Is $\Pr(X<a\mid X<Y)>\Pr(Y<a\mid X\geq Y)$?

Question

Here, I have two probabilities: $\Pr(X<a\mid X<Y)$ and $\Pr(Y<a\mid X\geq Y)$, where $X$ and $Y$ are two non-negative continuous random variables, and $a$ is a positive constant. If I know that $\Pr(X<Y)$ is a value close to 1, does it tell me that $\Pr(X<a\mid X<Y)>\Pr(Y<a\mid X\geq Y)$?

I did matlab simulations to check this. In my case, $X:= 1+\Gamma_1$, $Y:= (1+\Gamma_2)(1+\Gamma_3)$, where $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are independent Gamma random variables. Setting the means of $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ to make $\Pr(X<Y)=0.99$ for example, I found that indeed $\Pr(X<a\mid X<Y)>\Pr(Y<a\mid X\geq Y)$ by adapting $a$. I am wondering that does it hold always? How to interpret these two conditional probabilities by incorporating the fact that $X<Y$ happens at a high probability?

Anyone could pass some wisdom?

Thanks a lot in advance.

Answer 1

No

As a counterexample, suppose you have the following table of values and probabilities

 X   Y  prob
 3   2  0.01
 3 222  0.49 
33   2  0.01
33 222  0.49

Then $\Pr(X \lt 10 \mid X \lt Y) = \frac12$ but $ \Pr(Y \lt 10 \mid X\geq Y) = 1$