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Math Help - Prove the probability statement

  1. #1
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    Prove the probability statement

    Let events A, B and C be independent. Show that (A union B) and C are independent.

    Attempt at solution:

    Setup: If [A intersection B intersection C] = [A][B][C], then show that [(A union B) intersection C] = [A union B][C].

    I'm not sure how to go about doing this. Would using conditional probability theorem be a good start?
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  2. #2
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by My Little Pony View Post
    Let events A, B and C be independent. Show that (A union B) and C are independent.

    Attempt at solution:

    Setup: If [A intersection B intersection C] = [A][b][C], then show that [(A union B) intersection C] = [A union B][C].

    I'm not sure how to go about doing this. Would using conditional probability theorem be a good start?
    Suppose events A, B, and C are independent. Then,

    P(A\cap\\C)=P(A)P(C), P(B\cap\\C)=P(B)P(C), and

    P(A\cap\\B\cap\\C)=P(A)P(B)P(C) .

    Now, (A\cup\\B)\cap\\C=(A\cap\\C)\cup\\(B\cap\\C), by the distributive law.

    So,

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

    =P(A\cap\\C)+P(B\cap\\C)-P((A\cap\\C)\cap\\(B\cap\\C))

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

    =P(C)[P(A)+P(B)-P(A)P(B)]=P(C)P(A\cup\\B)=P(A\cup\\B)P(C), as was to be shown.

    All that was needed to prove this was the distributive law, multiplicative law of probability, and the additive law of probability.
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