Two identical decks consist of 6 white cards and 6 black cards each. The top card of each deck is flipped at the same time. If this is done repeatedly, what is the probability that at least one of the pairs of flipped cards is a match?
I figured the first card flipped didn't matter, and that the probability of drawing any given card from the second deck would be 6/12. The probability for the next draw would be 5/11 (assuming you're still looking for the same color). Following this pattern I got a total probability of (6/12)(5/11)(4/10)...(6/6)(5/5)...(1/1) = 1/252 but this seems highly unlikely given that there are only two unique cards to get in each deck. I don't think that derangement applies directly either since each card is not unique.