The mathematical underpinnings of blackjack are both interesting and not immediately obvious. Let’s wonder a bit about the casino game of blackjack.

A glance at the grandeur of a typical casino in, say, Las Vegas or Monte Carlo tells you that most people lose money when they gamble. There are sound mathematical reasons for a casino’s winning ways. Although there are many books and websites that purport to teach you how to sidestep the odds of almost every casino game, many are misguided at best and rubbish at worst.

Unlike in other casino games, as a blackjack player you can increase your bet in favorable situations

When it comes to the card game blackjack, however, there are some solid bits of mathematics that can help a player turn the tables on the gambling establishment. Unlike in other casino games, as a blackjack player you can increase your bet in favorable situations. You can also make decisions based on information you gather as you play. Both can turn the odds in blackjack in your favor.

First, a quick primer on blackjack, the game is played using one or more 52-card decks. The value of each card is either the number on the card, or 10 for face cards, or one or 11 for an ace. The goal of the game is to accumulate cards that total as close to 21 as possible without going over. Players get dealt two cards and can choose to take more. The casino—the dealer—also gets two cards, and must stand (stop drawing cards) or draw (request more cards) based on a set of rules.

Increasing Your Bet in Favorable Situations

Which situations are favorable, or not, in blackjack?

Consider getting dealt a hand that totals 16. If the dealer has a strong hand—say she has a face card showing—you’ll probably lose if you stand. But drawing is hardly better. Only a 2, 3, 4, or 5 prevents you from going over 21 and losing immediately. Draw, and you’ll probably lose. A hand of 16 is unfavorable for you.

What about a hand that totals, say, eight? More than half of the cards you might draw next are either aces or 10-valued, any of which would give you a strong hand. A hand totaling eight is favorable for you.

So let’s say you get dealt two 8s. With a value of 16, this hand is most unfavorable. However, when you have two cards that are the same—e.g., two 8s or two jacks—most casinos allow you to split them into two separate hands. In doing so, you also must double your wager. And splitting the two 8s into two separate, eight-valued hands … wow! Not only do you get to turn one bad hand into two good ones, you get to increase your bet.

The Rules of Blackjack Enable You to Increase Your Wager in Favorable Situations

In most casino games, roulette for example, every separate wager has the same probability of winning or losing. Not so in blackjack. If, for example, there are more 10-valued cards than small-valued cards left to be dealt, you’re more likely to get a winning hand. By keeping track of how many high-valued and how many low-valued cards have been played, you can predict the likelihood of the next hand being a winner. This method, called “counting cards,” is far easier than you might guess.

Counting can be as simple as tracking cards in three categories: those that are favorable for you, those that are unfavorable, and those that don’t have much effect on the likelihood that a hand will either win or lose.

Aces and 10-valued cards are favorable for you because they get you quickly up to (or close to) 21. The more aces and 10-valued cards that remain in a deck, yet to be dealt, the more likely it is that you will win the next hand.

Low-valued cards—2s, 3s, 4s, 5s, and 6s—are not favorable for you. Because there are so many 10-valued cards in the deck, if you get dealt a low-valued card, the likelihood is that your other card will bring your total into the low teens. Which means your next card will take you over 21 and you’ll lose. The more low-valued cards that remain in a deck, yet to be dealt, the more likely it is that you will lose the next hand. Cards valued 7, 8, or 9 don’t have as much of an effect on winning or losing as low- and high-valued cards do.

In the simplest card-counting system, aces and 10-valued cards are assigned a value of negative-one, low-valued cards are assigned positive-one, and 7s, 8s, and 9s are assigned a value of zero. The total for all 52 cards in a deck is zero. So start with a “count” of zero and subtract or add one (or zero) as you see cards played. A positive count means that more low-valued cards have been dealt—the rest of the deck is in your favor! A negative count means that fewer aces and 10-valued cards remain—not good for you. Increasing and decreasing your bets according to the count increases the chance that you will bring home some winnings from the blackjack table.

The role of mathematics in the game of blackjack means strategy—not luck—and can turn the tables in your favor

The role of mathematics in the game of blackjack means strategy—not luck—and can turn the tables in your favor.

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Questions to ponder:

Purely based on statistics, some casino gamblers get lucky and win money. Blackjack, however, can be beaten based on skill—no luck involved. Are there other casino games that can be beaten in this way? (The answer is surprising.)

A computer analysis of blackjack reveals that for every possible combination of a player’s hand and the dealer’s card there is exactly one play (hit, stand, split, and so on) that results in the highest probability of winning. For example, the player should draw a card when holding a hand of 12 against a dealer’s 10. Would there ever be a situation in which this is not the right play?