Independent Events and Bernstein Paradox for n events

Question

Is it possible to extend Bernstein Paradox example (about pairwise independence, but joint dependence of 3 events (color sides of tetrahedron)) to n events using the same reasoning?

Answer 1

I don't believe so. The Bernstein paradox seems to rely heavily on the fact that $\frac{2}{4}\cdot \frac{2}{4}=\frac{1}{4}$, while in general it does not hold that $\frac{2}{n}\cdot \frac{2}{n}=\frac{1}{n}$.

What you can do is this. Let $X_1,\dots,X_{n-1}$ be IID Bernoulli($1/2$), and let $X_n=\left(\sum_{i=1}^{n-1}X_i \right)\mod 2$. Then you can show that any $n-1$ of the $X_i$'s are mutually independent, but all of the $X_i$'s are not mutually independent. If you want to phrase things in terms of events rather than random variables, then let $C_i=\{X_i=0\}$. Then any $n-1$ of the events $C_i$ are mutually independent, but all of the $C_i$'s are not.