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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.

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We calculate the probability of no match. Whatever be the arrangement of the first deck, we get no match if the second deck has blacks precisely where the first has whites.

There are $\binom{12}{6}$ equally likely ways to place the blacks in the second deck. Only $1$ of them gives no match. So the probability of no match is $\dfrac{1}{\binom{12}{6}}$.

It follows that the probability of at least $1$ match is $1- \dfrac{1}{\binom{12}{6}}$.

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  • $\begingroup$ I believe you mean $\binom{12}{6}$, the number of ways to select the 6 positions for black among 12. $\endgroup$
    – vonbrand
    Commented Apr 21, 2014 at 5:14
  • $\begingroup$ Yes, thank you. Logic was OK, the numbers less so. $\endgroup$
    – André Nicolas
    Commented Apr 21, 2014 at 5:17
  • $\begingroup$ +1; numerically, the probability of at least one match is $923/924 \approx 0.9989$. $\endgroup$
    – ShreevatsaR
    Commented Apr 21, 2014 at 5:22

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