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Suppose I have 2 white bags A and B, and 2 black bags C and D. All these bags only contain a bunch of cool beans and bad beans.

Bag A has a greater % of cool beans than C. Bag B has a greater % of cool beans than D. I combine the two white bags into a giant white bag and black bags into a giant black bag. Is it possible for the % of cool beans in the giant white bag is less than that of the giant black bag?

I don't think this is possible because the individual components already maintain that the white bag will have a greater % of cool beans.

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2 Answers 2

up vote 4 down vote accepted

Consider the following scenario:

A has 1 cool bean out of 100 total. B has 1 cool beans out of 1 total. C has 1 cool bean out of 101 total. D has 99 cool beans out of 100 total.

A+B: 2 cool beans out of 101 total. C+D: 100 cool beans out of 201 total.

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Oh wow, you're right! What are the conditions under which this is possible? –  Minden Petrofsky Sep 26 '12 at 22:42
    
Intuitively: The reason this can happen is because D has many more total beans than B, so even though B has a higher percentage, D affects the overall percentage of C+D much more than B affects the overall percentage of A+B. –  only Sep 26 '12 at 22:44

Yes, this is possible: it’s called Simpson’s paradox.

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