How OddsOfBlackjack turns the exact remaining shoe into a mathematical decision.
OOB evaluates the current blackjack state using the exact remaining card composition,
game rules, player hand and dealer upcard — then calculates the Expected Value
of each available action.
Remaining shoe + player hand + dealer upcard + rules
Action EV
Hit · Stand · Double · Split · Surrender
Deterministic
Same state → same result
No Monte Carlo Core
Deterministic recursive evaluation
The mathematical state
The decision starts with the complete current state.
OOB evaluates each decision from the exact combination of remaining cards, player
hand, dealer upcard and active game rules.
Shoe
Exact Remaining Composition
The engine tracks how many cards of each rank remain in the shoe after every exposed card.
A · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10/J/Q/K
Player
Current Player Hand
The player's cards, hand type and current decision context determine which actions are available for evaluation.
Dealer
Dealer Upcard
The visible dealer card affects the probability distribution of future dealer outcomes and therefore the EV of player actions.
Rules
Active Game Configuration
Deck count, S17/H17, Peek or ENHC, doubling, splitting, surrender and payout rules are part of the state being evaluated.
A count is not the full state.
Different remaining-shoe compositions can map to the same Hi-Lo True Count while
still producing different card probabilities, action EVs and, in some cases,
different highest-EV decisions.
This is why OOB evaluates the exact remaining composition rather than using True
Count as the sole decision input.
From state to probability
Every possible next card is weighted by what actually remains in the shoe.
OOB calculates card probabilities directly from the current remaining composition.
As cards are removed, those probabilities change with the state.
P(r|S)=nrN
Where nr
is the number of remaining cards of rank r,
and N is the total number of cards remaining
in the shoe.
Count what remains
Exact Rank Counts
Each card rank is represented by its current remaining count in the shoe.
Weight the branch
Current Probability
The probability of drawing a rank is determined by its remaining count relative to the total cards left.
Update the state
Without Replacement
When a card is exposed, it is removed from the current composition and the probabilities for the next state are recalculated.
The probabilities are state-dependent.
For example, a rank with 32 cards remaining earlier in the shoe may later have
only 18. OOB uses the current composition at the moment of the decision, not the
starting-shoe distribution.
This changing probability distribution is the foundation for the action-EV
calculations that follow.
From probability to Expected Value
OOB compares actions by the value of their possible future outcomes.
For each available action, OOB evaluates the possible outcomes from the current
state, weights them by their probabilities and combines them into an Expected Value.
EV(a,S)=ΣiP(i|a,S)·Vi
Where a is an available action,
S is the current game state,
P(i | a, S)
is the probability of outcome i, and
Vi
is the value of that outcome.
Enumerate
Possible Outcomes
OOB evaluates the future outcomes that can follow from the selected action under the current shoe composition and rules.
Weight
Outcome Probabilities
Each outcome contributes according to its probability under the exact current state.
Compare
Highest-EV Action
The available actions are compared using their unrounded internal EV values, and the action with the highest Expected Value is selected.
Expected Value is not the same as the next result.
A higher-EV action can still lose the next hand. EV describes the mathematical
expectation across possible outcomes, not a guaranteed short-term result.
The same framework is applied to the actions available in the current state: Hit,
Stand, Double, Split and Surrender.
How each action is evaluated
Each action follows a different path through the same current state.
OOB evaluates only the actions available under the current rules and hand state, then
compares their Expected Values using the exact remaining shoe.
Stand
Hold the Current Hand
Stand EV is determined by comparing the player's current hand against the probability distribution of the dealer's possible final outcomes.
Hit
Draw and Re-Evaluate
For every possible next card, OOB updates the remaining shoe and evaluates the resulting player state, including further decisions where applicable.
Double
One Card, Additional Wager
Double EV evaluates the one-card draw constraint together with the additional wager, using the exact current-state probabilities and applicable rule restrictions.
Split
Create Multiple Dependent Hands
Split EV evaluates the resulting hands while accounting for card depletion, wager changes and the active rules for resplitting, split aces and doubling after split.
Surrender
Accept the Defined Loss
When surrender is available, its fixed outcome is evaluated against the Expected Values of continuing the hand under the applicable surrender rules.
Only available actions are compared.
The available action set depends on the current hand and rule configuration. OOB
does not compare actions that are unavailable under the selected game rules.
The highest-EV action is selected from the actions available in that exact state.
Recursive enumeration & determinism
Each possible card creates a new state. OOB evaluates the branches and works backward to the current decision.
Starting from the exact current shoe, OOB evaluates possible future card outcomes
recursively until the relevant terminal outcomes are reached, then combines those
branches into action EVs.
Branch
Evaluate Every Possible Next Rank
Each rank still present in the shoe creates a possible next state, weighted by its current probability.
Recurse
Continue Through Future States
When a branch requires another decision or dealer draw, the updated shoe state is evaluated again using the same probability and EV framework.
