Heads up - how often do both players have at least one card the same?

Darson

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My first time playing live poker was on Sunday night when a buddy and his wife came over for dinner and we threw some cards. This was my first time playing heads up all night (as opposed to the end of a tournament). Anyway, it was a lighthearted game with a lot of banter and a lot of calls to the river.

One thing that really surprised both of us was the number of hands where we both had at least one card that was the same. I'm gonna say that 90% of the time we got down to the river, we both had one card that was the same.

So can anyone calculate what the probability is of there being at least one same value hole card for both players? It's been 23 years since I had to do serious statistics so I've completely forgotten the methodology.
 

Kakeesh

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Well, I would think the cards you show at showdown should be close/similar, considering you both got a piece of the board. What it sounds like you are failing to "mentally account" for are all the times one of you folded on a flop when only one of you connected.

Here are basic probabilities, after a quick google:
https://www.primedope.com/texas-holdem-poker-probabilities-odds/
 

skylines

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Assuming you have 2 cards of separate rank, the odds of him having one of those in his hand is 6/50 + 6/49 (assuming the first card does not match). So almost a quarter of the time you should have 1 of the same card. If you get to the river there is better chance, since as Kakeesh pointed out, you probably both connected somehow with at least one of your cards.
 

WedgeRock

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Only lawyer I speak with is my a criminal defense attorney.

Otherwise, I'd agree with @Hornet's formula (assuming your opponent doesn't have pair).

The chance that your first hole card doesn't match one that the opponent holds is 44/50 and the chance that your second hole card doesn't match one that your opponent holds is 43/49. So the chance that neither hold card matches one of your opponent's hole cards is (44/50 * 43/49), or about 77.2% of the time. So the chance that at least one hole card matches your opponent is 1-0.7722, or about 22.78% of the time.

Trust @Hornet, he's smart.
 
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Darson

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Since we were just having fun and not playing too seriously, we often showed folded hands just for conformation bias and sure enough, more often than not, we both had one card matching. Bias confirmed!
 
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