Problem. On the Cartesian plane with origin O and x- y-axes, I randomly pick a point P. What is the probability that the line segment OP has a slope at least 1?
Is the answer 1/4 or 1/2?
answer = 1/4 : divide the whole plane into 8 "equal" triangles like those in the UK flag. 2 of the 8 "equal" triangles have slope >= 1.
answer = 1/2 : i know for infinite stuff, part of it CAN equal to the whole. i can make a bijection from the region with slope >= 1 to the region with slope <= 1 (divide the region with slope >= 1 into 3 parts and stretch clockwise). This bijection means that the probability of choosing a point in two of the 8 triangles is the same as choosing in 6 of the 8 triangles?
(There is a similar question here about probability on infinite plane, but i don't understand what they mean by translation-invariant measure...)
I would argue that each of the following are equally likely: slope greater than 1; slope between 0 and 1; slope between -1 and 0; and slope less than -1 (ignore the boundary cases as they don't affect the result). Each has probability 1/4.