Mathematics of Blackjack

Relatively simple to play, fast-paced and requiring more skill than luck, blackjack is one of the most exciting casino games in existence. It attracts various types of players, including complete novices, high-rollers, expert-level gamblers, and skilled card counters. While some rely on knowing the basic rules of the game to play for fun, others believe they need a deep understanding of the mathematics and principles behind blackjack.

This article aims at explaining the mathematics of the game, as well as some of the ideas that are fundamental to blackjack theory – deck composition, probability, and house edge are just some of them. Those who are interested in this classic casino game would benefit from expanding their knowledge, whether they are making their first steps in blackjack or consider themselves experienced.

Blackjack Deck Composition

Before looking at the odds of blackjack, players need to understand the importance of the decks used in every game – the number of decks, the way cards are being dealt, etc. Blackjack is usually played with 6 or 8 standard decks of 52 cards, but there are also games that use 1,2, or 4 decks. Since all face cards (Jacks, Queens, and Kings) are counted as 10, there will be more 10-value cards than any other type of cards in each deck – 16 to be precise.

In addition, each deck consists of 20 high (from 10 through Ace) and 32 low (from 2 through 9) cards. Other forms of categorization are typically used in card counting systems. All decks are shuffled together before a game and are dealt from a box called a “dealer’s shoe” or from a shuffling machine.

Independent vs Dependent Events
Deck Penetration

Blackjack Probability and Odds

Many people use the expressions “odds”, “probability”, and “likelihood” interchangeably. And while these three show basically the same thing, they are actually quite different from each other. For gamblers and blackjack players, it is particularly important to understand this difference. Both “odds” and “probability” are used to express the likelihood of an event happening. Both are basic statistics terms that estimate chance and while we often associate odds with gambling, probability also exists as a separate branch of mathematics and could be used in a variety of fields.


Blackjack Probability Charts

When beginning to use basic strategy, blackjack players need to understand that all moves suggested in the various strategy charts have been determined by probabilities. It is important to know the probability of a hand busting if the player decides to draw an additional card, as well as the probability of player or dealer blackjack in the initial deal.

Based on the calculations illustrated above, we can identify the probability of busting, getting a natural 21, or pulling any particular card in different situations. It is sometimes referred to as the “frequency” of a card or an outcome – the frequency or probability of a player blackjack is around 4.80%, for example. Most of the time – in 38.70% of the time to be precise, players would be dealt the so-called decision hands totaling 2 through 16.

To illustrate how we have estimated that, let’s see how many combinations there are for the first two cards in a single-deck variation – (52*51)/2, which is 1326. The possible combinations for blackjack are 64 (4 Aces multiplied by the 16 cards with a value of 10). So, the probability for having 21 in the initial hand is 64/1326 – this is 0.04826 or 4.83%. In a double-deck game, the probability for blackjack is 4.77%, in a 4-deck game it is 4.75%, and in a 6-deck blackjack variation, it is 4.74%.

Below, players will find several important probability charts that display the likelihood of various favorable or unfavorable outcomes.

Player First Hand Probability
Player Busting with a Hit on First 2 Cards
Dealer 2 Cards Probability
Dealer Bust Probability with Up Card

Blackjack House Edge and Variance

House Edge

Variance and Standard Deviation

Most casino patrons are well aware of the house edge in every game they play. The term refers to the mathematical advantage casinos have over their patrons and it is measured in percentages. The house edge is a direct result of the rules of the game and in blackjack, this includes the number of decks, the Hit/Stand on Soft 17 rule, the presence of a hole card, the bonus payout for player blackjack (3:2), and many others. Of course, the house edge is also determined by the probabilities.

More importantly, the dealer has a significant advantage in this game because he is the last participant to act. Typically, blackjack is considered to be the casino game with the lowest house edge at only around 0.50%. This means that the casino will keep 0.50% of all wagers in the game, whereas players can expect to lose on average $0.50 on their $100 bets. However, the said percentages can be achieved only when casino patrons use a perfect strategy when playing blackjack. Otherwise, casinos have an advantage of up to 8% in most variations of blackjack and a typical player would often play at around 2% or 3% disadvantage.

Another interesting thing players need to consider is the fact that the house edge and its counterpart, the return to player (RTP), are theoretical concepts. They could be true only after millions or even billions of blackjack hands have been played. Still, they are essential when we assess a certain variation of blackjack – the lower the house edge, the more favorable rules.

View more...