Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\{E\_i\}\_{i = 1}^N,$ with $E\_i \in\mathcal{F}$ be a set of events and let $i(X)$ be a R.V. assuming values in $\{1,...,N\}$


Is there a way to bound the following quantity?


$$\mathbb{P}\left[\bigcup\_{i\in[N]: i \neq i(X)} E\_i\right].$$


I am looking in an upper-bound that resemble the union bound: in fact, if the union would not depend on the R.V. $X$ we may use the union bound in the following way


$$\mathbb{P}\left[\bigcup\_{i\in[N]} E\_i\right] \leq \sum\_{i\in[N]}\mathbb{P}\left[ E\_i\right].$$



Union bound probability of random union
Let $n:=N$ and $J:=i(X)$. Then the probability to bound is
\begin{aligned}
P\Big(\bigcup\_{i\in[n]\setminus\{J\}}E\_i\Big)
&=\sum\_{j\in[n]}P\Big(\{J=j\}\cap \bigcup\_{i\in[n]\setminus\{j\}}E\_i\Big) \\ 
&=\sum\_{j\in[n]}P\Big(\bigcup\_{i\in[n]\setminus\{j\}}\big(\{J=j\}\cap E\_i\big)\Big) \\ 
&\le\sum\_{j\in[n]}\sum\_{i\in[n]\setminus\{j\}}P(\{J=j\}\cap E\_i) \\
&=\sum\_{j\in[n]}\sum\_{i\in[n]\setminus\{j\}}P(J=j)P(E\_i|J=j) \\
&=\sum\_{j\in[n]}P(J=j)\sum\_{i\in[n]\setminus\{j\}}P(E\_i|J=j).
\end{aligned}
If each event $E\_i$ does not depend on $J$, then we further have
\begin{aligned}
P\Big(\bigcup\_{i\in[n]\setminus\{J\}}E\_i\Big)
&\le\sum\_{j\in[n]}P(J=j)\sum\_{i\in[n]\setminus\{j\}}P(E\_i) \\ 
&=\sum\_{i\in[n]}P(E\_i)\sum\_{j\in[n]\setminus\{i\}}P(J=j) \\ 
&=\sum\_{i\in[n]}P(E\_i)P(J\ne i).
\end{aligned}


