Union bound probability of random union

Question

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

$$\mathbb{P}\left[\bigcup_{i\in[N]} E_i\right] \leq \sum_{i\in[N]}\mathbb{P}\left[ E_i\right].$$

Answer 1

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}