How We Compute the Odds
Every displayed win percentage and expected value comes from one deterministic probability engine. It does not simulate millions of hands and report a sample average. It starts with the actual remaining-shoe composition and evaluates the available actions analytically, using the documented approximations below.
Current model version: 2026.08.1 · Browser probability model · Released August 9, 2026
We derive a dealer-outcome distribution from the current shoe proportions (conditioned on the US peek rule), then evaluate standing, hitting, doubling, and splitting against that model. The process is deterministic—not Monte Carlo—but it is not a full card-removal recursion for every future draw.
The dealer's outcome distribution
The heart of the engine is a distribution for the dealer finishing at 17, 18, 19, 20, 21, or busting given the up-card. We compute it with memoized recursion over the dealer's hand state—total and whether it is soft. Draw weights start from the actual composition of the remaining shoe, with the unseen hole card added back, so the model responds to cards that have already left the shoe. Within that calculation, future draws use the starting proportions rather than decrementing the composition after every hypothetical card.
Crucially, the distribution is conditioned on the US peek rule: when the up-card is an ace or a ten, the dealer has already checked the hole card for blackjack, so we exclude the blackjack case and renormalize. This is why our dealer-bust numbers (a 6 busts 43.9%, an ace only ~20%) match the standard published figures for a peeked game.
Your expected value for each play
With the dealer distribution in hand, we evaluate every legal action as an expected value per dollar of your original bet:
- Stand — your total compared against the dealer distribution: win, push, or lose.
- Hit — draw one card from the shoe composition, then play the resulting hand optimally (the engine recursively chooses the better of hitting again or standing), so hit's value reflects correct continuation, not a single draw.
- Double — exactly one card, forced stand, at twice the stake (the EV is stated per original bet, which is why doubling can differ from hitting even at the same win rate).
- Split — valued as two independent post-split hands, with double-after-split allowed and split aces receiving one card each.
The live panel shows the modeled value of each action. Coaching uses a fresh-shoe baseline for the selected deck count and soft-17 rule, so a depleted shoe can occasionally make the live model prefer a different play. When that happens, the interface states the distinction instead of pretending the two concepts are identical.
The default game
Unless you change the settings, the trainer uses a common US shoe game: 6 decks, dealer hits soft 17, blackjack pays 3:2, double on any two cards, double after split, split to four hands, split aces get one card, no surrender. Settings change the shoe size, dealer behavior, payout, and probability inputs. The reference chart is the standard multi-deck H17/S17 card; rule-sensitive coaching is calculated separately from a fresh-shoe baseline.
The approximations — stated plainly
Two simplifications keep the model responsive enough to run in a browser:
- Fixed-proportion future draws. Within one EV calculation, future draw probabilities use the current shoe proportions without removing each hypothetical card. The approximation is usually small in a six-deck shoe but becomes more visible in short or heavily depleted shoes.
- Split valuation. A split is valued as two times one optimally played post-split hand, with double-after-split, but without a full tree of correlated hands and re-splits. Pair-hand values can therefore differ from a full rules-complete simulator.
Validation
Validation is based on named checkpoints, not on whether an output merely looks plausible. Each comparison must use the same deck count, soft-17 rule, peek rule, payout, and action assumptions. The current model is checked against the following published results:
- Insurance: an off-the-top 2:1 insurance wager has an expected return of −7.69% in the fixed-proportion baseline: four ten-valued ranks out of thirteen produce
(4/13 × 2) − (9/13) = −1/13. This is the standard insurance result discussed in Peter A. Griffin's The Theory of Blackjack. - Dealer 6: the model returns a dealer bust probability near 43.9% for a six-deck H17 game with a 6 showing. Wizard of Odds publishes 43.9144% for the same deck count and soft-17 rule in its dealer-bust table.
- Hard 16 vs 10: both hit and stand are close to −$0.54 per $1, with hitting only marginally less negative in the baseline model. The order and scale agree with Wizard of Odds' published action EV tables; exact decimals move with composition and rules.
- Rule-sensitive strategy: representative decisions and expected values are cross-checked against Wizard of Odds' deck- and rule-specific EV tables and Don Schlesinger's Blackjack Attack: Playing the Pros' Way.
- Counting foundation: the direction of count effects—high-card-rich shoes improve blackjacks, doubles, and insurance—is checked against Edward O. Thorp's Beat the Dealer and Griffin's treatment of card removal and betting effects. These sources validate the mathematical framework; they do not turn this browser model's documented approximations into full finite-deck recursion.
A checkpoint passes only when the rule assumptions match and the result agrees within the tolerance expected from the fixed-proportion and split approximations. If a result does not reconcile, send the hand, rule set, expected result, and source so it can be reproduced and corrected.
Model changelog
- 2026.08.1 — August 9, 2026: documented fixed-proportion future draws and approximate split valuation; separated live depleted-shoe estimates from fresh-shoe strategy coaching; added named validation checkpoints.
- 2026.06.1 — June 17, 2026: initial published browser model and methodology for dealer outcomes and action EV.
References and validation sources
- Wizard of Odds — Blackjack, including rule effects, strategy calculators, and appendices.
- Don Schlesinger, Blackjack Attack: Playing the Pros' Way, 3rd edition.
- Peter A. Griffin, The Theory of Blackjack, 6th edition.
- Edward O. Thorp, Beat the Dealer.
- G. Van Brummelen, “Finding Blackjack's Optimal Strategy in Real-time and Player's Expected Win”, an exact finite-deck dynamic-programming treatment.
References are used as validation targets and background. Figures labeled as ours are outputs of our model under the stated rules, not copied tables — and those outputs are published in full as free, machine-readable open datasets (CSV and JSON, CC BY 4.0) so anyone can check or reuse them.