---
title: "card games"
url: https://sacstat.org/card-games/
description: "Why card games behave differently from wheels and dice, how dependence between trials changes the arithmetic, and how baccarat's commission sets its edge."
published: 2026-08-17
updated: 2026-08-17
---

# card games

Published 2026-08-17

[sacstat](https://sacstat.org/) › Card and machine games › card games

trials = dependent

A wheel forgets. A shoe of cards does not. Removing a card changes what remains, which makes card games the one large family in this subject where the state of the game carries information — and the family where the arithmetic is correspondingly harder.

## Dependence between trials

In an independent game, the probability of every outcome is the same on every trial. In a dependent one it moves as cards are removed. Dealing one ace from a single deck lowers the chance of the next card being an ace from 4/52 to 3/51 — a small change, but a real one, and one that compounds as a shoe is depleted.

This is why card games are shuffled at intervals, why multiple decks are used, and why penetration — how deep into the shoe play continues before a shuffle — is an operational decision rather than a cosmetic one. Everything distinctive about the family follows from the single fact that the deck is a finite, changing population.

Dependence, immediately visible

```
single deck, no cards seen:      p(ace) = 4/52  = 0.0769
one ace already dealt:           p(ace) = 3/51  = 0.0588
all four aces already dealt:     p(ace) = 0/48  = 0

compare a wheel:
  after any history at all:      p(red)  = 18/37 = 0.4865
```

## Blackjack and composition

Blackjack's expected value depends on the rule set — how many decks, whether the dealer draws on a soft seventeen, what a natural pays, whether doubling and splitting are restricted — and on the composition of the remaining cards. Under common rules and accurate play the edge is a fraction of one percent, which makes it unusually sensitive to rule changes that sound minor.

The most consequential single change is the payout on a natural. Shortening it from three to two down to six to five raises the edge by well over a percentage point on its own, dwarfing the effect of any of the play decisions a table conversation is usually about.

*Rule changes and their direction of effect*

| rule | direction of effect on the bettor |
|---|---|
| natural pays 3:2 rather than 6:5 | substantially better |
| dealer stands on soft 17 | better |
| doubling allowed after a split | better |
| more decks in the shoe | slightly worse |
| dealer draws on soft 17 | worse |

## Baccarat and the commission

Baccarat removes decisions almost entirely: the drawing rules are fixed, and the only choice is which side to back. Over a full shoe the banker hand wins slightly more often than the player hand, which would make backing it profitable at even money. The rules therefore take a commission — conventionally five percent — from banker wins, and that commission is precisely what converts a positive expectation into a small negative one.

It is the cleanest example in the subject of an edge introduced as an explicit charge rather than hidden in a payout table, and comparing it to roulette's missing pocket is a good way to see that the two mechanisms do the same job.

Where the baccarat commission comes from

```
over a full shoe, ignoring ties:
  banker wins slightly more often than player
  paid at even money, backing banker would show a positive expectation

the rules therefore deduct 5% of the win:
  a 1-unit banker win returns +0.95 rather than +1.00

the deduction is set so the residual expectation is small and negative
the tie wager, paid short of its true price, is far more expensive than either
```

## What dependence does not give you

Commonly misread.

Dependence means information exists, not that it is usable. Tracking a shoe changes expectation only while the composition is favourable and only if stakes move with it, which is why continuous shuffling machines, shallow penetration and frequent shuffles exist. Against a freshly shuffled shoe the arithmetic is exactly the arithmetic of the rule set, and no record of previous hands alters it.

Return to the [Card and machine games](https://sacstat.org/#entries) index on the reference landing.

‹

[dice and craps](https://sacstat.org/dice/)

[reel machines](https://sacstat.org/slot-machines/)

›

Last modified 17 August 2026.