Sorry, I should have said more clearly...

I understand that it is impossible to defind a uniform probability on the plane. But this problem still has an answer.

First, any triangle is inscribed in a unique circle, and the triangle is obtuse if and only if the three points are located on a semi-circle.

Infinite plane is equivalently converted into finite circle.

It is not hard to show, if you drop three points randomly on a circle, the probability that they are ona same semi-circle is 3/4.