The following is a problem from PUMaC 2007:
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square
and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw
the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
I tried to find the probability by taking the general point $(x,y)$ and finding the region in which the second point should lie so that the line segments intersect, however, I am stuck in this step. My idea was to find the area of the region, divide by area of the square to get probability and then double integrate wrt y and x to get probability. I have failed at this after multiple attempts, If the region is a triangle (which is erroneous) I got the answer as $3/8$ , and if I try to do the general case, I find that the integral does not converge.
(Do note, over here vertices are adjacent, not opposite)
Best Answer
Let the square be $ABCD$ and the points $P$ and $Q$.
Consider the bent lines $APC$ and $BQD$. They intersect once, so by symmetry the chance that $AP$ and $BQ$ cut is $1/4$.