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.