Meaning of P ( X ∩ 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 ) &gt; 0. Then, what does it mean the ratio P ( X ∩ Y ) P ( X ) P ( Y ) ?I know that P ( X ∩ Y ) P ( X ) P ( Y ) = P ( X | Y ) P ( X ) = 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.

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Meaning of              P      (      X      ∩      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  )  &amp;gt;  0. Then, what does it mean the ratio             P      (      X      ∩      Y      )              P      (      X      )      P      (      Y      )      ?I know that             P      (      X      ∩      Y      )              P      (      X      )      P      (      Y      )        =            P      (      X              |            Y      )              P      (      X      )        =            P      (      Y              |            X      )              P      (      Y      )       and if that ratio is equal to 1 then X,Y are independent events, but I can&#039;t figure out what exactly it means... please give simple examples.I found this when reading about lift-data mining.

Samuel Vang · Accepted Answer

The ratio             P      (      X      ∩      Y      )              P      (      X      )      P      (      Y      )        =            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             P      (              X        c                    |            Y      )              P      (              X        c            )        =            1      −      P      (      X              |            Y      )              1      −      P      (      X      )        =            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 &amp;lt;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 &amp;gt;1, as Y is presumably associated with an increased probability of X.

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