If we look at the definition of independence for discrete random variables:
$$ \DeclareMathOperator{\P}{\mathbb{P}}
\P(X \mid Y) = \P(X)
$$ iff $X,Y$ independent.
isn't this then true for any random variable itself, i.e.:
$$
\P(X \mid X ) = \P(X)?
$$
Best Answer
If $X$ and $Y$ are independent, so that $$ \DeclareMathOperator{\P}{\mathbb{P}} \P(X=x,Y=y) = \P(X=x)\P(Y=y) $$ So, if $X$ is independent of itself, then $$ \P(X=x)=\P(X=x,X=x)= \\ \P(X=x)\P(X=x)=\P(X=x)^2 $$ Solving the equation $\P(X=x)=\P(X=x)^2$ gives $\P(X=x)= 0 ~\text{or}~ =1$ So $X$ is a constant random variable. (That is, constant with probability one).