Resolve
Work Back to the Current EV
Terminal outcomes are probability-weighted back through the decision tree until the Expected Value of the current action is determined.
Deterministic by design.
The same exact shoe composition, player hand, dealer upcard and rule
configuration produce the same calculated action EVs and highest-EV decision.
Core decision calculations do not depend on random sampling or repeated Monte Carlo
trials.
Dealer outcome model
Dealer probabilities are calculated from the same exact remaining shoe.
OOB evaluates the dealer's possible future draws using the active table rules and the
cards that actually remain in the shoe.
Draw rule
S17 or H17
Dealer drawing behavior follows the selected rule configuration, including whether the dealer stands or hits on soft 17.
Outcomes
Bust, 17, 18, 19, 20 or 21
Possible dealer terminal outcomes are evaluated from the current composition and weighted by their probability.
Blackjack
Dealer Blackjack Treatment
Dealer blackjack is handled according to the selected hole-card rule and the applicable wager treatment for the current game.
Hole card
US Peek or ENHC
OOB applies the selected Peek or European No-Hole-Card rule when evaluating dealer outcomes and player-action EVs.
The dealer model changes with the shoe.
Because dealer draws come from the same depleted shoe, removing cards changes the
probabilities of dealer busts, final totals and blackjack outcomes.
These dealer-outcome probabilities feed directly into the EV calculations for the
player's available actions.
Rule-sensitive decisions
Some decisions depend on more than the cards alone.
Split, surrender and dealer-blackjack treatment can change with the selected rules,
so OOB evaluates them within the exact rule configuration of the current game.
Split logic
Multiple Hands, One Depleting Shoe
Split EV accounts for the cards consumed by resulting hands, additional wagers and the active rules for resplitting, split aces and doubling after split.
Ace rules
Split Aces Are Evaluated Separately
Resplitting aces and hitting split aces are applied according to the selected configuration because these restrictions change the available future actions.
Surrender
Availability and Timing Matter
When surrender is supported by the selected game, OOB evaluates its defined outcome against the Expected Values of continuing the hand.
Blackjack treatment
Peek and ENHC Are Not Interchangeable
OOB applies the selected hole-card rule when evaluating dealer blackjack and the treatment of player wagers under that rule set.
The rule configuration is part of the calculation.
The same player cards and remaining shoe can require different evaluation logic
when table rules change. OOB therefore evaluates decisions using the rules
configured for that game state.
Rule-sensitive calculations are included in the reproducibility and validation
framework.
From static strategy to exact composition
Basic Strategy, True Count and exact composition answer different questions.
Basic Strategy provides a rule-specific baseline. Hi-Lo summarizes card removal into
a count. OOB evaluates the full remaining-shoe composition at the current decision
state.
Baseline
Basic Strategy
Basic Strategy provides the optimal baseline decision for a defined set of game rules under its reference shoe conditions. It does not recalculate from the exact depleted composition after every exposed card.
Rule-specific · Static baseline
Compression
Hi-Lo True Count
Hi-Lo converts card removal into a Running Count and adjusts it for the estimated decks remaining, producing a compact measure of the current shoe.
One count · Many possible compositions
Full state
Exact Composition
OOB retains the remaining count of every card rank and uses that complete composition when calculating the EV of the actions available in the current state.
Exact shoe · Composition-dependent EV
Same True Count does not mean same shoe.
Multiple remaining-shoe compositions can map to the same Hi-Lo True Count. Because
those compositions contain different cards, their outcome probabilities and action
EVs can differ — and, in some states, so can the highest-EV decision.
Basic Strategy→fixed baseline
Hi-Lo→compressed shoe information
OOB→exact current composition
OOB derives Hi-Lo True Count from the exact number of cards remaining rather than an
estimated deck count. True Count is displayed as a reference metric, while core
action decisions are calculated from the exact remaining composition.
The best decision and the right amount to risk are different calculations.
OOB first evaluates the mathematical quality of the current blackjack decision, then
applies bankroll and risk context separately to betting guidance and session-risk
metrics.
Bankroll
Current Bankroll Context
The current bankroll provides the financial context used by OOB's risk-aware betting layer.
Bet sizing
Bankroll-Aware Bet Guidance
Bet guidance adapts to the current player edge and bankroll rather than treating the same mathematical advantage as the same wager in every session.
Risk of ruin
Model-Estimated Risk
OOB estimates Risk of Ruin from the current bankroll and betting context so the player can see how estimated risk changes as bankroll and betting conditions evolve.
Variance
Variance & Session Risk
Variance and session-risk metrics provide context for the range of outcomes that can occur even when decisions are made according to Expected Value.
Positive EV does not remove variance.
A mathematically favorable state can still produce a losing hand, session or
longer run. Bankroll and risk metrics describe exposure to uncertainty; they do
not guarantee a financial outcome.
