In a lively $4-8 Hold 'Em game in late fall at my local poker room, I had been playing long enough to assign styles tentatively to the four players new to me. In middle position I peeked at my hand and saw both red Tens. I called and waited for the others to act. Alex, one of the new-to-me players, raised. The Button called Alex's raise cold, the Big Blind and I just called. Four-handed we saw the dealer put As-Jh-4d on the table.
From the texture of the Flop, there was little chance of a straight or a flush. I checked, Alex bet at the Flop, The Button and The Big Blind folded and it was back to me. There were two overcards on the Flop and I was in poor position. It was an easy fold, but something didn't seem right because Alex had bet straight away. I had seen enough of Alex's play to learn that he's a tricky player: he would have feigned weakness, pausing just a little before betting with Top Pair. Also, he was tricky enough to slow-play a Set or the top Two Pairs, and he hadn't checked. So, I ruled out his having Ace- Any, pocket Aces or Jacks.
He could have had pocket Kings, Queens, or another pair smaller than Jacks. Possibly he could have been betting with a ten-out hand like Kh-Qh, hoping to hit a King, Queen, or Ten on the Turn or River. Any of those holdings would be consistent with his behavior. I called.
The dealer put the Ac on the table. I checked. Again Alex bet quickly, representing Ace-Any or pocket Jacks. Still suspicious, and despite my poor situation, I called.
The dealer put the 8d on the River. I checked and he checked. Alex showed down the two black Tens. I showed my red Tens and the dealer split the pot.
As I stacked my half of the pot I wondered, in a ten-handed game what is the chance that anyone else has a pocket pair of the same rank?
Given that you have a pocket pair, there are fifty unknown cards before the Flop: the thirty-two cards remaining in the deck and the eighteen cards dealt to the nine other players. No one else could possibly hold a pocket pair of the same rank unless both cards were among those eighteen. That probability is C(48,16)/C(50,18), or .1249, about one time in eight.
There are 17!! ways to deal those eighteen cards into nine, two-card hands. If anyone else has a pair of the same rank, then there are 15!! ways to deal the other sixteen cards into the other eight twocard hands. The probability of anyone else having the pair is 15!!/17!!, one-seventeenth, or .0588.
The answer to the question of whether anyone else has a pair of the same rank is the product of the two probabilities, .007347, about one time in 136.
The chance of your being dealt any pocket pair is one in 17. The chance of anyone else's also having a pocket pair of the same rank is 1 in 136, for a combined chance of 1 in 17*136, about 1 in 2300. If, before the hand is dealt, you were to offer someone 100 to 1 odds that (a) you won't have a pocket pair, and (b) even if you do, no one else will have a pair of the same rank, then you'd have a huge edge. You can bet on that.
