Inegalitate cu probabilitatile unor evenimente

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Cezar Lupu
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Inegalitate cu probabilitatile unor evenimente

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Fie \( A_{i}, i=1,2, \ldots n \) evenimente independente intr-un camp \( (E, \mathcal{K}, P) \) de probabilitate \( P(A_{i})=p_{i} \) si \( A \) evenimentul ca sa se produca cel putin unul din evenimentele \( A_{i} \), \( P(A)=p \). Sa se arate ca

\( \sum_{i=1}^{n}p_{i}>p>1-e^{-\sum_{i=1}^{n}p_{i}} \).
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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