Winning with less than a pair.

What would the odds be that in a 9 handed NLH game, the winning hand would be less than a pair?

Assuming everyone stays in? Pretty damn unlikely - just the chance of no pair coming up on the board, and no one catching, is 0.000144%, and that doesn't account for the possibility of someone having pockets. But I don't think that's what you mean, or you wouldn't have specified "NLH." Frankly, the probability you want can only be found by observation.

OK, a more realistic view would be; 9 hands dealt in, all fold except 2, hand goes to showdown. My guess is that it would be somewhere in the 6% range.

Quote: kissapig

OK, a more realistic view would be; 9 hands dealt in, all fold except 2, hand goes to showdown. My guess is that it would be somewhere in the 6% range.

In a 9 handed game that actually goes to a showdown I would guess that number of times BOTH players were bluffing without even a pair is less than 1%. It is hard to imagine calling as the second player with less than a pair, or if you have balls and raise, then having the original bluffer calling your bluff.

I'm not sure what he wants for a scenario exactly. If it's a nine-handed game, but goes heads-up to the flop, it will happen some of the time, usually because players will check down postflop. But if we take all nine hands to showdown, it will happen very, very rarely.

Quote: tringlomane

I'm not sure what he wants for a scenario exactly. If it's a nine-handed game, but goes heads-up to the flop, it will happen some of the time, usually because players will check down postflop. But if we take all nine hands to showdown, it will happen very, very rarely.

Right. I'm just looking for a math answer. 9 player game, 2 players see the flop, both check it down to showdown. 18 cards dealt, four cards left at the end of the hand, 5 cards on the board. What are the odds that neither player has a pair or better. If player A has any two cards and player B has any two cards, and neither players cards match the other players cards, then player A could match either of the two of his cards to 6 other cards to make a pair... etc.

Quote: SOOPOO

In a 9 handed game that actually goes to a showdown I would guess that number of times BOTH players were bluffing without even a pair is less than 1%. It is hard to imagine calling as the second player with less than a pair, or if you have balls and raise, then having the original bluffer calling your bluff.

Way higher than that.
I have AKo , you have AQs. I'm 68% to win vs your 28% to win with a 4% chance to tie.
We both go all in pre flop. Neither makes a pair or better.
Happens a lot more than once every hundred hands.

Quote: kissapig

Right. I'm just looking for a math answer. 9 player game, 2 players see the flop, both check it down to showdown. 18 cards dealt, four cards left at the end of the hand, 5 cards on the board. What are the odds that neither player has a pair or better. If player A has any two cards and player B has any two cards, and neither players cards match the other players cards, then player A could match either of the two of his cards to 6 other cards to make a pair... etc.

Well, an exact answer to the full question is impossible since it would require hand ranges that are influenced by individual players. And if we assume non-random starting hands, simulation is your best bet. Even with random hands, solving via combinatorics wouldn't be easy.

The other 7 players in the hand don't mean much except help shape the range of hands the other 2 players play.

If we assume the 2 players only play the top 20% of hands here, then both having high card will happen a little over 5% of the time:

From: http://www.propokertools.com/pql

PQL Query:
select count(winningHandType() = highcard)
from game="holdem", hand1="20%", hand2="20%"
Results:
Trials COUNT 1
600000 30844 (5.14%)

If two players would have a random hand instead, it's closer to 6%:

PQL Query:
select count(winningHandType() = highcard)
from game="holdem", hand1="**", hand2="**"
Results:
Trials COUNT 1
600000 35956 (5.99%)

If the whole table agrees to a family pot and check it down just for kicks, it's much more rare. It's a poor sample size, but they only cycle 600k hands per analysis for free. :(
PQL Query:
select count(winningHandType() = highcard)
from game="holdem", hand1="**", hand2="**", hand3="**", hand4="**", hand5="**", hand6="**", hand7="**", hand8="**", hand9="**"
Results:
Trials COUNT 1
600000 3 (0.0%)

So for the top twenty percent of the hands it would happen 5.99% of the time.
If the hand1="20%", hand2="20%" was changed to 100% and examined all
possible hands, would the percentage go up or down?

Quote: kissapig

So for the top twenty percent of the hands it would happen 5.99% of the time.
If the hand1="20%", hand2="20%" was changed to 100% and examined all
possible hands, would the percentage go up or down?

No, for random hands, a heads up showdown with both players having no pair, was 5.99%. With top 20% hands, it was 5.14%. So with tighter ranges, the chances go down slightly since pocket pairs make up a much larger portion of the players' ranges.

Quote: SOOPOO

In a 9 handed game that actually goes to a showdown I would guess that number of times BOTH players were bluffing without even a pair is less than 1%. It is hard to imagine calling as the second player with less than a pair, or if you have balls and raise, then having the original bluffer calling your bluff.

I think what you're not considering is table stakes. Someone short-stacked might easily shove pre-flop without a pair, and get called. On the turn, depending how much was in the pot, someone might shove on a draw, and get called.

Even if everyone's stacks are adequate, at a weak table, bet-call on the flop, then two streets checked down, or the turn checked and a post oak river bluff that's called, isn't an unusual pattern with two drawing hands in.

The point is, though, there's not a "math answer," because it's a human problem. The closest is the two players heads-up, but then the number of initial players doesn't matter, except that it makes them more likely to hold pockets, but again, that's a human problem. If what he's looking for is a game-wide shortcut to what odds he's getting with his AK on the river, well, that just doesn't exist. "Play the man, not the cards."