Question

Determine whether each statement is true or false. If the statement is false, give an example...

Determine whether each statement is true or false. If the statement is false, give an example showing that it is false in general. a) If events A_1, A_2, A_3 partition a sample space, then the events A_1 and A_2 are independent. b) If events A_1, A_2, A_3 partition a sample space, then the events A_1, A_2 are disjoint. c) If the events A_1, A_2, A_3 are pairwise independent events, then A_1, A_2, A_3 are mutually independent events. d) For any two events A and B if P(A)>0 and P(B)>0 and A and B are independent, then A and B cannot be disjoint. e) For any two disjoint events A and B, P(A)+P(B)=1

Homework Answers

Answer #1

a) FALSE. For Example a and c

Mutual independence: Every event is independent of any intersection of the other events.

Pairwise independence: Any two events are independent.

A,B,CA,B,C are mutually independent if

P(A∩B∩C)=P(A)P(B)P(C)P(A∩B∩C)=P(A)P(B)P(C)

P(A∩B)=P(A)P(B)P(A∩B)=P(A)P(B)

P(A∩C)=P(A)P(C)P(A∩C)=P(A)P(C)

P(B∩C)=P(B)P(C)P(B∩C)=P(B)P(C)

On the other hand, A,B,CA,B,C are pairwise independent if

P(A∩B)=P(A)P(B)P(A∩B)=P(A)P(B)

P(A∩C)=P(A)P(C)P(A∩C)=P(A)P(C)

P(B∩C)=P(B)P(C)

b) TRUE

c)FALSE

d) TRUE

e) False

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