Link to Wikipedia article on the Bertrand paradox
There's another question asked recently that superficially looks like Bertrand's paradox. Both involve picking random points/chords and then calculating a property. Yet one leads to a not-well-defined situation, while the other apparently has a well-defined answer of 0.25.
Why? How does one tell if a question is well-defined or if Bertrand's paradox will apply?
The point of Bertrand's paradox is that you must properly specify the probability model for a "random" object in order to get a unique correct answer. In the "other" question you refer to, although the model is not quite explicitly specified there is an obvious choice: $X$ and $Y$ with joint distribution uniform on the square $[0,1]\times[0,1]$. Therefore there is no paradox here. But if somebody wanted, say, $X^2$ and $Y^2$ rather than $X$ and $Y$ to be uniform, they would get a different answer.