Question

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

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