Suppose we have 2 well shuffled decks and alternately draw 1 card from each deck without replacement, placing the cards in a 2 deep 52 wide pattern, and looking for exact card matches between the 2 decks. For example, if at the 3rd pair of cards, we see K of spades in both decks, that is a single match between the decks.
What I am looking for is if we do this for the entire deck, what is the probability that at least 2 pairs of cards will match exactly between the 2 decks? For example, if the 3rd pair of cards are both K of spades and the 51st pair of cards are both 4 of hearts, now we have a "winning" combo that we are trying to count.
Note that if more than 2 pair of exact matches are found that is still considered a "win" and we want to count that. For example, if we see 3 pair of matching cards between the decks, we count that the same as if we saw only 2 pair of matching cards.
So what is the probability of this happening?
If that is too easy to solve, then as a bonus 2nd question, what is the probability that EXACTLY 2 pair of cards match between the 2 decks? Realize that any # of card pairs can match, including possibly all 52 pairs, but we only want to count the cases where EXACTLY 2 card pairs match.