8)AnB=everything outside An B.An B did not occur,soanB did occur. 1)Yes 2)No 3)No 4)Yes 5)No 6)Yes 7)No 8)Yes Question:What is the event ?=0 Chanllenge:Can you express B using only a sign? Answer:A0B=(B). Limitations of Venn diagrams Venn diagrams are generally useful for up to 3 events,although they are not used to provide formal proofs.For more than 3 events,the diagram might not be able to represent all possible overlaps of events.(This was probably the case for our transport Venn diagram.) Example: (a)AUBUC (b)AnBnc 6125
6 / 25 did occur. 1)Yes 2) No 3)No 4) Yes 5) No 6) Yes 7) No 8) Yes. Chanllenge: Can you express A∩ B using only a ∪ sign? Answer: A∩ B = (A∪ B). Venn diagrams are generally useful for up to 3 events, although they are not used to provide formal proofs. For more than 3 events, the diagram might not be able to represent all possible overlaps of events. (This was probably the case for our transport Venn diagram.)
Properties of union,intersection,and complement The following properties hold. )万=2and=0. (ii)For any event A, AU万=2， and An万=0 (iii)For any events A and B. AUB BUA, and A∩B=BnA. (iv)(a)(AUB)=AnB. (b)AnB可=AUB () 2 Distributive laws: For any sets A,B,and C: AU(BnC)=(AUB)n(AUC). and An(BUC)=(AnB)U(AnC). More generally,for several events A and B1,B2,...,Bn., 4u(a）-4u (a)-gunm 7125
7 / 25 Properties of union, intersection, and complement The following properties hold. Distributive laws: For any sets A, B, and C:
Definition:Two events A and B are mutually exclusive,or disjoint,if AnB=0. This means events A and B cannot happen together.If A happens, it excludes B from happening,and vice-versa Note:Does this mean that A and B are independent? No,quite the opposite.A EXCLUDES B from happening,so B depends strongly on whether or not A happens. Definition:Any number of events 4.4..4 are mutually exclusive if every pair of the events is mutually exclusive:ie. AinA=0 for all i.j withi j. Definition:A partition of the sample space is a collection of mutually exclusive events whose union is That is,sets B,B....B form a partition of if BinBj 0 for all i,j withij, and B:B UB2U...UBk Q. =1 Examples: 8/25
8 / 25 Definition: Two events A and B are mutually exclusive, or disjoint, if This means events A and B cannot happen together. If A happens, it excludes B from happening, and vice-versa. Note: Does this mean that A and B are independent? No, quite the opposite. A EXCLUDES B from happening, so B depends strongly on whether or not A happens. Definition: Any number of events A A An , , , 1 2 are mutually exclusive if every pair of the events is mutually exclusive: ie. Definition: A partition of the sample space Ω is a collection of mutually exclusive events whose union is Ω
B.B2,Bs.B form a partition of: B....B partition B Important:B and B partition for any event B Partitioning an event A If B....B form a partition of,then (AB)....(AB) form a partition of A. 2 B B Bs We will see that this is very useful forfinding the probability of event A. This is because it is often easier to find the probability of small 'chunks'of A (the partitioned sections)than tdind the whole probability of A at once.The partition idea shows us how to add the probabilities of these chunks together:see later 9125
9 / 25 . form a partition of A. We will see that this is very useful for finding the probability of event A. This is because it is often easier to find the probability of small ‘chunks’ of A (the partitioned sections) than to find the whole probability of A at once. The partition idea shows us how to add the probabilities of these chunks together: see later