Probability on an infinite plane (I thought this is a popular problem, but sadly Google yields nothing.)
Three points are chosen at random on an infinite plane. What is the probability that they are on a line?
And a variant: the plane is discrete, that is, the point coordinates are both integers.
How do I even approach this?
 A: There is no such thing as "at random on an infinite plane", just as there is no "at random on an infinite line" or "at random on the integers".  That is, there is no translation-invariant probability measure on a line or a plane or an integer lattice.  There are lots of non-translation-invariant probability measures, but no particularly distinguished one.  However, for any probability measure that is absolutely continuous with respect to Lebesgue measure, the probability of the three points being collinear is 0.  For the discrete case, you will get a nonzero answer, but it will depend on which measure is chosen.
A: There is no translation invariant probability measure on the plane, hence one has to find a way to give a meaning to the question. 
In the continuous setting, one possibility is to choose three independent points uniformly distributed in a domain of the plane of finite area, then let the domain grow. For every given domain, conditioning on the locations of two points, the probability that the third point is on the line they make is the ratio of the area of the line (which is zero) to the area of the whole domain, hence it is zero, and the limit when the domain grows will be zero as well.
Similar reasoning applies to the discrete plane, except that the probability that the third point is on the line the two first ones make will not be exactly zero, but it will go to zero when the domain becomes large hence the conclusion is the same.
