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