poker_math self-test: 37/37 checks passed
ALL CHECKS PASSED50 Calculations & Verification
Every quantitative claim in this book — pot odds, minimum defense frequency, optimal bluff ratios, combinatorics, expected value, variance, and ICM — is backed by a small, tested calculation library rather than hand-typed numbers. This appendix executes that library live when the book is rendered, so the tables below are computed, not transcribed, and a self-test asserts the key identities every time. If a formula were wrong, this page would fail to build.
The source lives alongside the book in calc/poker_math.py, with a self-test in calc/verify.py. The functions are pure standard-library plus NumPy.
Self-test
The library ships with assertions for the identities the book relies on. This runs them at render time:
50.1 Pot odds → required equity
Facing a bet of a given fraction of the pot, the equity you need to break even on a call (required_equity = bet / (pot + 2·bet) when the bet is a fraction of a 1-unit pot):
| Bet size | Bet (×pot) | Equity needed to call | Pot odds |
|---|---|---|---|
| 1/4 pot | 0.25 | 16.7% | 5.00 : 1 |
| 1/3 pot | 0.33 | 20.0% | 4.00 : 1 |
| 1/2 pot | 0.50 | 25.0% | 3.00 : 1 |
| 2/3 pot | 0.67 | 28.6% | 2.50 : 1 |
| 3/4 pot | 0.75 | 30.0% | 2.33 : 1 |
| pot | 1.00 | 33.3% | 2.00 : 1 |
| 1.5× pot | 1.50 | 37.5% | 1.67 : 1 |
| 2× pot (overbet) | 2.00 | 40.0% | 1.50 : 1 |
50.2 Minimum Defense Frequency (MDF) and alpha
Versus a bet of a given size, the fraction of your range you must continue with so the bettor cannot profit by bluffing any two cards: MDF = pot / (pot + bet), and alpha = 1 − MDF is the fraction you may fold (and the success rate a pure bluff needs).
| Bet size | MDF (defend ≥) | Alpha (may fold) |
|---|---|---|
| 1/4 pot | 80.0% | 20.0% |
| 1/3 pot | 75.0% | 25.0% |
| 1/2 pot | 66.7% | 33.3% |
| 2/3 pot | 60.0% | 40.0% |
| 3/4 pot | 57.1% | 42.9% |
| pot | 50.0% | 50.0% |
| 1.5× pot | 40.0% | 60.0% |
| 2× pot (overbet) | 33.3% | 66.7% |
50.3 Polarized betting: value-to-bluff ratio and bluff frequency
For a polarized bet, the value:bluff ratio that makes a bluff-catcher indifferent, and the resulting fraction of the betting range that should be bluffs: value:bluff = (pot + bet) : bet and bluff fraction = bet / (pot + 2·bet).
| Bet size | Value : bluff | Bluffs (% of betting range) |
|---|---|---|
| 1/4 pot | 5.00 : 1 | 16.7% |
| 1/3 pot | 4.00 : 1 | 20.0% |
| 1/2 pot | 3.00 : 1 | 25.0% |
| 2/3 pot | 2.50 : 1 | 28.6% |
| 3/4 pot | 2.33 : 1 | 30.0% |
| pot | 2.00 : 1 | 33.3% |
| 1.5× pot | 1.67 : 1 | 37.5% |
| 2× pot (overbet) | 1.50 : 1 | 40.0% |
NoteThe pot-bet anchor
A pot-sized bet wants 2 : 1 value-to-bluff (one third bluffs) and lets the defender fold half their range (MDF 50%). Memorize the pot-bet row; everything else is “more bluffs / less folding as the bet shrinks, fewer bluffs / more folding as it grows.”
50.4 Drawing odds
The rule of 2 and 4 versus the exact probability of completing common draws (cards counted from the unseen deck):
| Draw | Outs | 1 card (rule / exact) | 2 cards (rule / exact) |
|---|---|---|---|
| Gutshot | 4 | 8% / 8.7% | 16% / 16.5% |
| Flush draw | 9 | 18% / 19.6% | 36% / 35.0% |
| Open-ended | 8 | 16% / 17.4% | 32% / 31.5% |
| Flush + gutshot | 12 | 24% / 26.1% | 48% / 45.0% |
| Two overcards | 6 | 12% / 13.0% | 24% / 24.1% |
50.5 Combinatorics & blockers
Combo counts for representative holdings, and how blockers on the board remove combinations:
| Holding | Combos | Removal |
|---|---|---|
| Pocket pair (e.g. QQ) | 6 | — |
| Suited (e.g. AKs) | 4 | — |
| Offsuit (e.g. AKo) | 12 | — |
| Any AK | 16 | — |
| AK with an ace on board | 12 | 16 → 12 |
| AKs with the A♥ visible | 3 | 4 → 3 |
| A set (e.g. 77 with a 7 out) | 3 | 6 → 3 |
50.6 Variance, confidence & risk of ruin
A 95% confidence interval for a measured win rate, and how many hands are needed before the lower bound clears zero — the reason small samples prove nothing:
| Hands | 95% CI on a 5 bb/100 win rate | Lower bound > 0? |
|---|---|---|
| 10,000 | -14.6 to +24.6 bb/100 | no |
| 50,000 | -3.8 to +13.8 bb/100 | no |
| 100,000 | -1.2 to +11.2 bb/100 | no |
| 500,000 | +2.2 to +7.8 bb/100 | yes |
| 1,000,000 | +3.0 to +7.0 bb/100 | yes |
Hands to 'prove' a 5 bb/100 win rate (SD 100): 153,664
Risk of ruin at 1000 bb bankroll: 36.79%
Risk of ruin at 2000 bb bankroll: 13.53%
Risk of ruin at 4000 bb bankroll: 1.83%50.7 ICM (Malmuth–Harville)
Dollar equities for a three-handed final table, showing that a chip is not a dollar — the chip leader’s equity is less than their chip share, and the short stacks’ equity is more:
| Stack | Chip share | ICM $ equity | Equity share |
|---|---|---|---|
| 5,000 | 50.0% | $38.39 | 38.4% |
| 3,000 | 30.0% | $32.75 | 32.8% |
| 2,000 | 20.0% | $28.86 | 28.9% |
TipWhy this appendix exists
Numbers in a strategy book are only as trustworthy as the arithmetic behind them. By generating the tables above from a tested module, the figures you study are guaranteed to be internally consistent with the formulas in the text — and any future edit that breaks the math breaks the build instead of silently shipping a wrong number.