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Thread: P(A n B) \leq P(A)

  1. #1
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    P(A n B) \leq P(A)

    Prove:

    $\displaystyle P(A\cap B)\leq P(A)$

    using the axioms of probability.

    $\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)\geq 0$

    $\displaystyle P(A\cap B)\leq P(A)+P(B)$

    Now, I am stuck with a P(B).
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint :

    $\displaystyle A\cap B\subset A$


    Fernando Revilla
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  3. #3
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    Suppose that $\displaystyle C\subseteq D$.
    Then $\displaystyle \mathcal{P}(D)=\mathcal{P}(C)+\mathcal{P}(D\cap C^c)\ge \mathcal{P}(C)$.
    Probability is monotone.
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