Cards: where skill changes the arithmetic

Counting a deck

The standard deck has 52 cards in four suits of thirteen ranks. The number of distinct five-card hands is the combination C(52,5) = 2,598,960. Poker hand rankings are simply this count sorted by rarity: four of a kind can be made 624 ways, a full house 3,744, a flush 5,108, a straight 10,200. The ranking order was settled by players long before it was verified by counting, and the counting confirmed it.

Draws without replacement

Dice and wheels reset entirely between plays. Cards do not. If four aces have been dealt from a shoe, the probability of an ace on the next card is zero, not 1/13. This dependence is why the state of the remaining deck carries information, and why any game dealt from a diminishing shoe is structurally different from a game of independent trials.

Blackjack and the boundary of skill

Blackjack is the clearest case. The dealer follows a fixed procedure; the player chooses. Because the choices interact with a known distribution, there is an optimal decision for every combination of hand and dealer upcard, and following it reduces the house edge to a fraction of a per cent under common rules. Departing from it raises the edge substantially. The skill here is entirely a matter of playing the arithmetic correctly rather than reading the game.

Rules move the number more than play does. The payout on a natural, whether the dealer draws on a soft seventeen, how many decks are used, whether doubling after splitting is allowed and whether surrender exists each shift the edge by a measurable amount. A table advertising a reduced payout on a natural can carry several times the edge of an otherwise identical table.

Games between players

Poker is a different structure again. Players stake against each other, and the house takes a rake -- a share of the pot or a time charge -- rather than holding a position in the outcome. The consequence is that the game is not negative-sum by design but by extraction: the pool of money shrinks by the rake at every hand, so the average participant loses even though no single hand is priced against them. Skill determines how that shrinking pool is distributed.

What card counting is and is not

Because removal changes the remaining composition, tracking roughly how rich the shoe is in high cards gives a small, genuine, slowly-accumulating statistical signal. It is a bookkeeping exercise, not prediction: it never says what the next card is, only how the average hand is currently priced. Continuous shuffling machines, frequent reshuffles and multiple decks exist largely to remove the signal, and they do.