Results 1 to 2 of 2

Thread: P(A)=\frac{n(A)}{n}

  1. #1
    MHF Contributor
    Mar 2010


    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

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

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

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

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

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

    Not sure what to do with 3.
    Last edited by dwsmith; Jan 25th 2011 at 12:34 PM. Reason: fixing latex
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Oct 2009
    For disjoint sets $\displaystyle A_1, ..., A_n$ it happens to be the case that, using your notation,

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

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: Mar 26th 2011, 06:05 AM
  2. [SOLVED] arg(\frac{1}{z})
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: Sep 24th 2010, 04:18 PM
  3. [SOLVED] arg(z)=\frac{\pi}{4}
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Sep 5th 2010, 04:50 PM
  4. \int \frac{x}{\sqrt{1+2x}}dx
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Feb 21st 2008, 02:19 PM
  5. \int{\frac{x^2}{\sqrt{4x-x^2}}}dx
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Feb 21st 2008, 06:15 AM

Search tags for this page

/mathhelpforum @mathhelpforum