How Card Counting Numbers Are Tested

Card counting comes with numbers attached: how big an edge a system produces, what an index number is worth, which of two systems performs better. Those numbers are quoted, compared and argued over, and they are usually quoted without any mention of how they were produced. That is a shame, because the production is the interesting part, and a reader who understands it can judge every counting figure they meet.

The core idea of this piece is simple. Every published counting number is the output of a computation, and every computation made assumptions. Two families of method produce essentially all of these figures, each family has habits and failure modes, and the arithmetic of sample size sets a hard limit on how fine a distinction a simulation can support. None of this requires mathematics beyond a square root.

Two families of method

The first family is exact, or combinatorial, analysis. The analyst fixes a deck composition, enumerates the possible outcomes, and adds them up with their probabilities. For a question about one hand from a known shoe, the enumeration is complete and the answer is exact, to the last decimal place, for exactly that composition. Change the composition and you have a new computation.

The second family is simulation. A computer deals a very large number of hands under stated rules, plays them by a stated strategy and betting method, and averages the results. The answer is not exact; it is an estimate with an uncertainty that shrinks as the sample grows. Simulation is how counting questions are usually answered, for a structural reason: the interesting quantities in counting, the running count, the bet on the table, and the cards remaining in the shoe, all change from hand to hand, and a simulation simply lives through those changes the way a player does.

The two families check each other. Exact analysis handles the corners a simulation would estimate badly, and simulation extends to territory an enumeration cannot reach. When the two agree on a question both can answer, confidence in both goes up.

The arithmetic of sample size

How uncertain is a simulated average? There is one number to start from. Per Wizard of Odds, the results of single blackjack hands vary with a standard deviation of about 1.15 betting units for flat bets, a figure our piece on session length works with. The uncertainty of an average over n flat bet hands, called its standard error, is that figure divided by the square root of n:

The square root is merciless. Going from 10,000 hands to a million, a hundredfold increase in effort, buys one decimal place of precision. Now ask what a comparison takes. A rough yardstick: for two results that differ by 0.1 percentage point to sit two standard errors apart, the standard error of each average needs to be at most 0.05 points. Writing 0.1 point as 0.001 of a unit, that takes about (2 x 1.15 / 0.001) squared, or about 5.29 million hands for each result. Telling apart differences of 0.01 point takes about 529 million. Both are floors, because the difference between two separately simulated averages is noisier than either average alone.

Two consequences follow. First, the differences between good counting systems are small, so only very large samples can separate them honestly, and a comparison run on tens of thousands of hands has not actually compared anything. Second, any figure produced from thousands of hands, which includes anyone's personal results, is mostly noise measuring the deal rather than the method. A counter who varies bets raises the spread of results per hand further, so the sample needed grows beyond these flat bet figures; the same square root rule keeps working, the numbers just get bigger.

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A worked example from this site

Our piece on how the count moves through a shoe makes a good test case, because it publishes its method. It simulated 200,000 six deck shoes counted with Hi-Lo, tracking the running count card by card and dividing by the exact number of decks remaining, unrounded, at the moment a given number of decks had been dealt. The percentages in its table, such as 23.9 percent of shoes sitting at a true count of +2 or higher after three decks, are shares of that sample.

How much do 200,000 shoes support a share near 24 percent? For a proportion p observed over n cases, the standard error is the square root of p x (1 minus p) / n. Here that is the square root of 0.24 x 0.76 / 200,000, which is about 0.001, or 0.1 percentage point. The table's percentages are therefore reliable to a few tenths of a point, which is plenty for the argument they carry. The article also cross checks itself against theory: its running count spreads matched an exact formula to within 0.01 at every depth, and that agreement between the two families of method is exactly what a published number should be able to show.

Why two sources disagree

When two credible sources publish different numbers for what sounds like the same question, the disagreement is usually in the assumptions, and the assumptions can be read off. A checklist:

• Rules and deck count. Six decks versus one, soft 17 rule, doubling and surrender rules: each moves the baseline.
• Penetration. How deep the shoe is dealt decides how often high counts appear at all, as our piece on deck penetration explains.
• Bet ramp and spread. How bets move with the count changes both the result being averaged and its variance.
• Number of players. More players at the table means fewer hands per shoe for the counter and different card flow.
• True count rounding. Floor, truncate or round: conventions produce different counts at the same moment, per our piece on rounding conventions.
• Index set. Which deviations are played, from the Illustrious 18 to extended lists, changes results at the same counts.
• Sample size. The square root arithmetic above sets what the comparison could possibly resolve.

Published figures on this site carry these qualifiers on purpose. Our piece on betting correlation and playing efficiency states that its decimals are computed under stated assumptions about rules and deck count, and that slightly different assumptions produce slightly different numbers. Our piece on where index numbers come from walks through why the same play can cross at a different count in a single deck game than in a shoe, and concludes that two sources often give different numbers because they are answering different questions. Neither article treats the disagreement as a scandal; it is arithmetic.

Reading a figure like a tester

Put together, the pieces give a reader a permanent habit. When a counting number appears, ask how it was produced: exact or simulated, under what rules and decks, at what penetration, with what bet ramp and rounding, over how many hands. A source that states those things can be checked and compared. A source that states none of them has published a mood.

The same lens applies to your own play, gently. A few thousand hands of personal results sit beyond the noisy end of the table above, which is why our pieces on the chance of being ahead after n hands and on variance keep saying the same thing: short runs measure luck. If the practice goal is counting skill rather than bankroll narrative, the app's Pro upgrade includes counting drills, index deviation training and a bankroll simulator with virtual chips for playing hands against a shoe, which trains the skills without lending the results any statistical weight they do not have.

One framing to close on. Counting with your head is legal in the United States. Casinos are private businesses that may refuse service to suspected counters, and no technique removes the risk from the game. Counting figures describe small effects that take millions of hands to measure, and the honest way to hold every one of them is with its assumptions attached, which is exactly what this article has tried to do.