The One2Six Advantage-Play Project, Part 5: Building and Testing the IID Baseline

25 June

Written By Paradis

Checking the card stream, blackjack engine and measurements before interpreting the shuffler

Part 4 put the first One2Six-style mechanism inside the simulation. The blackjack engine could deal and settle a complete game, physical cards could move through an ordered discard rack, and the source could recycle those cards through carousel shelves and an output buffer without losing or duplicating them.

That was an important milestone, although it only established that the machinery could run. A simulator can operate smoothly, produce polished tables and generate thousands of decimal places while still measuring the wrong thing.

Part 5 was therefore about calibration. Before interpreting an unusual return-time curve, blackjack rate or profit result from the One2Six model, I needed to apply the same measurements to a process whose expected behaviour was already known.

If an independent random source produced unusual recurrence, the problem would probably lie in the experiment. If blackjack frequency used the wrong denominator, an ordinary result could look catastrophic. If the settlement layer mishandled splits, doubles or pushes, any reported edge would inherit the error.

This was the first point where the project had to demonstrate that it could recognise normal behaviour before being trusted with anything unusual.

What Needed to Be Calibrated

Several parts of the system required separate checks.

The card source needed to produce the expected frequencies and recurrence patterns. The blackjack engine needed to turn those cards into plausible games. Settlement had to account correctly for initial wagers, doubles and splits, while the result tracker needed to calculate rates using denominators that matched the events being measured.

Those checks sit at different levels:

Validation layer	Question
Symbol-level source	Are ranks, suits and card symbols produced with the expected probabilities?
Recurrence measurement	Does the waiting time between target cards follow the expected distribution?
Physical-card null model	Does a labelled physical card recur as expected under memoryless sampling?
Blackjack engine	Do the rules and fixed strategy produce plausible event frequencies?
Settlement and accounting	Are wagers, payouts and rates calculated from the correct quantities?
Experiment framework	Can the same measurements later be applied to other sources without changing their definitions?

Keeping these levels separate was important because one part of the system could appear healthy while another remained wrong. The source might generate correct card frequencies while the recurrence code measured the wrong gap. The game could produce plausible win and loss rates while mishandling additional wagers. A calculation could also be perfectly accurate and answer a question nobody intended to ask.

Part 5 was designed to expose those problems while the expected answers were still available.

Why Start with IID?

An IID source produces independent and identically distributed card draws. Every symbol-level draw is independent of those before it, with each suit appearing with probability (1/4), each rank with probability (1/13), aces with probability (1/13), and ten-value cards with probability (4/13).

This is simpler than the physical processes used in a casino. A conventional shoe deals without replacement, so every card changes the population remaining. The One2Six is stateful in another way because physical cards pass through feeder, storage and output stages before returning.

The simplicity of IID is precisely what makes it useful. Its expected behaviour can be calculated directly, which means any meaningful departure gives me a short list of places to investigate: the source, the metric, the data collection or the assumptions behind the comparison.

IID provides the ruler against which the measurement framework can first be checked. Before studying a mechanism whose equation is only partially known, I wanted to confirm that the experiments could recognise one whose equation was straightforward.

Source-Level Validation Comes First

The first experiment draws directly from the card source without playing blackjack hands. This removes strategy, dealer logic, splits, doubles and settlement from the problem and leaves a sequence of card symbols to examine.

The basic checks are:

Metric	Expected IID behaviour
Rank frequency	Each rank approaches (1/13)
Suit frequency	Each suit approaches (1/4)
Rank-suit frequency	Each card symbol approaches (1/52)
Ace rate	Approaches (1/13)
Ten-value rate	Approaches (4/13)
Hi-Lo mean	Approaches zero
Target-card recurrence	Follows the relevant geometric distribution

The frequency checks are simple, but they remain necessary. A source producing the wrong number of aces or ten-value cards has failed before any more sophisticated analysis begins.

Correct frequencies alone do not establish independence. A sequence can contain the expected number of tens and fives while arranging them in a highly unusual order. Recurrence therefore needs to be checked separately because the One2Six project is concerned as much with timing and local structure as it is with long-run averages.

Waiting Times Are Geometric

Latency and recurrence are central to the One2Six investigation, so the correct waiting-time distribution needs to be defined carefully.

Suppose a target appears independently on each draw with probability (p). If (X) records the number of draws required to see the next target, including the successful draw itself, then:

P(X = k) = (1 - p)^(k - 1) × p

Here, (k) begins at 1.

If the metric instead records the number of non-target cards appearing between successive targets, then:

P(G = g) = (1 - p)^g × p

Here, (g) begins at 0 because two target cards can appear consecutively.

For a specific rank and suit, such as the ten of spades:

p = 1 / 52

For one rank, such as any five:

p = 1 / 13

For any ten-value card:

p = 4 / 13

The geometric distribution is the direct model for the number of draws until the next target. A Poisson model can approximate the number of target cards appearing inside a fixed window, but it answers a different question.

