Meaning of P ( X ∩ Y ) <

Question

Meaning of

\frac{P (X \cap Y)}{P (X) P (Y)}

Imagine that we have a set

Ω

and X and Y are events that can happen, I mean,

P (X), P (Y) > 0

. Then, what does it mean the ratio

\frac{P (X \cap Y)}{P (X) P (Y)}

?
I know that

\frac{P (X \cap Y)}{P (X) P (Y)} = \frac{P (X | Y)}{P (X)} = \frac{P (Y | X)}{P (Y)}

and if that ratio is equal to 1 then X,Y are independent events, but I can't figure out what exactly it means... please give simple examples.
I found this when reading about lift-data mining.

Answer 1

The ratio

\frac{P (X \cap Y)}{P (X) P (Y)} = \frac{P (X | Y)}{P (X)}

is called lift, and it is a measure of how good the occurrence of event Y is at predicting the occurrence of event X. We interpret the lift as the ratio in which the probability of X increases after the occurrence of Y. Notice that

\frac{P (X^{c} | Y)}{P (X^{c})} = \frac{1 - P (X | Y)}{1 - P (X)} = \frac{1 / P (X) - lift}{1 / P (X) - 1}

, which is 1 precisely when lift =1, and increases as lift increases. This is consistent with predicting the non occurrence of X after Y when lift <1.
Example: X is the event that this answer receives an up-vote. Y is the event that I provide an example. In this case we expect lift >1, as Y is presumably associated with an increased probability of X.