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I don't quite understand upper and lower bound. To explore, I am using the following problem. Since I am not able to post an image of Venn diagram, I will try my best to explain the problem:

70% like A and 80% like B, what is the upper and lower bound of liking both A and B?

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Upper bound: everybody who likes $A$ likes $B$, so $70\%$. Lower bound: As few as possible like both. Since (at most) $A\cup B$ is everybody, that is, $100\%$, as few as possible means $50\%$: make the $20\%$ who don't like $B$ like $A$. – André Nicolas Apr 25 '12 at 17:51
@AndréNicolas: Is it correct to assume that upper bound visually means you have 70% circle A within 80% circle B eliminating the extra 10% that only like B. And, how did you deduce 50% as few as possible? – Vivek Apr 25 '12 at 18:44
If $A$ and $B$ have as little in common as possible, the $20\%$ space outside $B$ must be filled with people from $A$, leaving $70-20$ as overlap. For a fancier version, use $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The biggest $P(A\cap B)$ can be is $1$, when we have $P(A\cap B)=P(A)+P(B)-P(A\cup B)=0.7+0.8-1=0.5$. – André Nicolas Apr 25 '12 at 18:48

The lower bound of liking both A and B is 50%, while the upper bound is 70%. In terms of Venn diagrams, I think of it like this: try to "push apart" the regions A and B. A can be pushed out of B, but only to a point, since B covers 80% of the space. In fact we can push 20% of A out of B. The other 50% of A remains in B, so then the lower probability 50%, this part of A remaining in B. For the upper probability, try to "push together" A and B. A can fit completely in B, and the upper probability is the area of A, 70%.

Another way to think of lower probability is as the "worst case scenario" for how much A and B overlap. At worst, A and B can be only partially apart, and the amount they must overlap is the lower probability, 50% of the total area. The upper probability then is the "best case scenario" that A sits completely in B.

Of course these intuitions can be made precise, but this is a way I found it helpful to think about it!

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