That distinction matters here because an incorrect reference curve could make ordinary recurrence look unusual. The baseline needs to be defined with the same care as the mechanism being tested against it.

Symbol Recurrence and Physical Recurrence

The first IID source generates card symbols independently. It can measure the waiting time until another ten of spades appears, but it does not represent a finite collection of physical cards.

A six-deck game contains 312 cards, including six separate tens of spades. Symbol recurrence cannot distinguish between the same physical ten returning and another copy of the symbol appearing.

The project therefore requires a second null model for physical-card recurrence. A physical-IID source assigns stable identities to 312 physical cards and samples independently from the entire labelled population on every draw.

This source does not model a shoe or a shuffling machine. It provides a memoryless reference for the return of individual physical cards.

For one specified card:

p = 1 / 312

The number of cards appearing between successive appearances of that card should follow:

P(G = g) = (1 - 1/312)^g × (1/312)

This gives later One2Six recurrence experiments a clean theoretical comparison. A manual shoe cannot provide the same null because depletion and reshuffling are built into its behaviour.

The physical-IID model asks what return times would look like if every physical card had the same chance of appearing on every draw and carried no memory from one appearance to the next.

The Denominator Problem

One of the first validation errors involved the player blackjack rate.

Suppose the simulation records 465 player blackjacks and $100,000 in initial wagers. Dividing one by the other produces:

465 / 100000 = 0.465%

The arithmetic is correct, but the result measures blackjacks per dollar rather than blackjacks per hand.

If those blackjacks occurred across 10,000 initial hands, the relevant calculation is:

465 / 10000 = 4.65%

That is an entirely plausible IID blackjack rate.

For two independent cards, the probability of a natural blackjack is:

P(blackjack)
    = P(ace followed by ten-value)
      + P(ten-value followed by ace)

    = 2 × (4 / 52) × (16 / 52)

    = 128 / 2704

    ≈ 4.7337%

This differs slightly from the finite-shoe probability because the second IID draw comes from the same unchanged distribution.

The mistake was useful because it exposed a broader accounting problem. The result layer needed to keep event denominators separate from monetary exposure.

The tracker therefore records:

initial_hands
initial_wagered
action_wagered
total_wagered
net_profit
edge_per_initial_wager
edge_per_total_wager

Natural blackjack frequency uses initial hands. Additional exposure from doubles and splits belongs in action wager, while total wager combines the initial and later commitments. Edge can be reported against either initial or total wager, provided the label makes the denominator clear.

The code had done exactly what I asked, which is one of the less comforting features of software.

A simulation becomes dangerous when it produces numbers more quickly than it produces definitions.

Game-Level Validation

Once the raw IID stream behaves properly, the source can drive the complete blackjack engine.

This introduces:

player decisions;
dealer drawing;
natural blackjacks;
ordinary twenty-ones;
doubles;
splits;
busts;
pushes;
settlement;
wager accounting;
profit and loss.

The purpose at this stage is to determine whether the engine behaves sensibly. The playing policy is a fixed published multi-deck strategy constrained by the implemented rules, rather than a solver-generated strategy optimised for every detail of the game.

The game-level output includes:

Metric	What it checks
Initial hands	Correct denominator for player naturals
Wins, losses and pushes	Overall settlement behaviour
Player blackjacks	Natural detection and payout handling
Doubles	Legal-action and additional-wager logic
Splits	Pair detection, strategy and hand creation
Busts	Hand resolution and discard collection
Initial wager	Original exposure
Action wager	Additional double and split exposure
Net profit	Final settlement
Edge measures	Profit normalised by clearly defined wager bases
Profit path	Drift and variance over time
Streak distribution	Serial patterns in completed-round outcomes

No single metric can certify the engine. Together they create overlapping checks that make major errors harder to hide.

Pushes and Streaks

Streak analysis required an explicit treatment of pushes.

For this project, a push leaves an existing win or loss streak open:

W W P W -> win streak of 3
L L P L -> loss streak of 3
W P L   -> win streak of 1, loss streak of 1

A push does not change the bankroll, so treating it as a new directional outcome would shorten streaks mechanically without a corresponding change in the financial result.

The tracker ignores pushes while a streak remains open and closes the streak only when the direction changes. It records current streaks, maximum streaks and the complete distribution of win and loss runs.

Later comparisons may ask whether One2Six outcomes show different run-length behaviour from IID or a manual shoe. That comparison would be difficult to interpret if the streak definition had not first been fixed and understood.

The First End-to-End Checkpoint

The first smoke experiment used:

source draws: 10,000
game rounds: 1,000

It produced:

player blackjack rate:      4.5000%
expected IID blackjack rate: 4.7337%

edge per initial wager:    -4.2000%
edge per total wager:      -3.7534%

The edge estimates carry almost no information at that scale. One thousand blackjack rounds can finish heavily in either direction through ordinary variance.

