Janson-type inequality, limited dependence So I am trying to figure out an upper bound on the probability of the following... 

This is a question related to a problem I am working on (not for a class, just for fun)

Let $\Omega=\{X_{1},\dots,X_{n}\}$
  where each $X_{i}$
  is i.i.d
  Bernoulli with success probability 1/2.
  Let $Y=\{Y_{1},\dots,Y_{N}\}$
  where $Y_{i}\subset\Omega$
  and $|Y_{i}|=t$.
  Let $Z_{i}$
  indicate the event that $\sum_{X\in Y_{i}}X\geq k$,
  where $k$
  is some value close to $t$. Let $Z=\sum_{i=1}^{N}Z_{i}$

Say that $I_{r}=
  \{\{Y_{i},Y_{j}\}:|Y_{i}\cap Y_{j}|=r\}$,
  that is, $I_{r}$
  is the set of all pairs from $Y$
  that share $r$
  elements from $X$.
  I am looking for an upper bound on $Pr[Z\leq\mathbb{E}[Z]-t]$.

I've seen Janson's inequality and notice that it looks quite similar to this, but I'm not sure where to go from here. Any help or any references would be greatly appreciated.
I'm posting a suitable answer that I found for this problem. The approach linked here does not exploit the fact that no $t$-tuples of elements of $\Omega$ are present in too many subsets $Y$ (for arbitrary $t$), but it does end up giving me a good enough bound anyway.