Simple One

[AdamL]

What are the odds of any player having a pair by the river, with x number of players.

x being 2 players (HU)
x being 3 players

I think it's 62% for one player to pair.

[AaronBrown]

It's much better than that. 82.6% of the time a hand played to the river will have a pair or better.

The chance of two or three players all having no better than high card hands is pretty low. 49.9% of the time, there will be a pair or better on the board, in which case all players will obviously beat a pair. If there is no pair on the board, there are 15 cards out of 47 that make a pair. The odds of drawing one hand without pairing is (32/47)*(31/46) = 45.9%, if that happens the odds are (30/45)*(29/44) = 43.9% for the second hand and (28/43)*(27/42) = 41.9% for the third. That means the chance of two players with only a high card hand at the river is less than 7.6% (because I didn't subtract for the times the second player makes a straight or flush). For three players it's less than 3.2%.

[AdamL]

Aaron,

Thanks for your help with this, much appreciated.