Take two decks of ordinary playing cards. Keep the packs separate. Shuffle each of them. Now place both decks face down side-by-side and begin simultaneously drawing one card off each deck and revealing them.
Odds are better than 50/50 that at some point before you reach the bottom of the decks, you will eventually draw the exact same card of suit and rank from either deck at the same time.
I originally saw this in a recreational math book years ago, but I'll give the credit to this fascinating poker odds wizard, since seeing the trick on that page reminded me of it.