Facing Bets and Check-raising

Three questions, in this order

Everything so far was written from the bettor's chair. Most of your hands will be played from the other one. Facing a bet, ask three questions and keep them in order, because the cheap one settles a lot of spots before you need the expensive one.

1. What is the price? One division. Half pot is 25.0%, three-quarters 30.0%, a full pot 33.3%.
2. What is this hand worth against what is betting? Not against the nuts, and not against one hand you imagined.
3. Does raising beat calling? A separate question with a separate threshold, and it is the next two pages.

Answer the price question first and take the fold it hands you. Within 3 points of the line, treat the spot as genuinely close rather than forcing a confident answer.

Any street, heads up. In a multiway pot the price question still works, but question two needs the player count from the previous lessons.

The order matters because the answer can flip on the price alone. J♠T♥6♣5♣ on 9♥8♦2♣ — 8 outs, all 8 of them nut outs — is worth 28.0% against top set. Facing three-quarters of the pot it is short by 2.0 points, which is inside the band and genuinely close. Facing a full pot it is short by 5.3 and it is simply a fold. Same hand, same board, same opponent.

How often a pure bluff needs you to fold

Turn the table around. A bet that will lose every showdown risks its own size to win the pot. That is a break-even calculation, and it gives you the other half of the defending picture — how often you can fold before folding is what is costing you.

• Quarter pot: needs you to fold 20.0% of the time. You can continue with 80.0%.
• One-third: 25.0% / 75.0%. Half pot: 33.3% / 66.7%.
• Three-quarters: 42.9% / 57.1%. Full pot: 50.0% / 50.0%.

Use this arithmetic to notice when you are folding too much against small bets, and nothing more. It is a statement about a bet with no equity at all, not a target frequency to hit.

Only when the bettor's hand genuinely cannot win at showdown. That case is rare here: four cards on a coordinated board almost always retain something, which is exactly why we present this as arithmetic rather than as a strategy.

Apply it to a semi-bluff and it lies to you. J♠T♥6♣5♣ with 8 nut outs is worth 28.0% against top set — it does not need you to fold 50.0% of the time to make a pot-sized bet work, because it also wins some of the pots where you call. Judging that bet against the pure-bluff threshold declares a profitable bet unprofitable.

Both columns are arithmetic we do on this page, not an equilibrium frequency and not a solver output: the first is the break-even fold rate for a bet with zero equity, the second is one minus it. No solver was involved and none of it describes what anybody should actually do with a range.

Check-raising is an arithmetic commitment

Out of position, checking and raising is the strongest weapon this game gives you, and in pot-limit it is also the least ambiguous — because the size of it is not up to you.

Pot 100, he bets 100, and the largest raise available is to 400. He then owes 300 and is being asked for 33.3%, the same ceiling as always. But look at your side: if your stack was 400 — a stack-to-pot ratio of 4 — that single action was all of it.

Rule of the game: a raise may not exceed the pot after your call, which fixes the maximum at 400 here. In no-limit you would choose a raise size; in pot-limit the biggest one is computed for you.

The amounts come from js/core/plo-pot.js, the legal-raise calculator behind the pot-odds and SPR lessons, verified against three independent public rule pages (docs/M2-SOURCES.md, section T20).

At a ratio of 4 or below, decide whether to check-raise as if you were deciding whether to put the stack in, because the maximum raise is the stack.

Pot-limit, out of position, facing a pot-sized bet on the flop.

Deeper it is a different action with the same name. At a ratio of 13 the same raise to 400 leaves most of your stack behind — that is not a commitment, it is pot building, and the hand it wants is different. The hand that fits the committing version is one like J♠T♥7♦6♣: dominated share 3.4%, 54.9% against the range that would call, 20 outs and 14 nut outs. Our threshold for raising as an attack is that dominated share stays under 45%.

The claim we are not going to put a number on: that check-raising is correct far more often here than a hold'em player expects. We think that follows from the two things we did measure — draws in this game reach equities that make raising with them profitable, and pot-limit caps how badly a raise can be punished — but that is reasoning, not a measurement. We have never measured hold'em check-raise frequencies in this project, so we will not quote a comparison, and neither should anyone who cannot say how they measured it.

What to take away

• Price first. Facing one bet you never need more than 33.3%, and inside 3 points of the line nobody knows.
• Know what a pure bluff needs. Against a quarter pot you can keep 80.0% of your hands; against a full pot, 50.0%. Use it to catch over-folding, not as a target.
• The maximum check-raise is computed, not chosen. Into a pot-sized bet it is 400, which at a ratio of 4 is the whole stack.
• Raise with the hands that survive the call. Dominated share under 45%; the model hand was at 3.4%.

These are rules of this course, not an equilibrium solution. One claim on the previous page — how often check-raising is right compared with elsewhere — is explicitly reasoning rather than measurement, and carries no number for that reason.