Is $P(X=x|Y=y)\le P(X=x)$?

Question

I know that

$$P(X\,|\,Y)\le P(X)\quad\Leftrightarrow\quad \frac{P(X , Y)}{P(Y)}\le P(X)\quad\Leftrightarrow\quad P(X , Y)\le P(Y)P(X)$$

is a proof for $P(X=x|Y=y)\le P(X=x)$ but i don't get the intuition.

If we just consider marginal probability of $X$ and probability of $X=x$ be $p$ , it isn't possible that probability of $X=x$ become greater than $p$ after revealing that $Y=y$?

Answer 1

You should read the inequality $P(X,Y) \leq P(X)P(Y)$ as: 'the events $X$ and $Y$ are negatively correlated'.

If $X$ was independent from $Y$, then $P(X)P(Y)$ would have been the probability of the two events happening together. What the inequality tells you is that there is some dependence between the events in such a way that they have a lower chance to happen together.

Now the intuition should be clear. If you know that $Y$ happened then $X$ now has a smaller chance of happening as well, so $P(X|Y) \leq P(X)$.