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Suppose we have a deck of card with numbers $1,\ldots,13$ totally $k$ times. (I mean if $k=4$ we have a standard deck). On every turn we check one card from the top of the pile and at the same time we increase the counter or set it to 1 if the counter is 14. How can we count the probability than at least once our counter shows the same value as the card? Or more generally, how can we count the probability than exactly $n$ times our counter shows the same value as the card?

I have no idea how this can be solved. I tried to find some useful recursion without lucky. I also read some basics of generating functions but found nothing useful.

Curious
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  • If you change the numbers, you can use the answers here: http://math.stackexchange.com/questions/414023/probability-of-winning-the-game-1-2-3 –  Oct 28 '13 at 12:15
  • Easier questions would be the expected number of matches at $k$, or an approximation to the distribution for large $k$ which would be close to Gaussian with mean $k$ and variance about $\frac{12}{13}k$ – Henry Oct 28 '13 at 12:17

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I suggest to try solve for simple cases first. Like for k=1. You will see that it is a known problem of derangement, see http://en.m.wikipedia.org/wiki/Derangement

From here you will get some further insight for increasing values of k.

aaa
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