Poker Odds Calculation Methodology & Simulation Whitepaper

1. Introduction & Theoretical Foundation

River Pairs was created to serve as a mathematically exact, transparent, and authoritative reference for Texas Hold’em heads-up probabilities.

In modern game theory and computer poker research, evaluating the strength of two starting hands requires determining the probability of each hand making the highest-ranking 5-card poker hand at the river across all possible community board run-outs.

This document describes the computational model, combinatorial algorithms, and mathematical definitions used to generate every equity, win rate, and showdown frequency displayed across our 14,365 matchup articles.

2. Combinatorial Deck Mathematics

Standard Texas Hold’em is played with a standard 52-card French deck:

4 suits: Spades ($\spadesuit$), Hearts ($\varpi$), Diamonds ($\diamondsuit$), Clubs ($\clubsuit$)
13 ranks: $2, 3, 4, 5, 6, 7, 8, 9, 10, \text{Jack}, \text{Queen}, \text{King}, \text{Ace}$

2.1 Starting Hand Categories (169 Hands)

Ignoring specific suits, any starting hand falls into one of three structural forms:

Pocket Pairs: $\binom{13}{1} = 13$ types (e.g., $AA, KK, \dots, 22$). Each has $\binom{4}{2} = 6$ unique suit combinations ($13 \times 6 = 78$ combos).
Suited Cards: $\binom{13}{2} = 78$ types (e.g., $AKs, 76s$). Each has 4 suit combinations ($78 \times 4 = 312$ combos).
Offsuit Cards: $\binom{13}{2} = 78$ types (e.g., $AKo, 76o$). Each has $4 \times 3 = 12$ suit combinations ($78 \times 12 = 936$ combos).

\[\text{Total Pre-Flop Combinations} = 78 + 312 + 936 = 1,326\text{ combos}\]

Compressing isomorphic suit permutations leaves exactly 169 canonical starting hand categories.

2.2 Heads-Up Matchup Universe (14,365 Matchups)

Comparing every canonical starting hand against another produces:

\[\frac{169 \times (169 + 1)}{2} = 14,365\text{ unique heads-up confrontations}\]

River Pairs catalogs complete data for all 14,365 matchups.

3. Board Enumeration & Card Removal Engine

3.1 Community Board Possibilities

Once four hole cards (2 for Hand 1, 2 for Hand 2) are dealt from a 52-card deck, 48 cards remain in the stub.

A complete Texas Hold’em board consists of 5 community cards (Flop: 3 cards, Turn: 1 card, River: 1 card). When evaluating non-overlapping hands, the total number of distinct 5-card boards that can be dealt is:

\[\binom{48}{5} = \frac{48 \times 47 \times 46 \times 45 \times 44}{5 \times 4 \times 3 \times 2 \times 1} = 1,712,304\text{ unique board run-outs}\]

For overlapping hands (e.g., $AA$ vs $AA$ where 4 Aces are removed), the simulation computes over all remaining cards in the deck, adjusting the denominator accordingly.

3.2 7-Card Hand Evaluation

For each of the $1,712,304$ possible boards, each player has a 7-card pool (2 hole cards + 5 community cards). The evaluator extracts the best 5-card subset:

\[\binom{7}{5} = 21\text{ combinations per player}\]

Each 5-card combination is classified into the standard 9 hand tiers:

Straight Flush (including Royal Flush)
Four of a Kind (Quads)
Full House (Boat)
Flush
Straight
Three of a Kind (Trips/Set)
Two Pair
One Pair
High Card

The absolute integer rank of Hand 1’s best 5-card hand is compared against Hand 2’s best 5-card hand:

If $\text{Rank}_1 > \text{Rank}_2 \implies \text{Hand 1 Win Counter} \gets \text{Win}_1 + 1$
If $\text{Rank}_2 > \text{Rank}_1 \implies \text{Hand 2 Win Counter} \gets \text{Win}_2 + 1$
If $\text{Rank}_1 = \text{Rank}_2 \implies \text{Tie Counter} \gets \text{Tie} + 1$

4. Equity and Probability Formulations

4.1 Win and Tie Probabilities

$\text{Win}_1\% = \left( \frac{\text{Wins}_1}{\text{Total Boards Evaluated}} \right) \times 100$ $\text{Win}_2\% = \left( \frac{\text{Wins}_2}{\text{Total Boards Evaluated}} \right) \times 100$ $\text{Tie}\% = \left( \frac{\text{Ties}}{\text{Total Boards Evaluated}} \right) \times 100$

In poker game theory, Equity represents a hand’s mathematical share of the pot at showdown. Because tied pots are split equally between both contestants:

$\text{Equity}_1 = \text{Win}_1\% + \left( \frac{\text{Tie}\%}{2} \right)$ $\text{Equity}_2 = \text{Win}_2\% + \left( \frac{\text{Tie}\%}{2} \right)$

\[\text{Equity}_1 + \text{Equity}_2 = 100.00\%\]

5. Limit Rating Models & Modern GTO Context

In addition to pure combinatorial equity, River Pairs incorporates classical and modern hand valuation frameworks:

5.1 David Sklansky’s Hand Groups

Classifies starting hands into Groups 1 through 8 based on expected win rate in Limit Texas Hold’em. Read our Sklansky Hand Groups Guide.

5.2 Bill Chen’s Formula

A numerical point system developed by Dr. Bill Chen and Jerrod Ankenman in The Mathematics of Poker. Evaluates high cards, pair multipliers, suitedness, and connector gaps. Read our Chen Formula Guide.

5.3 Phil Hellmuth’s 7 Tiers

Codified in Play Poker Like the Pros, Hellmuth’s tiers focus on tournament survival and minimizing post-flop reverse-implied odds. Read our Hellmuth Tiers Guide.

5.4 Expected Value (EV)

Quantifies the net expected profit or loss per hand:

\[EV = (P_{\text{win}} \times \text{Pot Won}) - (P_{\text{lose}} \times \text{Bet Risked})\]

Read our Expected Value Guide.

6. Verification & Data Integrity Standards

Deterministic Exhaustion: For all heads-up matchups, odds are determined by exhaustive combinatorial enumeration rather than random sample approximations, guaranteeing mathematical reproducibility.
Card Removal Fidelity: Blockers and suit conflicts (e.g. holding an Ace that prevents an opponent from making a nut flush) are fully accounted for on every iteration.
Open Access: All data is freely available for educational, strategic, and machine learning research.

For mathematical questions, verification inquiries, or data collaboration, contact Reece Williams at contact@riverpairs.com.