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Math Help - Mathematical induction and Boole's Law

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    Mathematical induction and Boole's Law

    Use the additive law of probability to establish, using mathematical induction, Boole's Law:

    P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})
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    Quote Originally Posted by acevipa View Post
    Use the additive law of probability to establish, using mathematical induction, Boole's Law:

    P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})
    Common proofs like this one can be found online.
    Here is one:

    http://www.andrew.cmu.edu/course/21-228/lec7.pdf

    There are many others that may better suit your needs, though.
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  3. #3
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    Hello acevipa
    Quote Originally Posted by acevipa View Post
    Use the additive law of probability to establish, using mathematical induction, Boole's Law:

    P(A_{1}\cup A_{2}\cup...\cup A_{n})\leq P(A_{1}) + P(A_{2})+...+P(A_{n})
    Since you are asked to use induction, the proof will be something like this:

    Suppose that the proposition is true for n = k. So
    P(A_{1}\cup A_{2}\cup...\cup A_{k})\leq P(A_{1}) +  P(A_{2})+...+P(A_{k})
    Then
    P\Big((A_{1}\cup A_{2}\cup...\cup A_{k})\cup A_{k+1}\Big) =P(A_{1}\cup A_{2}\cup...\cup A_{k})+P(A_{k+1})-P\Big((A_{1}\cup A_{2}\cup...\cup A_{k})\cap A_{k+1}\Big), using the addition law of probability
    \leq P(A_{1}\cup A_{2}\cup...\cup A_{k})+P(A_{k+1}), since P\Big((A_{1}\cup A_{2}\cup...\cup A_{k})\cap A_{k+1}\Big) \geq 0
    \Rightarrow P(A_{1}\cup A_{2}\cup...\cup A_{k}\cup A_{k+1}) \leq P(A_{1}) +  P(A_{2})+...+P(A_{k})+P(A_{k+1}), using the Induction Hypothesis

    When n = 1, the hypothesis is clearly true. So it is true for all n \ge 1.

    Grandad
    Last edited by Grandad; June 2nd 2010 at 12:02 AM. Reason: Corrected typo
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