Are these the same? P(X,Y=y_1) and P(X|Y = y_1)

Question

A simple question,

$P(X,Y=y_1)$ and $P(X|Y = y_1)$

are the same? (here, $X$ and $Y$ are random variables, and $y_1$ is a fixed variable of $Y$.)

The left is a marginal pdf from a joint pdf, and the right is a conditional pdf. The notations are different but to me, their conceptual meanings are the same.

Answer 1

No, they are not same.

If $X$ and $Y$ are discrete random variables then

$P(X=x,Y=y)$ := Probability of the random variable $X$ taking a value $x$ and random variable $Y$ taking a value $y$.

$P(X=x~|~Y=y)$ := Probability of the random variable $X$ taking a value $x$ given that the random variable $Y$ has picked a value $y$.

The statement,

The left is a marginal pdf from a joint pdf

is true only if random variable $Y$ takes a value $y_1$ with probability $1$, basically $Y$ is deterministic.