The observed blackjack rate is also too noisy to validate the engine by itself, although it sits in the expected neighbourhood.

The useful result was that the complete pipeline operated from beginning to end:

source generation
    -> game simulation
    -> settlement
    -> metric calculation
    -> recurrence analysis
    -> plot generation
    -> output files

The smoke test showed that the components connected and that the results could be inspected. It did not complete the validation.

What Full Validation Requires

The next stage was to increase the scale and make each result harder to dismiss as a convenient seed.

At source level, a large IID run should show:

rank, suit and rank-suit frequencies close to expectation;
ace and ten-value rates near their theoretical values;
Hi-Lo values centred around zero;
symbol waiting times following their geometric reference curves;
physical-IID recurrence following the (p = 1/312) null model.

At game level, the engine should produce stable event frequencies across independent seeds, including player blackjacks, doubles, splits and the main outcome categories.

Profit needs more caution because its variance is much larger than the variance of simple card counts. A source can produce almost exact rank frequencies while the corresponding blackjack result remains noisy over a substantial number of rounds.

Independent seeds also matter. Millions of rounds inside one stream provide scale, but replication across separately generated streams offers a stronger check on whether a result is stable. This becomes particularly important for the One2Six, where consecutive rounds share internal machine state and should not automatically be treated as independent observations.

The baseline needed to be difficult to fool before the shuffler was allowed to produce anything interesting.

Why Profit Still Comes Later

The first One2Six smoke run happened to finish in profit. That result was expected to attract attention because profit is the most immediate output and the easiest one to understand.

A short profitable run under a losing process is completely ordinary. Blackjack players demonstrate this every day, sometimes with great confidence.

Observed profit sits at the end of a long chain:

card source
    -> card sequence
    -> game rules
    -> player strategy
    -> wagers
    -> settlement
    -> variance
    -> observed profit

The early measurements should sit much closer to the card source because that is the part of the system being investigated.

The first useful One2Six comparisons therefore concern:

physical-card return times;
very short returns;
discard-to-return latency;
same-batch reappearance;
output-buffer delay;
recurrence against the physical-IID null;
sensitivity to recycle-batch size.

These measurements can describe what the mechanism is doing without requiring profit to act as a noisy proxy. Once a structural difference is established, the project can examine whether the effect is visible at the table and whether it changes the expected value of future deals.

The Role of the Manual Shoe

IID is the first reference point, although it is not the only one.

A conventional shoe naturally differs from IID. Cards are dealt without replacement, recent cards remain unavailable, and the composition changes as the shoe is depleted. A cut card creates a reshuffle boundary after which the full physical population returns.

Those features produce ordinary recurrence and dependence.

The manual-shoe comparator therefore helps answer whether an apparent pattern comes from the One2Six mechanism or from using a finite population of physical cards.

The three main comparisons are:

IID
    -> memoryless mathematical reference

manual shoe
    -> conventional physical-card reference

One2Six
    -> stateful continuous-shuffler model

A departure from IID is only the beginning of the analysis because a manual shoe also departs from IID. The shape of the difference and the process producing it are more informative than the fact that a difference exists.

What Part 5 Established

By the end of this stage, the project had:

separate source-level and game-level experiments;
known IID expectations for rank, suit and card-symbol frequencies;
geometric reference distributions for target recurrence;
a clear distinction between symbol and physical-card recurrence;
corrected denominators for blackjack and wager statistics;
explicit treatment of pushes in streak calculations;
a manual-shoe comparison path;
an end-to-end smoke test of the experiment framework;
a defined route from calibration into large-run validation.

The large simulations still needed to be completed and compared across independent runs. Part 5 did not establish anything about whether the One2Six could be exploited.

It established the conditions under which a later result would deserve to be taken seriously.

Where the Project Stood After Part 5

The simulator could run, and the experiment framework now had explicit expectations for ordinary output.

The next step was to compare the return of individual physical cards. Long-run card frequencies can look perfectly normal while the distribution of return times changes underneath them, particularly when cards pass through compartments, group ejection and a finite output buffer.

Part 6 therefore examines physical-card recurrence through the One2Six-style source and compares it with the memoryless physical-IID model.

The mean is only one part of that comparison. The shape of the return-time distribution is where the mechanism begins to show itself.

References

mathematical-ev/shufflemaster-simulation. Public repository containing the card sources, blackjack engine, experiment framework, recurrence measurements and validation runners described in this article.
Dimitri P. Bertsekas and John N. Tsitsiklis, Introduction to Probability. Relevant to independent trials, geometric waiting times and the distinction between inter-arrival distributions and fixed-window event counts.

Paradis