card games

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.

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.

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.

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