How we compute the numbers
Almost everything on this site is calculated exactly, not estimated. Here is the method behind each game, and what we check it against.
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Three kinds of numbers
- Exact enumeration We list every possible deal, add up what each one pays, and divide. Nothing is sampled, so the result is exact; the only rounding is in how many decimals we show.
- Combinatorial analysis Exact probability math that doesn’t need to list every deal one by one: closed-form formulas, recursions over the cards left in the shoe, small Markov chains. Also exact.
- Seeded simulation When exact counting would take too long, we deal a very large number of rounds from a fixed random seed, so anyone rerunning it gets the same result, and we report a 95% confidence interval.
Only two things on the site are simulated: the pai gow win, push and loss rates, and the betting system simulator. Everything else is exact. Every engine has automated tests that compare its results with the published figures below. The engines are open source under the MIT license, so anyone can check or reuse the math: github.com/casinogamessimulator/engines.
| Game | Method | Checked against |
|---|---|---|
| Video poker | Exact enumeration | Returns for 5 pay tables, to 6 decimals |
| Blackjack | Combinatorial analysis | House edge for 22 rule sets, dealer bust odds, strategy charts |
| Roulette | Combinatorial analysis | Edge of every bet and zero rule |
| Baccarat | Exact enumeration | Outcome counts (exact match) and bet edges |
| Craps | Combinatorial analysis | Edge of every bet |
| Pai gow poker | Exact enumeration + seeded simulation | Hand counts (exact match), house edge inside our 95% interval |
| Hold’em vs. Dealer | Exact enumeration | House edge, element of risk, trips pay tables |
| 3-Card vs. Dealer | Exact enumeration | 9 ante bonus and 16 pair plus tables, outcome counts |
| Betting systems | Seeded simulation | The exact house edge, shown next to every result |
Video poker
Exact enumeration Each pay table’s return and hand frequencies come from playing all 2,598,960 possible deals with each of the 32 ways to hold them, and counting every possible draw exactly (by inclusion–exclusion, not by sampling). The Easy and Best strategy charts are scored the same way, over all 2,598,960 deals. In the game and the hand analyzer, your browser does the same for the hand on screen: all 32 holds against every possible draw, at the coins you actually bet.
Checked against Wizard of Odds’ published returns at the 5-coin bet, to six decimals: Jacks or Better 9/6 99.5439%, Jacks or Better 8/5 97.2984%, Bonus Poker 8/5 99.1660%, Double Bonus 10/7 100.1725%, Double Double Bonus 9/6 98.9808%. Bonus Poker Deluxe 9/6 (99.6417%) is computed the same way but isn’t on that list of published checks.
Video poker strategyStrategy cardHand analyzer
Blackjack
Combinatorial analysis The dealer’s chances of ending on 17 to 21 or busting are computed exactly for the real shoe, with the right number of decks (not an infinite-deck shortcut). The value of every hit, stand, double, split and surrender is then found by exact recursion over the cards left. The house edge we show is for the basic strategy chart, the best play that depends only on your total; we also compute perfect, card-by-card play. Common rule sets are computed in advance; any other rules you pick are computed in your browser in a second or two.
Checked against Wizard of Odds’ blackjack house edge calculator for 22 rule sets, all within 0.005 percentage points (for example, 6 decks, dealer hits soft 17, double after split, late surrender: 0.53%), against its tables of the dealer’s bust chances, and against its strategy charts.
What we approximate, and how much it matters:
- After a split, each hand is valued without the cards dealt to the other split hands, and resplits use an expected count. This moves the house edge by thousandths of a percent.
- House edges assume a full shoe every round, like a continuous shuffler. The effect of the cut card isn’t included.
- The game deals from a shoe until the cut card, but the trainer’s values use a full shoe minus the cards in the current round. They don’t track cards from earlier rounds, so they aren’t card-counting advice.
We also cross-check the edges with a seeded simulation of a million rounds, but no simulated figure is shown on the site.
Basic strategy chartsHouse edge calculator
Roulette
Combinatorial analysis Every pocket is equally likely, so each bet’s chance is the numbers it covers divided by the pockets on the wheel, and its edge follows from what it pays. La partage and the en prison variants have exact formulas too. The trainer and the odds page use the same functions.
Checked against Wizard of Odds’ roulette pages: 2.70% on a single-zero wheel, 5.26% on double zero, 7.69% on triple zero, 7.89% for the double-zero five-number bet, 1.35% with la partage and 1.39% or 1.37% with the en prison variants.
