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.