I had a question about the probability of unions to infinity.
1) Everyone in a group of $N > 3$ people writes their name on a slip of paper and drops the slips into an urn. Then, one at a time (say, in alphabetical order– it doesn’t matter), each person draws a slip of paper, until someone gets their own name or until all the slips of paper are taken, at which point they stop. What is the probability that this process stops after the second person draws a slip of paper?
2)You roll a 6-sided die repeatedly until you roll a 6, at which point you stop. Let $A_n$ be the event that this process stops after exactly n rolls. What is $P(\mathop{\cup}\limits_{n=1}^\infty A_{2n})$?
For question 1, I assume the set up is the probability of the union from n=4 to infinity. I just wanted to understand how one would go about solving the probability of solving for infinity.