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Probability Proof Help

  1. Mar 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Let A, B and C be any three events. Show that

    i) P(A) = P(B) if and only if P(A U Bc) = P(Ac U B)

    ii) Given P(A) = 0.5 and P(A U (Bc ∩ Cc)c = 0.8
    determine P(Ac ∩ (B U C))

    2. Relevant equations
    the probability axioms?

    3. The attempt at a solution

    i) not sure where or how to start

    ii) P(A U (Bc∩Cc)c) = P(A U B U C) = 0.8

    then, P(Ac ∩ (B U C)) = P(B U C) - P(A) = 0.8 - 0.5 = 0.3

    I think I'm wrong... though I'm not sure...
     
    Last edited by a moderator: Mar 16, 2013
  2. jcsd
  3. Mar 16, 2013 #2

    tiny-tim

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    hi a little lost! :smile:

    for (i), try taking the complement of the second equation :wink:

    (and (ii) looks fine :smile:)
     
  4. Mar 18, 2013 #3
    @tiny_tim: oops, i just realised i wrote the iff "P(A U Bc) = P(Ac U B)" wrong
    it should have been the complement as you said ^^"
    -have been staring at this question for the past few days wondering what i should do next...

    so, would i then substitute
    P(A ∩ Bc)= P(A) - P(A ∩ B)
    and likewise for P(Ac ∩ B) ?

    if so, how does event C appear in/affect the proof?
     
  5. Mar 18, 2013 #4

    tiny-tim

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    hi a little lost! :smile:
    yes

    A = A ∩ (the whole space) = A ∩ (B U Bc) = (A ∩ B) U (A ∩ Bc) :wink:
    i'm confused …

    are we talking about question i) or ii) ? :confused:
     
  6. Mar 18, 2013 #5
    i mean i) i just assumed since it was stated as an event it may have to appear in the proof for part i)
     
  7. Mar 18, 2013 #6

    tiny-tim

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    then no, it's not mentioned in i), so you needn't bother with it until ii) :smile:
     
  8. Mar 18, 2013 #7
    ok thank-you very much for the help :D
     
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