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Thread: Pairwise Independence vs Idependence help

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    Member diddledabble's Avatar
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    Pairwise Independence vs Idependence help

    Okay. I have search the internet over for independence in probability and all I get is pairwise examples. I am trying to figure out exactly what the difference is. For example, what would the formula be to find the $\displaystyle P(A\cup B\cup C) $ Does it differ from pairwise independence too just finding indepence?
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    Quote Originally Posted by diddledabble View Post
    Okay. I have search the internet over for independence in probability and all I get is pairwise examples. I am trying to figure out exactly what the difference is. For example, what would the formula be to find the $\displaystyle P(A\cup B\cup C) $ Does it differ from pairwise independence too just finding indepence?
    Here is a standard example.
    Toss a coin twice.
    $\displaystyle \mathcal{A}$ is the event that the first toss yeilds heads.
    $\displaystyle \mathcal{B}$ is the event that the second toss yeilds heads.
    $\displaystyle \mathcal{C}$ is the event that both tosses match (two heads or two tails).

    You can show that $\displaystyle \mathcal{A}~\&~\mathcal{B}$, $\displaystyle \mathcal{A}~\&~\mathcal{C}$, and $\displaystyle \mathcal{C}~\&~\mathcal{B}$ are all independent: Pair-wise independence.

    But these are not mutually independent: $\displaystyle P\left(\mathcal{A}\mathcal{B}\mathcal{C}\right)\ne P(\mathcal{A})P(\mathcal{B})P(\mathcal{C})$
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