Given $\{-1,1\}^n$ we randomly choose a hyperplane in $\mathbb{R}^n$. Now given an integer $p \in [1,n]$ and a number $\epsilon \in [0,1]$, I want to ask, "*How likely is it that at least one of the $p-$dimensional faces of the discrete hypercube gets sliced by this plane and has $< \epsilon$ fraction of points on either of the linear fragments so produced on that face?*"

- Is this question well defined? 
- Do we know anything about this? 


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Somewhat related previous MO discussions, https://mathoverflow.net/questions/117912/faces-in-the-discrete-cube and 
https://mathoverflow.net/questions/98516/partitioning-the-vertices-of-an-n-cube-with-random-hyperplane-cuts?rq=1 and https://mathoverflow.net/questions/195403/directed-hypercube-minimal-cuts?rq=1