1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability of unions/intersections

  1. Mar 14, 2008 #1
    Is it true that

    Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
    = Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )

    where A_1, A_2, ... is any sequence of sets.

    Well, for the (k=m+1) case I am convinced since I can see they are equal after expanding both sides out, so for example I can see that
    Pr((A∩(B^c))∪(B∩(C^c))) = Pr(A∪B) - Pr(B∩C)

    but I can't manage to do the same for the (k>m) case in general, so overall I'm not convinced.
  2. jcsd
  3. Mar 15, 2008 #2
    I'm sorry I can't make much sense out of the formula, but assuming its correct, try induction. Assume the proposition is true for k <= m, and add one more term to it and use the truth of the m previous propositions to prove it for (m+1). Will require some grouping and basic properties of sets and cardinalities under union and intersections.
  4. Mar 15, 2008 #3
    That (induction) is exactly what I've been attempting to use to convince myself that it is true. I've got the base step (k=m+1) which was (for me) expand-able to see that both sides are equal.
    However I couldn't get through the inductive step. Perhaps it is false then? Or it could also just mean that I got totally lost within the messy algebra?
  5. Mar 15, 2008 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    You have a "special symbol" right at the beginning of that formula that will not show up on my (or Maverick280857's) browser.
  6. Mar 15, 2008 #5
    Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
    = Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )


    the probability of [ the union (where n goes from m to k) of [ A_n intersect (A_(n+1) compliment) ] ]

    is equal to

    the probability of [ the union (where n goes from m to k) of A_n ]
    the probability of [ A_k intersect A_(k+1) ]
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Probability of unions/intersections
  1. Unions and Intersections (Replies: 15)