At a certain time, 101 ants are placed on a one-meter stick, with one of them, Alice, placed at the exact center. The positions of the other 100 ants are random, as are the directions they face. All ants start crawling in whatever direction they are facing, always traveling at one meter per minute. When an ant meets another ant or reaches the end of the stick, it immediately turns around and continues going in the other direction. What is the probability that after 1 minute Alice is at the exact center of the stick?
Extending Charlie's brilliant approach, it is easily shown that after 2 minutes all ants will have reached their initial position !!!
I find this a very surprising fact, considering that this holds for any initial distribution of the ants along the stick .