Methodology

The datasets

• Preflop hand equity — all 169 starting hand classes simulated against random opponents at every player count from 2 to 9.
• Hand-type frequency — how often each poker hand category (pair through straight flush) shows up at a table, both assuming everyone plays to the river and under a more realistic folding model.
• Positional profitability — simulated bb/100 win rate by seat, showing how much of poker’s “position edge” comes from range selection versus postflop advantage.
• Postflop outs — not a dataset. Given a number of outs and how many cards are still to come, the chance of hitting is exact combinatorics over the unseen cards, so the page computes it directly rather than looking it up.
• Catan resource odds — also not a dataset. Given the numbers and resources touching your settlements and cities, production probability is closed-form two-dice math, computed directly as you build the board.
• Monopoly square odds — the long-run probability of landing on each of the 40 board squares, solved exactly as the stationary distribution of a Markov chain over dice rolls, doubles, Jail, and the Chance/Community Chest decks (not simulated, and not closed-form enough to compute live, so it ships as a small precomputed dataset like the poker and blackjack figures).
• Scrabble tile and rack odds — a 7-tile rack drawn from a 100-tile bag is a multivariate hypergeometric problem, so every figure is a closed form or a small exact count over all 16,007,560,800 possible racks — nothing is dealt. The tile distribution itself was not written from memory: it was taken from two independent published datasets and cross-checked letter by letter, and they agree on every count and face value. These pages are about what comes out of the bag, not about which words are legal, so no dictionary is involved.
• Full-game-optimal Yahtzee — the expected value of the rest of a 13-round game from every one of 536,448 reachable scorecard states, solved exactly by backward induction, not simulated. A full per-roll decision table for every state would run to roughly half a gigabyte, so only that compact expected-value table ships; the actual keep-or-reroll and category recommendation for whatever you just rolled is worked out live in your browser from the table, the same closed-form arithmetic the keep-or-reroll guide above already does for one category at a time. Optimal play averages 254.6 points a game, within 0.01 of the figure most commonly cited for this exact problem (Verhoeff, 1999).
• Tennis — an exact model of the scoring system in which every point is an independent trial at a fixed serve strength, and every loop (deuce, a tiebreak’s win-by-two stage, an advantage set) closes in a formula. The format, deuce and point-importance figures ship as static data. The chance of winning from a particular score is the exception: there are far too many scores to precompute for every format and strength, so the one score you set is solved live in your browser, by the same engine the offline checks run against. Match-level importance is checked against the figures in Klaassen and Magnus (2001), a study of 86,298 points at Wimbledon.

Frequently asked questions

How are the odds calculated?

It depends on the game, and the difference matters. The poker equity figures come from Monte Carlo simulation run offline: millions of random hands are dealt and evaluated with a standard hand evaluator, and the resulting win/tie/loss rates are stored as static data. Everything else is exact rather than sampled — blackjack from a recursive expected-value engine, dice and Catan from straight combinatorics, Monopoly from the stationary distribution of a Markov chain over the board, Scrabble from hypergeometric tile-draw counts, tennis from an exact chain over the scoring system in which every loop — deuce, the win-by-two stage of a tiebreak, an advantage set — closes in a formula. Either way the work happens offline, so results load instantly and stay consistent between visits.

Which Scrabble dictionary do the word figures use?

ENABLE, and every Scrabble page says so. It matters more than it sounds: a bingo rate or a word list is meaningless without naming the lexicon behind it, and the three in common use genuinely differ. ENABLE is public domain, which is why it is the one that can ship here. It holds 96 two-letter words against NWL2023's 107 and Collins CSW24's 127. The 11 that NWL has and ENABLE does not are DA, EW, FE, GI, KI, OI, PO, QI, TE, ZA and OK — two of them QI and ZA, which is why "C, Q, V, Z appear in no two-letter word" is true of the figures here and false at a rated game, where the dead letters are C, V under NWL and just V under Collins. Pages making claims like that name the list they describe rather than implying all three agree.

Why is this more than "beat the odds"?

The goal is to make the underlying math visible and explorable, the way a stats-explainer site would -- not to suggest a way to gain an edge over a casino or another site. Poker outcomes are governed by probability, and that probability is what this site visualizes.

What is the Chen Formula?

The Chen Formula is a well-known quick-scoring heuristic (by Bill Chen) for estimating a starting hand’s strength from its two hole cards. Odds The Law uses it as a simplified decision rule to model realistic folding behavior and positional ranges in a couple of the simulated datasets -- it is not the source of the equity numbers themselves, which come from full Monte Carlo simulation.

How accurate are the numbers?

Preflop equity figures are generated from at least 50,000 simulated hands per starting-hand-class-and-player-count combination (more for pairs and commonly searched hands), which keeps sampling error well under a percentage point for most figures. The generation scripts run automated sanity checks (e.g. win/tie/loss percentages summing to ~100%) before any data ships, and results are cross-checked against published reference values where they exist. The blackjack, dice, Catan, Monopoly, Scrabble and tennis figures are computed exactly, so they carry no sampling error at all — only the modelling assumptions stated alongside each tool.