In Information theory, the conditional entropy of a Random variable $Y$, conditioning on another random variable $X$, is the average entropy of a random variable conditional on another random variable

$H(Y|X) = \sum_{x,y} p(x,y) \log{p(y|x)}$

Conditional entropy video

Some results:

$H(X,Y) = H(X) + H(Y|X) = H(Y) + H(X|Y)$

Where we use the Entropy and Joint entropy of the random variables.

Proof