint and AAA A A A S =  . 1 23 Note that , , are disjoint BA  . 11 2 2 3 3 Therefore [ ] [ ][ ] [ | ][ ] [ | ][ ] [ | ][ ] PB A PB A PA = ++ =  1 A 2 A 3 A B

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6 Example. Defect A store gets ½ of its items from Factory 1, ¼ from Factory 2, and ¼ from Factory 3.
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Probability Conditional - Conditional Probability P A B P B...

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