Berkson's paradox

Berkson's paradox is a result in conditional probability and statistics which is counter-intuitive for some people, and so has been described as a paradox.

The result is that two independent events become conditionally dependent given that at least one of them occurs. Symbolicly:

if 0 < P(A) < 1 and 0 < P(B) < 1,

and P(A|B) = P(A), i.e. they are independent,

then P(A|B,C) < P(A|C) where C = A∪B (i.e. A or B).

As an example, suppose I have 1000 postage stamps, of which 300 are pretty and 100 are rare, with 30 being both pretty and rare. 10% of all the stamps are rare and 10% of the pretty stamps are rare, so prettiness tells me nothing about rarity.

I put the 370 stamps which are pretty or rare on display. Just over 27% of the stamps on display are rare, but still only 10% of the pretty stamps on display are rare. If I only consider stamps on display, I will observe a spurious negative relationship between prettiness and rarity as a result of my selection bias.