Given two decks of \(n\) cards what is the probability that at least one of a pair of cards matches ?

Imagine the two decks of cards lined up, consider the event \(A_i\) to be that the \(i\)-th cards in the two decks match.

- Compute \[ \mathbf{P}(\bigcup_{i=1}^n A_i)\].
- (**)compute \[ \lim_{n\rightarrow \infty} ( \mathbf{P}( \bigcup_{i=1}^n A_i )\]