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I was playing around with a deck of cards and I wanted to try out what the odds for a identic pair of cards during a strip-like shuffle were. What I mean by strip like shuffle is the shuffle of the deck where you remove the top and bottom card from the deck, and continue to do so until the deck is consumed.

I noticed that every time I tried it, there were at least one identical pair, e.g. 2 Kings or 2 Jacks. I got curious and after a quick test with python, I noticed that the probability for such a pair to be present was around 78-79% for 1 million iterations.

As my probability understanding is not the best, I am asking this question if someone can "prove" or explain in a mathematical way the reason for such a high probability.

To better rephrase the question: "What is the probability so that in a shuffled deck of cards, 2 identical cards (ignoring suit) are in mirror positions in the deck? (e.g. first and last, third and third to last, etc.)"

Thanks in advance for anyone who takes the time and effort for this problem!

infinitedreamer666
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    An equivalent question is this: when a shuffled $52$-card deck is dealt out to $26$ people, two cards per person, what the is probability at least one player has a pair? This is answered at this MSE question. In the notation there, the answer is $1-P(26,0)$. – Mike Earnest Jun 16 '23 at 20:05

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