Say I have a conditional probability bound $$ \Pr\left(X \le Y + f_1(\delta_1) \mid Y = y\right) \ge 1-\delta_1, \tag{1}\label{one} $$ for some function $f_1$ and some $\delta_1 \in (0,1)$.
Further, I know that $Y$ is close to its expectation with high probability, that is, $$ \Pr\left(|Y-\mathbb EY| \le f_2(\delta_2)\right) \ge 1-\delta_2, $$ for some function $f_2$ and $\delta_2 \in (0,1)$.
Can I somehow use this knowledge to bound $\eqref{one}$ in terms of $$ \Pr\left(X \le \mathbb EY\right)? $$
In other words, is there a method where "knowing that $Y$ and $\mathbb EY$ are rather similar (with large probability)" lets one "replace" $Y$ by $\mathbb EY$?
Sorry for the somewhat fuzzy question. I am open for any solutions, tips, and hints where to look at.