Given $m \geq 2$ subsets of $\{1,...,n\}$, say $A_1,...,A_m$, each of size $r$, we pick $B \subset \{1,...,n\}$ such that for any $i \in [n]$, $\displaystyle Pr\left[i \in B\right] = m^{-\frac{1}{r+1}}$, independently of the other choices. I want to show that there is a constant $\gamma_r$ (dependent only on $r$) such that
$Pr\left[A_1\not\subseteq B \wedge ... \wedge A_m \not\subseteq B\right] \leq \gamma_r \left(1-m^{-\frac{r}{r+1}}\right)^m$
It seems that Janson's inequality should be used here, however I can't get that to give me a constant which depends only on r. I'd appreciate any idea, and if this is at all correct.
Thanks.
Edit: I had mistakenly written that I used Azuma's inequality, however I meant Janson's.