Why card counting is a probability idea
Blackjack differs from roulette in one way that matters to the maths. Roulette spins are independent: the wheel has no memory. In blackjack, cards that have been dealt stay out of the shoe or deck until the next shuffle. So the composition of what remains changes, and with it the chance of future cards. Card counting is the practice of tracking that change.
This article explains the arithmetic and its limits. It is not legal advice and it is not a recommendation to attempt it. Casinos may refuse service to people they suspect of counting, and rules and attitudes vary by place and by operator, so anyone curious about the practice should treat that as a real risk.
What a dealt-down deck does
Start with a single 52 card deck. It contains 16 ten-value cards (the tens and the three kinds of face card in four suits) and 4 aces. The chance that a random card is ten-valued is 16/52, about 30.8%.
Now suppose, as an illustration, that the cards dealt so far have been unusually low ones, and 12 low cards have left the deck with no tens removed. The remaining deck has 40 cards, still with all 16 tens. The ten-value share is now 16/40 = 40%, up from about 30.8%.
A richer share of tens and aces generally favours the player slightly. Roughly speaking, a dealer is more likely to bust when forced to draw, a player’s strong hands such as a natural (an ace and a ten-value card) arise more often, and doubling down becomes more attractive. Conversely, a deck rich in small cards slightly favours the dealer. This is an application of conditional probability: the chances are conditional on what has already been seen.
How a simple count works
One widely described scheme is the Hi-Lo count. Each card seen changes a running tally:
- Low cards (2 to 6): add 1. These leaving the deck makes the remainder richer in high cards.
- Neutral cards (7 to 9): add 0.
- High cards (10 to ace): subtract 1. These leaving the deck makes the remainder poorer.
A positive count suggests a deck that is richer in tens and aces than usual. Since a count that is +4 means something different with one deck left than with six, players divide the running count by the estimated number of decks still to be dealt to get a “true count”. For example, a running count of +6 with 3 decks remaining gives a true count of +2.
The point is not that the count predicts the next card. It reflects a small shift in the odds, as in the 30.8% to 40% illustration above, only slowly and noisily.
Why the advantage, if any, is small
Even in a favourable situation, any edge is generally described as small, often a fraction of one percent to around one percent, and it depends heavily on the rules of the game. The ordinary disadvantage of standard play also depends on the rules, so a modest shift in the deck can be enough to cancel it, or not.
Several features of real play limit the effect:
- Penetration. Only the cards that are actually dealt before a reshuffle provide information. If the dealer shuffles after a small part of the shoe, there is little to track.
- Shuffling methods. Some arrangements, such as continuous shuffling machines, keep the deck composition essentially constant, which removes the information entirely.
- Rules. Payouts for a natural, the number of decks, whether the dealer hits on a soft 17, and many other rules change the baseline.
- Online and software dealt games. Where each hand is dealt from a fresh random shuffle, there is no depletion to track.
Variance and the long run
A small positive expectation does not mean steady winning. Blackjack outcomes are highly variable, and a player with a slight advantage can still lose for long stretches. The standard deviation of a single hand is much larger than any plausible edge, so many hands are needed before results reflect it. Short run results are dominated by luck either way. Hands are dependent events only through the cards removed, and that effect is small.
Bankroll maths matters for the same reason. A person who needs to win within a few sessions is exposed to variance, and a bet size that is too large relative to funds can end the experiment through gamblers ruin before any average is reached. The law of large numbers describes where averages settle, but only if you can afford to survive long enough to reach them.
Practical limits beyond the maths
- Counting accurately under real conditions, with noise and conversation and time pressure, is difficult, and a small error can erase a small edge.
- Casinos may ask a player to leave, change the rules, or shuffle more often, as mentioned above.
- Rules and law differ between places. Nothing here is legal advice.
- The technique does not apply to online blackjack with fresh shuffles, or to tables with continuous shuffling.
The takeaway
Card counting is a real example of probability that depends on changing information: removing cards changes the odds. It also shows how limited that effect is. It does not make blackjack a guaranteed source of money, and the ordinary player who does not count faces a house edge that depends on the table’s rules. Gambling carries a risk of losing money and of harm. If you are struggling to control it, contact a local support service.