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We know that the probability of the intersection of two independent events is equal to the product of their probabilities? Are there any conditions under which the probability of the intersection of two events is less than or equal to the product of their probabilities? In other words what are the known bounds for the probability of the intersections in terms of the product of the probabilities in general ?

Thanks

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Optimal universal bounds: $$\max\{0,P(A)+P(B)-1\}\leqslant P(A\cap B)\leqslant\min\{P(A),P(B)\}$$

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  • $\begingroup$ Thank you! But I am mostly interested to know when I have the inequality in terms of the product of the probabilities? $\endgroup$
    – user24175
    Commented Aug 28, 2014 at 14:44
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    $\begingroup$ You have it when you have it, meaning there are various reformulations of the condition $P(A\cap B)\leqslant P(A)P(B)$ (such as $\mathrm{cov}(\mathbf 1_A,\mathbf 1_B)\leqslant 0$ or $P(A\mid B)\leqslant P(A)$ or $P(B\mid A)\leqslant P(B)$) which bring nothing new. Let me add that this is not what you asked in your question (and that what you did ask, to which I answered, is more interesting). $\endgroup$
    – Did
    Commented Aug 28, 2014 at 15:00

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