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Copyright © Columbia Pictures (2008)

Tips & Tricks

How To Count Cards

An Homage & Walk-Through Of The Infamous Hi-Lo System

Jesus Najera
Nov 18, 2019 · 6 min read

How Is It Played?

Blackjack is a game played against the dealer, not other players. To start, the dealer deals two cards to both the player & themselves; all cards except for one of the dealer’s cards are face up (revealed). Each card is worth the number value attached to it — 2 is worth 2, 3 is worth 3, 4 is worth 4, etc. Face cards (J,Q,K) are worth 10, & Aces are worth either 1 or 11 (dealers announce whether Aces are high (11) or low (1)). After the first two cards, each player has the option to ask for another card (to “hit”) or not (to “stand”). The objective of Blackjack is for the sum of the player’s cards to be as close to 21 as possible without going over 21. The player loses if either the cards add up to more than 21 (“busting”), or if the sum of the dealer’s cards is closer to 21.

But What About The House Edge?

It’s not obvious, but the house-edge creeps in from two asymmetries. First, the dealer’s hidden (aka “hole”) card — knowing this would complete the player’s informational map (though the obvious argument here is that dealer also doesn’t know this card). The second asymmetry, however, is where the edge really creeps-in: forcing the player to make the first move (hit or stand). By hiding one card & forcing the player to decide with incomplete information, the player is susceptible to unnecessarily busting.

However, There’s An Additional Principle To BlackJack That Quick Mathematicians Can Leverage To Increase Their Odds…

Blackjack games have a memory — they’re played with a finite set of cards (one or many decks). By knowing which cards were already dealt, it’s possible to calculate which cards are to come.

Crash Course To The Hi-Lo System

Card counting is a strategy constructed on simple addition/subtraction & conditional probability to determine what kind of card (high or low — thus the name) is likely to be dealt next. In the Hi-Lo card-counting system, each card has one of the three following points: -1,0, & 1. As seen in the diagram below, cards 2–6 are worth +1, cards 7–9 are worth 0, and 10s, face cards, & Aces are worth -1:

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The Underlying Mechanics

The Hi-Lo system works because it’s constructed on pure probability. The system itself is essentially a clever shortcut that circumvents the more-complex mental tasks involved with calculating percentages. For a comparative example, let’s consider that if instead of the system above, we started with the more basic question: after the initial round of dealing in a single-handed game, what’s the probability of the next card being a high-value card?

That’s It? That’s All It Takes To Count Cards?!?

Believe it or not, that is indeed all the math required in the Hollywood-famed Hi-Lo card counting system. It’s deceptively basic, however, most people are overlooking two additional points of complexity that rear their head in the theory-to-implementation translation:

  1. Blackjack is almost never two-handed (only player vs. dealer)

Cantor’s Paradise

Medium’s #1 Math Publication!

Jesus Najera

Written by

Owner @ SetDesign, NightKnight & CryptoSpace | Product Designer | Hobbyist Mathematician | VR Developer | MS in Finance @ UF

Cantor’s Paradise

Medium’s #1 Math Publication

Jesus Najera

Written by

Owner @ SetDesign, NightKnight & CryptoSpace | Product Designer | Hobbyist Mathematician | VR Developer | MS in Finance @ UF

Cantor’s Paradise

Medium’s #1 Math Publication

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