Evan Chen's recount of the Taiwanese IMO team's journey recorded a game the team members played at their free time, which runs as the following:
There are $n$ team members (in the actual case $n=6$ but here we simply take $n\geq2$)Every team member points at another team member (whom must be different from him/herself) and thus we obtain a (directed) graph with $n$ vertices and $n$ edges. It has one more edge than a tree and therefore must contain a cycle. Any member who is a vertex of any cycle in this graph loses the game. (So it is possible that everyone loses but it's impossible that no one loses.)
Assume everyone chooses the person s/he points at randomly, what is the probability of a player losing the game?
Reference: Evan Chen's recount