Compare two random variables and their probability with respect to same number

Question

Problem

Given random variables $X$, $Y$, if I know $X\geq Y$, what can I say about $\Pr[X\geq a]$ and $\Pr[Y\geq a]$ where $a\geq 0$.

What I Have Done

$\Pr[X\geq a]$ and $\Pr[Y\geq a]$ could be evaluated with integral but since I do not know the pdf of $X$ and $Y$,this approach could not give me any result.

Answer 1

Note that $$ Y\geq a\implies X\geq Y\geq a $$ whence $$ (Y\ge a)\subseteq(X\geq a) $$ and so by monotonicicity of measure (namely, $A\subseteq B\implies P(A)\leq P(B)$) we have that $$ P(Y\geq a)\leq P(X\ge a) $$