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Math Help - 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 P(A \cap B) = P(AB). It makes it easier.
    Basically we know that P(AB) = P(A|B)P(B).
    Consider: P(ABC) and let X=BC.
    The we know that
    \begin{array}{rcl}   {P(ABC)} &  =  & {P(AX)}  \\   {} &  =  & {P(A|X)P(X)}  \\<br />
   {} &  =  & {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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