Appropriate Level: Calculus (1 or 2, depending on how your courses are divided).
The Question: You start with a unit square (say a piece of paper). Choose two adjacent sides of the square and randomly pick a point on each. Connect the two points with a line and cut along that line, thus removing a triangle (it could be a degenerate triangle) from the piece of paper. Let A be the area of the triangle that has been removed.
What is the smallest A can be?
What is the largest A can be?
What is the probability that A is less than 1/4? In particular, is this probability equal to 50%? (after all, 1/4 is half way between 0 and 1/2).