New statement of the Partition Theorem The Law of Total Probability) The Multiplication Rule gives us a new statement of the Partition Theorem (Total Probability Theorem) If B,...,Bm partition S,then for any event A, F(A)=SP(AB)=>P(A B:)F(B). Both formulations of the Partition Theorem are very widely used, but especially the conditional formulation ∑1P(AB)P(B). Examples of conditional probability and partitions Example 3.A news magazine publishes three columns entitled"Art"(A). "Books"(B),and"Cinema"(C).Reading habits of a randomly selected reader with respect to these columns are Read A BCAOBAOC BOC AOBOC regularly Probability0.140.230.370.080.090.130.05 We thus have P(4IB)-P(4B)_0.08 P90230348 P4BU0=P4n8UC》_004+005+003.0255 P(BUC) 0.47 P(Alreads atr least one)-P(ALAUBUC)-(A(BC) P(AUBUC) “n40G-85-02c P(A P4U81C=P4UnC》.004+0o5+008-0459 P(C) 0.37New statement of the Partition Theorem (The Law of Total Probability) The Multiplication Rule gives us a new statement of the Partition Theorem (Total Probability Theorem) Both formulations of the Partition Theorem are very widely used, but especially the conditional formulation Examples of conditional probability and partitions Example 3. A news magazine publishes three columns entitled “Art”(A), “Books”(B), and “Cinema”(C). Reading habits of a randomly selected reader with respect to these columns are Read regularly A B C A  B A  C B  C A  B  C Probability 0.14 0.23 0.37 0.08 0.09 0.13 0.05 We thus have 0.459 0.37 0.04 0.05 0.08 ( ) (( ) )) ( | ) 0.286 0.49 0.14 ( ) ( ) ( ) ( ( )) ( | ) ( | ) 0.255 0.47 0.04 0.05 0.03 ( ) ( ( )) ( | ) 0.348 0.23 0.08 ( ) ( ) ( | )                                    P C P A B C P A B C P A B C P A P A B C P A A B C P A reads at least one P A A B C P B C P A B C P A B C P B P A B P A B