Let $S_1,\dots, S_n$ be Bernoulli random variables which are $4$-wise independent. We have that for each $i$, $P(S_i = 1) = p$ for some fixed probability $0 < p < 1$. What can we say about $P(\forall i\; S_i= 1)$ in terms of upper ~~ and lower ~~ bounds?

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Clearly $P(\forall i\; S_i= 1) \geq p^n$ but what is the largest it can be?~~

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