Decision EV determines which available action is mathematically preferred. Bankroll
and risk calculations determine how that decision fits into the player's current
financial context.
Precision, rounding & borderline decisions
A displayed tie does not always mean the underlying EVs are equal.
OOB compares actions using their unrounded internal EV values. Display rounding is
applied only after the action comparison has been made.
Internal values
Compare Before Rounding
Action ranking is determined from the internal EV values before those numbers are rounded for display.
Display
Readable EV Output
Displayed EV values are rounded for readability, so two actions can appear equal on screen even when their internal values differ slightly.
Borderline
Near-Ties Are Flagged
When the top available actions are very close in EV, OOB can flag the decision as borderline so the player can see that the mathematical difference is small.
Small EV differences are still small.
Selecting the highest-EV action does not imply that every difference is
practically large. OOB distinguishes the mathematically preferred action from the
magnitude of the EV gap.
Validation comparisons use defined numerical tolerances so display rounding is not
mistaken for a mathematical discrepancy.
Validation & reproducibility
A result is only useful as proof if someone else can reproduce it.
OOB validation is designed around fully specified game states, deterministic reruns,
independent comparison and a documented process for resolving discrepancies.
Specify
Publish the Complete State
A reproducible state includes the exact remaining shoe, player hand, dealer upcard, active rules, available actions, engine version and methodology version.
Rerun
Same State, Same Output
The same fully specified state can be rerun to confirm that OOB produces the same action EVs and highest-EV decision.
Cross-check
Compare Independently
Selected states can be reproduced with an independent method or external tool using the same rules, state and EV convention.
Investigate
Resolve the Difference
If outputs disagree, the state, rules, EV normalization, numerical precision and methodological assumptions are checked before a conclusion is recorded.
A match only means something when the inputs match.
Two calculations are comparable only when the game rules, exact shoe composition,
player and dealer state, available actions and EV convention are equivalent.
Match→result reproduced within the defined tolerance
Mismatch→difference requires investigation
Unsupported→independent method cannot reproduce the exact state
Inconclusive→available evidence is insufficient
Validation records are versioned so later corrections do not erase what was
originally tested.
View Validation Records Coming soon
Limitations
Exact mathematics still has defined boundaries.
OOB evaluates the game state it is given. Its outputs depend on accurate card
tracking, the selected rules and the mathematical assumptions documented in the
methodology.
Input accuracy
The State Must Be Correct
If exposed cards or table rules are entered incorrectly, the calculated state no longer represents the actual game.
No next-card prediction
Composition Is Not Prediction
Knowing the exact remaining composition determines current card probabilities; it does not reveal the order of future cards or predict the next card.
Expected Value
EV Is Not a Guaranteed Outcome
Expected Value describes mathematical expectation across possible outcomes. A higher-EV decision can still lose the next hand or session.
Rule dependency
Results Depend on the Game
Action EVs and optimal decisions can change when deck count, dealer rules, doubling, splitting, surrender or blackjack payout rules change.
What OOB claims
OOB is designed to select the highest-EV available action for the exact game
state and rule configuration being evaluated.
What OOB does not claim
OOB does not guarantee wins, profit, positive session results or knowledge of
future card order.
Mathematical proof evaluates the engine. Realized gambling outcomes remain subject to
variance.
Reproducibility & technical record
Every published result should be traceable back to the state, method and version that produced it.
OOB proof assets are designed to preserve the technical context behind each published
calculation, comparison and validation record.
State ID
A Permanent Reference
Published proof states receive a unique identifier so the same game state can be referenced across proof pages, validation records and later reviews.
Versioning
Engine and Methodology Version
Each published record identifies the OOB engine version and methodology version used for the calculation or experiment.
Input record
The Full Calculation Context
The exact remaining shoe, player hand, dealer upcard, active rules and available actions are preserved with the published state.
Validation history
Checks, Discrepancies and Corrections
Validation status, independent comparisons, identified discrepancies and subsequent corrections remain linked to the relevant published record.
Corrections do not erase history.
If a published result is later corrected, the original version, affected engine
version, reason for the correction and updated result remain part of the technical
record.
Reproducible States
Inspect individual published states and their calculation context.
Follow independent checks, discrepancies and resolutions.
View Validation Records Coming soon
Technical Whitepaper
Read the full mathematical methodology and supporting definitions.
Read the Whitepaper Coming soon
The objective is not to make OOB beyond question. It is to make published results
possible to question precisely.
From method to evidence
Read the method. Inspect the proof. Then test the workflow yourself.
The methodology explains how OOB evaluates the game. The proof library shows how
published results are tested. The free trial lets you experience the workflow
directly.
Method
Understand the calculation framework.
Review the state model, EV logic, rule handling, precision and validation process.
You are here
Proof
Inspect published evidence.
Explore reproducible states, controlled comparisons and validation records as they become available.