Intersection of unindependent events

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Homework Statement

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-0.25 <= P( X \cap Y ) - P( X )P( Y ) <= 0.25

for any events X, Y

Homework Equations

P( X \cap Y ) = P( X )P( Y | X )
Bayes' theorem
Anything I missed?

The Attempt at a Solution

Obviously if X and Y are independent
P( X \cap Y ) = P( X )P( Y )
so
P( X \cap Y ) - P( X )P( Y ) = 0

but if they are not then I hit a wall. I've done pages of math but I go round in circles. I think there's some trick but I can't figure it out. Can anyone tell me what I'm missing?

 

[I like Serena]

If the events are totally independent, as you surmised, the result is 0.

What if, they are totally dependent, that is, P(X)=P(Y)?
Or if they are totally anti-dependent, that is, P(X)=1-P(Y)?