Poker Math¶
The handful of formulas that actually drive decisions. All of these are verified identities — the arithmetic checks out.
Pot odds & required equity¶
Pot odds = the price you're getting to call (pot : cost-of-call). To turn that into a decision:
Required equity to call = call / (pot + call)
Example: calling $10 into a $30 pot → 10/(30+10) = 25%. If your hand has ≥25% equity, calling is profitable. This is the threshold a bluff-catcher must clear.
MDF & Alpha (the defensive pair)¶
When you face a bet, how much of your range must you continue with so the bettor can't just print money bluffing?
MDF (Minimum Defense Frequency) = pot / (pot + bet) Alpha (required fold frequency) = bet / (pot + bet) MDF + Alpha = 1
Derivation: a 0%-equity bluff has EV = f·pot − (1−f)·bet. Set to 0 → fold freq f = bet/(pot+bet) = Alpha; so you must defend pot/(pot+bet) = MDF.
| Bet (into pot of 100) | MDF (you defend) | Alpha (you fold) |
|---|---|---|
| ½ pot (50) | 67% | 33% |
| ⅔ pot (66) | 60% | 40% |
| Pot (100) | 50% | 50% |
| 2× pot overbet (200) | 33% | 67% |
Bigger bets → you fold more. That's the mathematical engine behind overbetting with a polarized range.
MDF is a baseline, not a law
MDF assumes bluffs have zero equity and ignores rake and blockers. In practice you defend based on your hands' actual equity and removal — MDF is the floor that stops pure bluffs from auto-profiting, and a starting point you deviate from when you have reads.
Bluff-to-value ratio¶
A polarizing bettor includes enough bluffs to make the caller's bluff-catchers indifferent. Mechanically, the bluff share of the betting range = Alpha = bet/(pot+bet) — the same fold-equity threshold. On the river, by size:
| River bet | Value : Bluff |
|---|---|
| ½ pot | ~2 : 1 |
| Pot | ~1 : 1 |
| 2× pot | ~1 : 2 |
Bigger bets → more bluffs allowed. On earlier streets you can bluff more because your "bluffs" still have equity/outs to improve (they're not pure bluffs), so the mix is more bluff-heavy on the flop/turn and tightens toward the river as equity crystallizes.
These are the textbook ratios
The 2:1 / 1:1 / 1:2 figures are the standard consequence of Alpha — the Alpha/MDF math is exact, but real solver river ratios deviate slightly due to card removal and range composition. Use them as anchors, not gospel.
EV, equity, fold equity¶
- EV (Expected Value) — the probability-weighted average outcome; everything else is in service of making +EV decisions.
- Equity — your share of the pot = chance of winning given both ranges.
- Fold equity — extra EV from the chance your bet folds out a better hand. Total bet EV ≈ (fold equity) + (equity when called).
- Implied odds — money you expect to win later when you hit (improves a draw's price). Reverse implied odds — money you expect to lose later with a second-best hand.
Combinatorics¶
Counting combos is the basis of hand reading and bluff/value selection.
- Pocket pair = 6 combos · Unpaired = 16 (4 suited + 12 offsuit) · 1,326 total starting combos.
- Card removal (blockers) reduces combos multiplicatively:
- Unpaired = (copies of card A left) × (copies of card B left)
- Paired =
n(n−1)/2fornunseen copies - Examples: an Ace out → AQ =
3×4 = 12; holding KQ on K-T-4 → AK =4×2 = 8; TT on K-T-4 →(3×2)/2 = 3.
SPR (stack-to-pot ratio) & commitment¶
SPR = effective stack ÷ pot at the start of a street. The equity you need to profitably stack off rises with SPR as risk/(risk+pot):
| SPR | Equity to stack off | What can commit |
|---|---|---|
| 1 | ~33% | almost any pair / draw |
| 2 | ~40% | any top pair (+ most 2nd pairs, a few lower pairs) |
| 3 | ~43% | top pair, strong draws |
| 5 | higher | top pair gets dicey; OESDs become pure folds |
| 16 (deep) | — | even top pairs & nut flush draws are only indifferent |
So 3-bet pots (low SPR) play for stacks with one pair; deep/limped pots (high SPR) demand much stronger hands and reward implied odds + playability.
Exploitability & Nash distance¶
Nash Distance (dEV) measures how far a strategy is from equilibrium = the most EV the best counter-strategy could win off it (0 = perfect equilibrium). At equilibrium, every mixed action has equal EV (the indifference condition). - Reported as a % of pot → convertible to bb/hand (e.g., 0.3% of a 5.5bb pot ≈ 0.017 bb/hand exploitable). - Solvers converge to ~0.2–0.3% of pot — far more precise than any human. The EV-loss of one of your decisions is this same idea applied to a single action: how much you concede vs the best response. (It's exactly what the GTO Wizard trainer scores.)
Rake¶
Rake is paid by the player closing the action, which worsens the caller's pot odds (like virtually enlarging the bet). - Effect: the defender tightens (the BB calls less everywhere, shifting to folds and check-raises), while the aggressor can bluff more — e.g. an NL100 BTN-vs-BB flop where BTN bets trash 45% raked vs 36% unraked. - The rake cap limits this once pots grow large, so the distortion is biggest in small/marginal pots — which is why rake quietly tightens opening ranges and trims the profit of steals, light 3-bets, and thin pots.
Sources: Pot odds · MDF & Alpha · Combinatorics · Card removal · SPR · Nash distance · Rake