Assume that $X$ and $Y$ are random variables and they are independent. They are following their own distributions, for example, $X$ follows $\mathcal{D}_{X}$.
Let me draw a sample from the distribution $\mathcal{D}_{X}$ and call it $X_1$. I know that the probability $\mathbb P(X_1 < Y_1) = a$.
However, now I want to pick more samples from $\mathcal{D}_{Y}$, for example, I sample $n$ instances, $Y_1,..., Y_n$.
I would like to make an upper bound of $\mathbb P(X_1<Y_1, X_1<Y_2,..., X_1<Y_n)$ in terms of $a$.
Any suggestions will be appreciated.
One of my trial $$\mathbb P(X_1<Y_1, ..., X_1<Y_n) = \sum_{x} \mathbb P\left(X_1<Y_1, ..., X_1<Y_n \mid X_1=x\right)\mathbb P(X_1=x) = \sum_x {\mathbb P(x<Y)}^n \mathbb P(X_1=x),$$ but hard to link it to $a$.
For an upper bound, the best that you can do is $$ \Pr(X_1 < Y_1 , \dotsc, X_1 < Y_n) \leq \Pr(X_1 < Y_1) = a. $$
To see that this is tight for any $a$, let $X \sim \mathrm{Bernoulli}(1 - a)$ and $Y_n = 1/2$ so that $$ \Pr(X_1 < Y_1 , \dotsc, X_1 < Y_n) = \Pr(X_1 = 0) = a. $$
More interesting is a lower bound, that follows from Jensen's inequality. Since $X_1$ is independent from the $Y_i$, we have \begin{align*} \Pr(X_1 < Y_1 , \dotsc, X_1 < Y_n) &= \mathbf{E}[\Pr(X_1 < Y_1| X_1)^n ] \\ &\geq \mathbf{E}[\Pr(X_1 < Y_1 | X_1)]^n \\ &= \Pr(X_1 < Y_1)^n \\ &= a^n . \end{align*}
This bound is also tight, which follows from taking $X_1 = 1 - a$ and $Y_n \sim \mathrm{Uniform}(0, 1)$.