Here is a question that my friend shared:
n people put their name in a hat.
Each person picks a name out of the hat to buy a gift for.
If any of the first people pick out themselves they put the name back into the hat.
What is the probability of the last person has his name in the hat?
For , the probability is 0.
For , the cases are , and so the probability is .
As gets larger, the probability should be about , since any name "should" be equally likely. However, this is not a rigorous argument, and the probability could differ.