suppose we have subsets $I_1,I_2 \subseteq \{1,\dots,n\}$ of size $n_1,n_2 <= n$ of n i.i.d Bernoulli trials $X_1,\dots,X_n$ and let $Y=\sum_{i \in I_1}X_i, Z=\sum_{i \in I_2}X_i$.
Any ideas on how to calculate
$\mathbb{P}(\{Y\leq k\}\cap\{Z\leq k\})$ in terms of the distribution of sums of some $X_i$ for some $k\leq n$?
I assume it will be useful to describe $Y = C + \tilde Y$ and $Z = C + \tilde Z$, where
$C = \sum_{i \in I_1 \cap I_2} X_i$
since then $C, \tilde Y, \tilde Z$ are independent.
I appreciate your help :)