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Thread: Proof of product rule for sucessive joint events

  1. #1
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    Proof of product rule for sucessive joint events

    Hi there,

    Does anyone know how to prove the following...

    P(E1nE2nE3nE4)=P(E1)P(E2\E1)P(E3\E2nE1)P(E4\E3nE2n E1)

    apparently it can be proved by induction

    Any better ideas
    Thanks
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  2. #2
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    Letís agree that notation wise $\displaystyle P(A \cap B) = P(AB)$. It makes it easier.
    Basically we know that $\displaystyle P(AB) = P(A|B)P(B)$.
    Consider: $\displaystyle P(ABC)$ and let $\displaystyle X=BC$.
    The we know that
    $\displaystyle \begin{array}{rcl} {P(ABC)} & = & {P(AX)} \\ {} & = & {P(A|X)P(X)} \\
    {} & = & {P(A|BC)P(BC)} \\ {} & = & {P(A|BC)P(B|C)P(C)} \\ \end{array} $

    That is idea you could use in an inductive proof.
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