the whole method, small enough to check by hand
A quick example
Every bet page on this site runs the same eight steps. On a real game each step takes millions of simulated rounds. Here the shoe has six cards, so you can check every number on a napkin, and the toy still carries the two facts that decide real bets: why counting works, and why some bets can't be counted with per-card tags at all.
1The rules, as data
The shoe holds six cards: two deuces, two sevens, two kings. They're dealt one at a time, face up, no reshuffle. You see every discard, and that's the premise of counting. A card that has left the shoe is information.
The house offers two bets before every draw. They look like cousins. One falls to the oldest trick in the book. The other doesn't fall to any per-card count at all.
| bet | wins when | pays |
|---|---|---|
| High Draw | the next card is a king | 9 to 5 |
| The Match | the next two cards share a rank | 3 to 1 |
2Reproduce the published figures
On a real game this step means matching the published house edge for the exact rules before trusting anything else. The toy casino publishes its edges too, and you can check them exactly. High Draw off the top: two kings in six cards, so the win probability is 1/3, and at 9-to-5 the expected value is 1.8 × 1/3 − 2/3 = −6.67%. The Match off the top: whatever the first card is, exactly one of the remaining five matches it, so the win probability is 1/5, and at 3-to-1 that's −20%. Both check out. Now the machinery has earned the interesting question.
3Build the oracle
The oracle knows the exact remaining composition and prices each bet perfectly. For High Draw, with k kings left among n cards, the expected value per unit is 2.8k/n − 1. For The Match, what matters is how many pairs survive: with rank counts nr, the win probability is Σ nr(nr−1) over n(n−1), and the bet is worth taking when that clears one quarter. That's the whole ceiling. Nobody at the table can know more than these two formulas do.
4Enumerate the states
A real game needs millions of simulated shoes here. The toy's states fit in one table. High Draw's oracle, by kings remaining and cards remaining, with player-positive states in green:
| kings left | n = 6 | n = 5 | n = 4 | n = 3 | n = 2 | n = 1 |
|---|---|---|---|---|---|---|
| 2 | −6.7% | +12% | +40% | +86.7% | +180% | — |
| 1 | — | −44% | −30% | −6.7% | +40% | +180% |
| 0 | — | — | −100% | −100% | −100% | −100% |
The bet starts negative, and the shoe's departures push it around, hardest near the bottom of the shoe. That's why penetration will matter in step seven.
5Fit a model a person can carry
The standard move is the effect of removal: how much does the first departure of each card move the bet's expected value? For High Draw, a departing deuce or seven moves it from −6.67% to +12%, a shift of +18.67 points. A departing king moves it to −44%, a shift of −37.33 points. The ratio is exactly 2 to 1, so the fitted tags are +1 for every small card, −2 for every king. And since this bet's value depends only on the king ratio, the running count plus cards-remaining recovers the oracle state completely. The fit has zero error.
Run the same procedure on The Match and it fails. Remove any single card and recompute: surviving pairs drop from 6 to 4, ordered two-card combinations drop from 30 to 20, and the win probability is 4/20 = 1/5, unchanged. Every single-card effect of removal is exactly zero. The least-squares fit dutifully puts a zero tag on every rank, and the best per-card count in the world captures none of this bet's value.
The value is still there, though. Look at two removals instead of one. If the two departed cards shared a rank, the survivors are two intact pairs among four cards and the bet swings to +33%. If they differed, one pair survives and it swings to −33%. The information was never in which cards left. It's in how they left together. So the model has to grow past per-card tags to a pairwise statistic: track the intact pairs, bet when Σ nr(nr−1) is large against the cards remaining. That statistic equals the oracle, and nothing simpler captures anything.
The real pages apply this same rule constantly: fit linear first, score it against the oracle, grow the model only when the leftover error or the bet's structure demands it. The 21+3 side bet needed a four-suit system after rank-only counts stalled. War Blackjack's tie leg is The Match wearing a suit, priced by this same pair statistic, and no high-low count can touch it.
The removal arithmetic, in full
The Match win probability from rank counts (n2, n7, nK) summing to n: draw two without replacement, and the chance both land in rank r is nr(nr−1)/n(n−1). Fresh shoe: (2·1)·3 / (6·5) = 6/30 = 1/5. Remove one card of any rank, say a deuce, leaving (1, 2, 2): (0 + 2 + 2)/(5·4) = 4/20 = 1/5. The numerator always loses exactly 2, because the removed card's rank goes from contributing 2 to contributing 0, while the denominator falls from 30 to 20. The ratio holds for every rank, so every effect of removal is exactly zero. After two removals the symmetry breaks: a same-rank removal leaves (2, 2) → 4/12 = 1/3 (EV +33.3%), a split removal leaves (1, 1, 2) → 2/12 = 1/6 (EV −33.3%). A per-card count is a sum of per-card contributions, and once those are identical it can't tell the two futures apart. The bet's value is a second moment of the composition, and no first-moment statistic carries it.
6Replay the human system
On a real bet this step replays the exact card, on fresh shoes it was never fitted to, and reports the capture fraction against the oracle. The toy's capture fractions are exact. High Draw's count-plus-depth system pins down the oracle state completely: capture 100%. The Match's best per-card count is flat: capture 0%. The pair statistic, which a person can keep by counting broken pairs, is the full oracle again. Real games land between the extremes. The 21+3 system keeps 78% of its ceiling, the baccarat scorecard about 87%, and one no-division count took 106.6% of what its divided benchmark took. That fraction is what the rest of the analysis prices.
7Put dollars on it
Give the toy its scenario card: one unit flat, bet only when the oracle is positive, full shoe dealt. A blind bettor who takes High Draw before every draw loses 6.67% of a unit each time, because with no information every undealt position is a uniform draw from the full shoe. Six draws, −0.40 units per shoe. The counter bets only in green states, which show up about 2.2 times per shoe, and clears +1.35 units per shoe exactly.
Then let the house respond the way real houses respond: cut off the last two cards and deal only four of six. The counter's take falls to +0.41 units per shoe. Cutting one third of the shoe removed 69% of the edge, because the violent states cluster at the bottom. That's penetration, and it decides more real verdicts than any other number in the game.
The per-shoe expectation, term by term
Before the draw with n cards remaining, the chance that j kings have already left is hypergeometric, and the counter collects max(0, EV) in each state. Five cards left: both kings remain with probability 2/3, worth 2/3 × 0.12 = 0.080. Four left: 6/15 × 0.40 = 0.160. Three left: 4/20 × 0.8667 = 0.173. Two left: 1/15 × 1.80 + 8/15 × 0.40 = 0.333. One left: 1/3 × 1.80 = 0.600. Total +1.347 units per shoe. The first three terms are all a two-card cut leaves standing, and they sum to 0.413.
8Deliver a verdict
High Draw: beatable, capture 100%, worth +1.35 units per fully dealt shoe, and two thirds of that disappears at a shallow cut. A real verdict would hinge on where this imaginary house puts the cut card, which is how real verdicts usually die. The Match: beatable too, through a second moment, and a conventional count, however lovingly tuned, captures nothing. Every page on the bet index is this same walk at full scale, with the millions of rounds and the dollars filled in.