So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the switch, some subset of the lightbulbs turn on. This subset can vary between flips.
Your guess is that each of the lightbulbs has an intrinsic probability of being illuminated when you flip the switch, and that this probability is pairwise independent of the other lightbulbs. In other words, for each lightbulb $i$ and $j$, you predict that $$\operatorname{P}\left(i\text{ and }j\text{ both illuminated}\right)=\operatorname{P}\left(i\text{ illuminated}\right)\operatorname{P}\left(j\text{ illuminated}\right)$$
To test this, you flip the lightswitch a whole bunch of times, each time recording the subsets of activated lightbulbs as $S_m$ (if you're on the $m^\text{th}$ flip). Afterwards, you record the number of times each lightbulb lit up as $$c_i=\left|\{m:i \in S_m \}\right|,$$ and, similarly, for each pair of lightbulbs $i$ and $j$, you set $$c_{ij}=\left|\{m:\{i,j\}\subseteq S_m \}\right|.$$
Note that if the lightbulbs really are illuminated pairwise independently of one another, we would see $$\ell_{ij}:=\frac{c_ic_j}{c_{ij}}\longrightarrow 1$$ for every $i,j$ as the sample size grows large. Naturally, we consider the matrix $\ell\in M_{n\times n}$ with entries $\ell_{ij}$.
Suppose that you have flipped the switch $N$ times and, from looking at $\ell$, you believe that some pair of lightbulbs $i$ and $j$ are not statistically independent. With what confidence can you assert, in terms of $N$, $n$, $c_i,c_j,$ and $c_{ij}$, that $$\left|1-\frac{\operatorname{P}\left(i\text{ and}j\text{ illuminated}\right)}{\operatorname{P}\left(i\text{ illuminated}\right)\operatorname{P}\left(j\text{ illuminated}\right)}\right|>\alpha\hspace{30pt}?$$
You may assume that $c_i$ and $c_{ij}$ are nonzero for each $i$ and $j$. (or, if you do not wish to assume this, please explain why and compensate by modifying the definition of $\ell_{ij}$.)