The Second Best Hand

Non-nut is not a verdict

"If it is not the nuts it is a bluff-catcher" is the single most expensive habit anybody brings to this game. It is imported from hold'em, where the ranking of made hands really is that flat. The fix is not a nuance — it is a missing argument. The strength of a non-nut hand is a function of how many people are in the pot, and the function is steep.

Same river K♥8♥3♥2♠7♦ as the previous lesson, opponents restricted to hands with two pair or better:

• Q♥T♥A♠4♣ — rank 11: 92.5% against one opponent, 68.1% against four.
• 6♥4♥A♠9♣ — rank 41, near the bottom of the flushes: 72.5% · 49.9% · … · 17.7%.
• Q♥A♥2♣2♦ — rank 1: 100%, and it does not care how many people are here.

Grade a non-nut hand with the player count attached. Heads up, the eleventh-best hand on the board is a value bet; four-handed, the forty-first is a fold to serious action.

River, against opponents who are actually betting. Against passive opponents you are being shown a narrower set of hands than these numbers assume, which moves the answer further in your favour, not less.

The trap runs the other way too. Rank 11 is still 68.1% against four opponents — above the 33.3% you would need to call a pot-sized bet — so "multiway, so I fold my second-best flush" is throwing away a profitable call. The rule that fails is not "be tight" or "be loose"; it is any rule that answers without knowing how many people are left.

Hitting your draw is how you get to be second best

The second-best hand is usually not a hand you were dealt. It is a draw that came in. So the number that matters for a draw is not only how often it arrives, but how often it arrives and is still the best hand. Enumerated over all 990 run-outs on 9♥8♦2♣:

• J♠T♥7♦6♣ (20/14): finishes with a straight or better 69.7% of the time; of those times it is still the absolute best hand 35.5%.
• J♠T♥6♣5♣ (8/8): arrives only 37.1% of the time, but when it does it is the nuts 46.6%.
• 7♣6♥5♦4♠ (13/3): the worst of the three at 11.3%.

Ask two things of a draw: how often it comes in, and how often coming in is enough. The second question is the one that decides how much money you should be putting in on the flop.

Flop, heads up, unpaired board with three suits. The quantity is defined as a conditional — given that you got there — so it says nothing about hands that never arrive.

More outs, less safety. The 8-out draw stays best 46.6% of the times it hits and the 20-out one only 35.5%, because many of the big wrap's outs make the nuts on the turn and get overtaken on the river. If you rank draws by out count you will consistently overpay for exactly the hands that turn into second best.

Two pair and sets: ahead is not the same as safe

Now the made hands. Flop 9♥8♦2♣, nut shift 77.6% — the most dynamic board in this camp. Against the stack-off proxy range — 800 sampled hands that already have two pair or better here, or a big draw of their own:

• 9♣9♠A♦K♥, top set: 76.7%.
• 8♣8♠A♣K♦, middle set: 65.8%.
• 9♦8♥Q♦J♥, top two pair: 63.8%.

All three are ahead. None of them is ahead enough to slow down: the biggest wrap on this board is worth 54.9% against the same range, which means a large slice of what you are "beating" is a hand that will pass you on a large fraction of run-outs.

On a dynamic board, a made hand that is ahead but not the nuts is worth money now and less money later. Put the chips in while the number is still 76.7%.

Flop, heads up, dynamic texture. On a static board the same hands keep their value and there is no hurry; the urgency comes from the board, not from the hand.

Rank them by category and you get the order wrong in practice: top two pair at 63.8% and middle set at 65.8% are almost the same number, while "a set beats two pair" suggests a gap. The gap that matters on this board is between all of them and the 20/14 draw at 54.9%, which is much closer to them than a category label implies.

One naming discipline, because it is the difference between a measurement and an invention: that is a stack-off proxy range, not "the opponent's range". This project has no postflop range data at all. The proxy is a crude, completely reproducible stand-in, and calling it anything grander would be describing data we do not have.

What to take away

• "Non-nut" needs a player count. The same flush wins 72.5% against one opponent and 17.7% against four.
• A draw has two numbers. How often it arrives, and how often arriving is enough: 46.6% for the small all-nut draw, 35.5% for the big wrap.
• Ahead is not safe. Top set 76.7%, top two pair 63.8%, and the draw chasing them 54.9% — all on one flop.
• Second best is a position you arrive at, not a hand you are dealt. Most of the money lost in this game is lost by hands that were winning at some point.

Every number here is an enumeration or a seeded sample from our own engine, and every threshold is a rule of this course, not an equilibrium solution. Next: the same board from the other chair — what to do when somebody bets at you.