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I'm currently reading an Introduction to Probabillity Theory and pretty early one comes across the definition of a Probabillity Space and a discrete Probability Space and after a little while one gets the definition of stochastically independence/Independence. And i asked myself a Question which is somewhat mind boggling me since i can't find a answer with a formal proof. So I want to find the smallest natural Number N sucht that we can define a discrete Probabillity Space $(\Omega ,P)$ with $\left| \Omega \right| =N$ and stochastically independent events. Or maybe to put it in another way What is the smallest number of Elements a Sample Space must have to define n independent (non trivial) events Would be nice if someone could help me with this problem and maybe even come up with a nice proof.

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