# Meaning of P ( X &#x2229;<!-- ∩ --> Y ) <

Meaning of $\frac{P\left(X\cap Y\right)}{P\left(X\right)P\left(Y\right)}$
Imagine that we have a set $\mathrm{\Omega }$ and X and Y are events that can happen, I mean, $P\left(X\right),P\left(Y\right)>0$. Then, what does it mean the ratio $\frac{P\left(X\cap Y\right)}{P\left(X\right)P\left(Y\right)}$?
I know that $\frac{P\left(X\cap Y\right)}{P\left(X\right)P\left(Y\right)}=\frac{P\left(X|Y\right)}{P\left(X\right)}=\frac{P\left(Y|X\right)}{P\left(Y\right)}$ 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.
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Samuel Vang
The ratio $\frac{P\left(X\cap Y\right)}{P\left(X\right)P\left(Y\right)}=\frac{P\left(X|Y\right)}{P\left(X\right)}$ 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 $\frac{P\left({X}^{c}|Y\right)}{P\left({X}^{c}\right)}=\frac{1-P\left(X|Y\right)}{1-P\left(X\right)}=\frac{1/P\left(X\right)-\text{lift}}{1/P\left(X\right)-1}$, 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.