Re: Daily question

Posted by varun on Feb 27, 2009; 10:21am
URL: http://discussion-forum.276.s1.nabble.com/Daily-question-tp2328964p2394843.html

2- Given three events A, B and C such that P(A ∩ B ∩ C) ≠ 0, and P(C|A ∩ B) = P(C|B), show that P(A|B ∩ C) = P(A|B)

  since  P(C|A ∩ B) = P(C|B)
                     P(C∩A∩B) / P(A∩B) = P(C∩B) / P(B)
                     P(A∩B∩C) = P(A∩B) P(B∩C) / P(B)      .....(1)
   therefore,
                P(A|B ∩ C) = P(A∩B∩C) / P(B∩C)
                               = P(A∩B) P(B∩C) / P(B)  /  P(B∩C) .......(from 1)
                               = P(A∩B) / P(B)
                               = P(A|B)