A hiker climbs all day up a steep mountain path and arrives at the mountain top where he camps overnight.?

This is a simplified version of the Fixed Point Theorem in topology. The answer is A), and it's actually 100%. To see why, first draw a graph of the hiker's ascent on the trail as a function of the time of the day, distance from the trailhead, not elevation. Then draw the graph of his descent on the following day. The two graphs will cross somewhere, unless he either 1) reaches the bottom before he started out the day before, or 2) he starts down after the time of his arrival at the top the day before. We'll assume that neither 1) nor 2) is true, otherwise it would be impossible to decide any probability at all.