I think there are two ways to look at this question:

1. The "fixed point in space" approach. Pick any point along the path, what is the likelihood the nun will cross that point at the exact same time the next day? The likelihood of that is almost zero.

2. The 'infinite points in space" approach. There are infinite points along the path. What is the likelihood the nun will cross at least one point at the same time the next day? 100% (I say at least one because she may stop for breaks at different times ascending and descending.)

Thanks to Biff for posting the graphs that helped me get it. To the others that tried to explain it but I argued with, I apologize for being so thickheaded.