At 6 a.m. a man starts hiking a path up a mountain. he walks at a variable pace, resting occasionally, but never actually reversing his direction. at 6 p.m. he reaches the top. he camps out overnight. the next morning he wakes up at 6 a.m. and starts his descent down the mountain. again he walks down the path at a variable pace, resting occassionally, but always going downhill. at 6 p.m. he reaches the bottom. what is the probability that at some time during the second day, he is in the exact same spot he was in on the first day?
The probability is 100%. the easiest way to see it is, consider that on the second day when the man is going down the mountain, a ghost follows his original pace up the mountain. so even if he varies his pace as he goes down the mountain, at some point in time, he will be in the same spot as the ghost, and therefore, the same spot he was in the day before.