Re: Daily question

Posted by Khushboo on Mar 02, 2009; 6:46am
URL: http://discussion-forum.276.s1.nabble.com/Daily-question-tp2328964p2407774.html

Show that P(A ∩ B ∩ C) = P(A).P(B).P(C) does not necessarily imply that A, B and C are all pairwise independent.


It is prooved by an example:
Suppose 8 tickets numbered 111, 121, 122, 122, 211, 212, 212, 221 are placed in a hat nd stirred. One of them is then drawn at random. The events are:
 
   A : the 1st digit on the ticket drawn will be 1,
   B : the 2nd digit on the ticket drawn will be 1,  
   C : the 3rd digit on the ticket drawn will be 1.

then, A = 111, 121, 122, 122
       P(A) = 4/8

       B = 111, 211, 212, 212
      P(B) = 4/8

      C = 111, 121, 211, 221
      P(C) = 4/8

     (A∩B∩C) = 111
     P(A∩B∩C) = 1/8

So,  P(A∩B∩C) = P(A).P(B).P(C) = 1/8

P(A ∩ B ∩ C) = P(A).P(B).P(C) does not necessarily imply that A, B are pairwise independent.

 As,
     P(A∩B) is not equal to P(A).P(B)  [ 1/8 is not equal to 4/8 . 4/8 ]