Let P(A)≥P(B), notice that  P(AB)≤P(B), then we have
    P(AB)- P(A)P(B)≤P(B)-P(A)P(B)≤P(B)-P(B)P(B)=P(B)(1-P(B))≤1/4,
vice versa. It is trival to show that the equation holds iff
    P(A)=P(B)=1/2

but how to prove  P(AB)- P(A)P(B)≥-1/4

Let  A+A'=Ω，then
P(AB)- P(A)P(B)=P(AB)- P(A)(P(AB)+P(A'B))=P(AB)(1-P(A))-P(A)P(A'B)≥-P(A)P(A'B)≥-P(A)P(A')≥-1/4

classic random variable example

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