Roulette oddsBetting system simulator
Baccarat
Exact enumeration Every coup is counted as an ordered draw of six cards from the shoe: 4,998,398,275,503,360 of them with 8 decks. Each one is played out with the drawing rules, for 8, 6 and 1 deck.
Checked against Wizard of Odds’ baccarat appendix: the counts of banker wins, player wins and ties match exactly, and so do the edges: banker 1.06%, player 1.24%, tie at 8:1 14.36% with 8 decks. The odds in the game panel are for a full shoe; they don’t change as cards are dealt.
Craps
Combinatorial analysis Each bet is a small Markov chain over the 36 rolls of two dice: a roll wins it, loses it, moves a line bet to a point, or leaves it alone. Solving the chain gives the exact edge per bet resolved, per roll, and excluding pushes. The trainer solves the same chains live for the bets on your table.
Checked against Wizard of Odds’ craps appendix: pass line 1.41% (7/495), don’t pass 1.36% (3/220), place 6 and 8 1.52%, field 2.78% (paying double on 2 and triple on 12), any seven 16.67%, and the other standard bets.
Pai gow poker
Exact enumeration The chances of each seven-card hand come from all 154,143,080 hands in a 53-card deck with the joker, and match Wizard of Odds’ “Probabilities in Pai Gow Poker” exactly.
Seeded simulation The win, push and loss rates and the house edge need every player hand against every dealer hand, about 8.25 × 1015 deals, so we simulate instead: 400 million rounds from seed 20261019, with 95% confidence intervals. Playing the house way, the house edge is 2.72% (95% interval 2.713% to 2.728%), which contains the exact 2.7212% published by Wizard of Odds. The house way is the Foxwoods house way, as Wizard of Odds publishes it.
The trainer’s best setting scores each way to set your hand against the exact distribution of the dealer’s hands, but it ignores that your seven cards can’t be in the dealer’s hand. That makes it slightly worse than truly perfect play: 2.55% against the published 2.51%.
Hold’em vs. Dealer
Exact enumeration The house edge, element of risk and preflop table come from every starting hand, flop, turn, river and dealer hand (the 1,755 flops that differ up to suits, each weighted by how often it happens). The trips side bet is counted over all 133,784,560 seven-card hands. In the trainer, your browser enumerates every remaining turn, river and dealer hand on the flop, and every dealer hand on the river.
Checked against Wizard of Odds’ page on the game: a house edge of 2.185% of the ante with optimal play and an element of risk of 0.526%, and the four trips pay tables. For Wizard of Odds’ simple strategy, it quotes a 2.43% house edge and a 0.58% element of risk, and enumerating that strategy exactly gives 2.433% and 0.584%. Where its wording leaves room, we read “two pair or better” on the flop literally (trips on the board count) and a “hidden pair or better” at the river as one that improves on the board.
3-Card vs. Dealer
Exact enumeration Every player hand against every dealer hand from the other 49 cards: 22,100 × 18,424 = 407,170,400 deals. That gives the house edge of each pay table, the chance of each outcome and the value of every hand, including the queen-6-4 rule.
Checked against Wizard of Odds’ page on the game: all 9 ante bonus tables and all 16 pair plus tables within 0.01 percentage points (ante bonus 5/4/1: 3.37% house edge, 2.01% element of risk; pair plus 1/3/6/30/40: 7.28%), and all 17 outcome counts exactly.
Betting system simulator
Seeded simulation It plays whole sessions of a system (Martingale, Paroli, Fibonacci, D’Alembert, Labouchère or flat betting) on a real wheel or a real eight-deck shoe: 10,000 sessions by default, or 1,000 to 50,000. Each run has a seed, shown on the page and kept in the link you share, so the same link always replays the same sessions.
It doesn’t show confidence intervals. Instead, the exact house edge of the bet appears next to the simulated loss per chip wagered, so you can see how close the run came. Our tests require a million flat bets to land within 0.4 percentage points of the exact edge.
Shuffles and dice
Cards and dice come from your browser’s cryptographic random number generator. Video poker, roulette, baccarat, craps and pai gow use it directly; blackjack (one seed per shoe), Hold’em vs. Dealer and 3-Card vs. Dealer (one seed per round) use it to seed a fast generator. Nothing is weighted or adjusted, and since the chips are virtual, nobody gains anything either way.
Our source
The published figures we check against come from Wizard of Odds, Michael Shackleford’s long-running reference on casino game math. We aren’t affiliated with it. When our figure differs from a published one, as with the simple Hold’em strategy, we say so on the page.
Spot something that looks wrong? Please tell us at admin@casinogamessimulator.com. More about the site on our About page.