Given $p>0$, I'm trying to find a collection of events that are pairwise independent and the probability of each event is $\le$ $p$ but the probability that none of them occur is zero.
For $p=\frac{1}{2}$, we can do the following. Let $X_1, X_2$ be independent random variables that take values in $\{0,1\}$ with equal probability and let $X_3 = X_1 +X_2 \mod 2$. Then the events $E_1:(X_1=X_2)$, $E_2:(X_2=X_3)$ and $E_3:(X_3=X_1)$ each have probability $\frac{1}{2}$ and are pairwise independent since $P(E_i \wedge E_j) = \frac{1}{4}$.
Also, $P(\bar{E_1}\wedge \bar{E_2}\wedge \bar{E_3}) = 0$.
How do we proceed for $p<\frac{1}{2}$? Any ideas are appreciated.
(I tried using independent $n$-binary strings $X, Y, Z$ such that $z_i = x_i+y_i \mod 2$ for $n$ large enough such that $\frac{1}{2^n} < p$ and then finding events that are pairwise independent and satisfy what I want. But the plan didn't work so far)