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Math Help - [SOLVED] Help Needed to Prove Probability Theorem

  1. #1
    Rhymes with Orange Chris L T521's Avatar
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    [SOLVED] Help Needed to Prove Probability Theorem

    I'm doing fine in all the other probability stuff I'm doing, but I got stuck in proving the following theorem:

    If A, B and C are any three events, then

    P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)
    Source: Probability and Statistical Inference, 7e, by Hogg and Tanis

    Here is my work thus far:

    P(A\cup B\cup C)=P(A\cup(B\cup C))

    This implies that
    P(A\cup(B\cup C))=P(A)+P(B\cup C)-P(A\cap(B\cup C))

    Now,
    P(B\cup C)=P(B)+P(C)-P(B\cap C)

    So, substituting this into the equation I get:

    P(A\cup(B\cup C))=P(A)+P(B)+P(C)-P(B\cap C)-{\color{red}P(A\cap(B\cup C))}

    I have no clue how to break up the part in red. Instead of working it out for me, give me a hint or two to help me get started.

    Thanks in advance for any help!!

    --Chris
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  2. #2
    MHF Contributor

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    It is easy. Take what you have in red.
     P\left( {A \cap \left[ {B \cup C} \right]} \right) = P\left( {\left[ {A \cap B} \right] \cup \left[ {A \cap C} \right]} \right) = P\left( {\left[ {A \cap B} \right]} \right) + P\left( {\left[ {A \cap C} \right]} \right) - P(A \cap B \cap C)
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Plato View Post
    It is easy. Take what you have in red.
     P\left( {A \cap \left[ {B \cup C} \right]} \right) = P\left( {\left[ {A \cap B} \right] \cup \left[ {A \cap C} \right]} \right) = P\left( {\left[ {A \cap B} \right]} \right) + P\left( {\left[ {A \cap C} \right]} \right) - P(A \cap B \cap C)
    Ahh...I didn't apply the intersection properly. Thank you very much. Its a lot clearer now!

    --Chris
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