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Math Help - P(A)=\frac{n(A)}{n}

  1. #1
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    P(A)=\frac{n(A)}{n}

    Suppose that an experiment is performed n times. For any event A of the experiment, let n(A) denote the number of times that event A occurs. The relatively frequency definition of probability would propose that

    \displaystyle P(A)=\frac{n(A)}{n}.

    Prove that this def. satisfies the three axioms of prob.

    1 and 2 0\leq P(A)\leq P(S)=1\Rightarrow 0\leq P(A)\leq 1

    \displaystyle P(A)\in [0,1]=\frac{n(A)}{n}\Rightarrow n\in [0,1]=n(A)\Rightarrow 0\leq n(A)\leq n=1

    3 A_i\cap A_j=\emptyset, \ \ i\neq j, then \displaystyle P\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1}  ^{\infty}P(A_i)

    Not sure what to do with 3.
    Last edited by dwsmith; January 25th 2011 at 12:34 PM. Reason: fixing latex
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  2. #2
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    For disjoint sets A_1, ..., A_n it happens to be the case that, using your notation,

    <br />
n\left(A_1 \cup \cdots \cup A_k\right) = \sum_{i = 1} ^ k n(A_i).<br />

    Note that because n < infinity, you can have at most n non-null disjoint events so it suffices to show finite additivity as opposed to countably infinite.
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