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How to proof P(A U B U C) without using Venn Diagram

  1. Oct 4, 2008 #1
    Do you know how to proof

    P(A U B U C) = P(A) + P(B) + P(C) - P(A^B) - P(B^C) - P(C^A) + P(A^B^C)


    ^ is intersection.

    Do you know how to find P(A U B U C U D)

    Thank you very much.
     
  2. jcsd
  3. Oct 4, 2008 #2

    statdad

    User Avatar
    Homework Helper

    Let

    [tex]
    D = B \cup C
    [/tex]

    and note that

    [tex]
    A \cup B \cup C = A \cup D
    [/tex]

    then

    [tex]
    \begin{align*}
    \Pr(A \cup B \cup C) & = \Pr(A \cup D)\\
    & = \Pr(A) + \Pr(D) - \Pr(A \cap D) \\
    & = \Pr(A) + \Pr(B \cup C) - \Pr(A \cap D)\\
    & = \Pr(A) + \Pr(B) + \Pr(C) - \Pr(B \cap C) - \Pr(A \cap D)
    \end{align*}
    [/tex]

    The rest of the proof comes from realizing that

    [tex]
    \Pr(A \cap D) = \Pr(A \cap \left(B \cup C\right)) = \Pr((A \cap B) \cup (A \cap C)),
    [/tex]

    using the Addition Rule for probability to expand the final term, and being very careful with positive and negative signs.
     
  4. Oct 4, 2008 #3
    Thank you so much Statdad. I would like to ask another question.

    How to proof P(A U B) = P(A) + P(B) - P(A ^ B) ?

    Thank you again.
     
  5. Oct 4, 2008 #4

    statdad

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    Homework Helper

    This proof isn't needed for the problem you posted above - is there a reason you need it here?
     
  6. Oct 4, 2008 #5
    Sorry. I'm just curious. :)
     
  7. Oct 4, 2008 #6

    statdad

    User Avatar
    Homework Helper

    No - I was interrupted by someone at the door.
    Here is one method - there are others.
    First, note that

    [tex]
    A \cup B = (A-B) \cup (A \cap B) \cup (B - A)
    [/tex]

    and the three sets on the right are pair-wise disjoint. Now

    [tex]
    \begin{align*}
    \Pr(A \cup B) & = \Pr(A-B) + \Pr(A \cap B) + \Pr(B - A)\\
    & = \left(\Pr(A-B) + \Pr(A \cap B) \right) + \left(\Pr(B-A) + \Pr(A \cap B)\right) - \Pr(A \cap B) \\
    & = \Pr(A) + \Pr(B) - \Pr(A \cap B)
    \end{align*}
    [/tex]

    Again, sorry for the abrupt end to my previous post - I'm getting really tired of our election season.
     
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