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The other day my friend gave me this problem:

Say you have the cartesian plane (R^2), and you take some infinite line that divides the plane up into two half-planes. Say I now randomly throw a point down on the plane: what is the probability that the point will land on one side of the plane?

My friend said I'd be surprised at the answer because it's not 1/2, but then my question is, what would be the answer? It sounds like it would be something from measure theoretic probability, but I've never studied that (nor basic probability, actually).

The reason I thought to post this was that there was a question similar to this one that was posted in this forum a while back. Would the answer to this problem be zero, because there are infinitely many points?

What kind of assumptions would have to be made to make this problem solvable, if it is not solvable already?

Thanks for the help.