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The chance of being dealt a pair in Hold'em is exactly 1/17.
At a FR game, there are 9 players. On average, you would expect to see 1 PP every other deal. 2 deals means that 18 people received cards (well, 9 people twice).
Since you expect to see 1 person out of 17 to be dealt a pair, then every 2 deals at a FR table, you'd expect to see 1 PP. Actually a bit more than that, since we're multiplying 1/17 by 18, we get 18/17 = 1 1/17... more than 1.
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The event (of drawing a pair) is random, so sometimes there might pass a few deals where no one gets dealt a pair, then later there may be a deal where more than 1 person is dealt a pair.
Ultimately, it's all going to even out to ~1 PP every 2 deals at FR games.
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Realistically, the odds for being dealt a pair will be different for an incomplete deck.
I mean, when the cards are dealt, they are dealt without replacement.
If we had knowledge of the specific cards as they are dealt, we could refine the estimate of 1/17 for each player, but it would not change the basic fact that everyone at the table expects to be dealt a pair 1/17 of the time, over the long term.
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