TITLE: Bound for probability of the intersection of a set of events
QUESTION [6 upvotes]: There are $N$ random variables $X_1,\dots X_N$ and $Pr(X_i=1)=p$ $\forall i\in N$. Can we upper bound the probability  that all random variables are $1$, i.e., $Pr(X_i=1,\forall i\in N)$. Note that the random variables are not independent.
Edit:
How about a lower bound?
Looking for answers other than $0,1$.
My attempt: I am thinking the product (as if independent) is a upperbound, but not sure.

REPLY [2 votes]: I guess a lower bound on the probability of the intersection can be obtained by the union-bound:
\begin{align}
\textrm{Pr}[\cap_i X_i] &= 1-\textrm{Pr}[\cup_i \overline{X_i}]\\
&\geq1-N\cdot\textrm{Pr}[\overline{X_1}]\\
&=1-N\cdot(1-p)\enspace,
\end{align}
which gives a nontrivial bound for $p>1-\frac{1}{N}$.