linguistic:paraphrasing, structural:reformatting, information:classification
For random variables

Intuitively, two random variables X {\displaystyle X} and Y {\displaystyle Y} are conditionally independent given Z {\displaystyle Z} if, once Z {\displaystyle Z} is known, the value of Y {\displaystyle Y} does not add any additional information about X {\displaystyle X}. For instance, two measurements X {\displaystyle X} and Y {\displaystyle Y} of the same underlying quantity Z {\displaystyle Z} are not independent, but they are conditionally independent given Z {\displaystyle Z} (unless the errors in the two measurements are somehow connected).

The formal definition of conditional independence is based on the idea of conditional distributions. If X {\displaystyle X}, Y {\displaystyle Y}, and Z {\displaystyle Z} are discrete random variables, then we define X {\displaystyle X} and Y {\displaystyle Y} to be conditionally independent given Z {\displaystyle Z} if

for all x {\displaystyle x}, y {\displaystyle y} and z {\displaystyle z} such that P (Z = z) > 0 {\displaystyle \mathrm {P} (Z=z)>0}. On the other hand, if the random variables are continuous and have a joint probability density function f X Y Z (x, y, z) {\displaystyle f_{XYZ}(x,y,z)}, then X {\displaystyle X} and Y {\displaystyle Y} are conditionally independent given Z {\displaystyle Z} if

for all real numbers x {\displaystyle x}, y {\displaystyle y} and z {\displaystyle z} such that f Z (z) > 0 {\displaystyle f_{Z}(z)>0}.

If discrete X {\displaystyle X} and Y {\displaystyle Y} are conditionally independent given Z {\displaystyle Z}, then

for any x {\displaystyle x}, y {\displaystyle y} and z {\displaystyle z} with P (Z = z) > 0 {\displaystyle \mathrm {P} (Z=z)>0}. That is, the conditional distribution for X {\displaystyle X} given Y {\displaystyle Y} and Z {\displaystyle Z} is the same as that given Z {\displaystyle Z} alone. A similar equation holds for the conditional probability density functions in the continuous case.

Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability given no events.