| Updated: September 25, 2025Relatively simple to play, fast-paced, and requiring more skill than luck, blackjack is one of the most exciting casino games in existence. It attracts all types of players, including complete novices, high rollers, expert-level gamblers, and skilled card counters. While some players rely only on the basic rules when they play for fun, others believe that a deep understanding of the mathematics and principles behind blackjack is essential.
This article aims to explain the mathematics of the game, along with several concepts fundamental to blackjack theory – deck composition, probability, and house edge, to name a few. Anyone interested in this classic casino game will benefit from broadening their knowledge, whether they are taking their first steps in blackjack or already consider themselves experienced.
Blackjack Deck Composition
Before looking at blackjack odds, players need to understand the importance of the decks used in every game – the number of decks, how the cards are dealt, etc. Blackjack is usually played with 6 or 8 standard 52-card decks, but there are also games that use 1, 2, or 4 decks. Because all face cards (Jacks, Queens, and Kings) count as 10, there are more 10-value cards than any other card in each deck – 16 to be precise.
In addition, each deck contains 20 high cards (from 10 through Ace) and 32 low cards (from 2 through 9). Other forms of categorization are commonly used in card-counting systems. All decks are shuffled together before the game and are dealt from a box called a “dealer’s shoe” or from a shuffling machine.
Independent vs Dependent Events
Deck PenetrationIndependent vs Dependent Events
Many casino games, such as slot machines or roulette, rely heavily on randomness – in roulette, for example, the outcome of each spin is independent of the previous one or the next. The probability of any particular number being hit remains the same on every spin of the roulette wheel. The numbers 7 and 13, for instance, are equally likely to win or lose, even though 7 is widely considered lucky whereas many people believe 13 brings bad luck.
On every spin of the roulette wheel, the likelihood of the ball landing on 13 (or any other number, for that matter) is exactly 1 in 37. In blackjack, however, each event is influenced by the previous one and affects the likelihood of the next. In other words, when one card is dealt onto the table, it changes the composition of the remaining cards in the shoe and, consequently, the probability of certain hands being formed.
If this explanation sounds too vague, consider this example – in single-deck blackjack we have four Aces, which means the likelihood of drawing an Ace is 4 in 52. The dealer deals 3-4 to the player and draws an Ace for himself. Suddenly, the likelihood that the player will receive an Ace decreases to 3 in 52 because there are only three Aces left in the deck. As we can see, the probabilities of winning or losing change with every card dealt by the dealer.
Deck Penetration
Another expression players will likely encounter in blackjack guides is deck penetration. The term describes the percentage of cards that have been dealt before the dealer reshuffles. Remember that not all cards in the shoe will be dealt – the dealer usually leaves a small portion at the bottom of the shoe. For example, if 34 cards have already been used in a single-deck game before the dealer reshuffles, 65 percent of the 52 cards have been dealt. The deck penetration is therefore 65%, and it is determined by the casino’s rules.
This principle has little value to players who rely solely on basic blackjack strategy, but it is crucial for card counters. They need to track the cards that have been dealt so they can estimate which cards are likely to appear next. Consequently, they look for games with higher deck penetration. Typically, it is around 70% or 75%, and if the variation offers even more – say 80% – the count will be more accurate.
Of course, the number of decks matters greatly when we discuss penetration. While 65% penetration is sufficient for an accurate true count in a 6-deck game, players prefer at least 75%. If penetration is lower, they cannot accurately track the remaining cards and will not know when to increase their bets.
Blackjack Probability and Odds
Many people use the terms “odds,” “probability,” and “likelihood” interchangeably. Although all three relate to the same concept, they are actually quite different. For gamblers and blackjack players, understanding this distinction is particularly important. Both “odds” and “probability” express the likelihood of an event occurring. They are basic statistical terms that estimate chance, and while we often associate odds with gambling, probability is also a separate branch of mathematics that can be applied in many fields.
Probability OddsProbability
Probability refers to the likelihood of an event occurring and ranges from 0 to 1, where 0 means the event is impossible and 1 means its occurrence is certain. If we toss a coin, for example, the probability of getting heads is exactly 0.50. The probability of tails is the same and can also be expressed as ½ or 50%. In blackjack, players need to understand the probability of winning or busting in every scenario if they want to be successful.
Assume we are playing a single-deck blackjack variation in which there is no hole card. The player receives 9-6 while the dealer shows a Queen. In this situation, basic strategy recommends a Surrender, but the game does not allow this option. Therefore, the best solution must be determined by the probabilities of busting and winning. To calculate them, we simply compare the number of favorable cards to all cards still left in the deck.
Because three cards have already been dealt, the shoe now contains 49 cards. A hand totaling 15 will bust with cards of 7 or higher – the 7s, 8s, and 10s, the three 9s still in the deck, along with the four Jacks, four Kings, and three Queens. Thus, all cards with a value of up to 6 plus the Aces are favorable – 23 in total. The probability of getting a good card is therefore 23/49, which equals 0.4693 or, expressed as a percentage, 46.93%.
The probability of busting the hand if we draw another card is 26/49, or 53.06%. Since hitting is more likely to result in a bust, the best option is to Stand. By calculating the probabilities for each situation, we can develop the optimal strategy for every variation of blackjack.
Odds
In gambling, we often see chance expressed as odds, but this term is somewhat different from probability. It represents the ratio of occurrence to non-occurrence or, in blackjack, the ratio between winning and losing outcomes. In fact, odds in gambling are usually shown in reverse, expressing the player’s chances against winning. The reason is simple – casino patrons are almost always more likely to lose than to win.
Using the same example in which it is best to Stand on a hard 15 against a dealer Queen, the odds of busting this hand if we Hit are 26 to 23 (26/23).
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Blackjack Probability Charts
When starting to use basic strategy, blackjack players need to understand that every move suggested in strategy charts is derived from probability. It is important to know the probability of a hand busting when the player draws an additional card, as well as the probability of player or dealer blackjack on 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. This is sometimes referred to as the “frequency” of a card or outcome – for example, the frequency or probability of a player blackjack is around 4.80%. Most of the time – 38.70% to be precise – players are dealt the so-called decision hands totaling 2 through 16.
To explain how we reached that number, let’s see how many combinations there are for the first two cards in a single-deck game – (52*51)/2, which equals 1,326. The possible combinations for blackjack are 64 (four Aces multiplied by the 16 cards valued at 10). So, the probability of having 21 in the initial hand is 64/1,326, which is 0.04826 or 4.83%. In a double-deck game, the probability of blackjack is 4.77%; in a 4-deck game it is 4.75%; and in a 6-deck variation, it is 4.74%.
Below, players will find several important probability charts that display the likelihood of various favorable and unfavorable outcomes.
Player First Hand Probability Player Busting with a Hit on First 2 Cards Dealer 2 Cards Probability Dealer Bust Probability with Up CardPlayer First Hand Probability
| Player First Hand Probability | |
|---|---|
| Hand Total | Probability (%) |
| Blackjack | 4.80% |
| Standing Hand (17-20) | 30.00% |
| Decision Hand (2-16) | 38.70% |
| No Bust | 26.50% |
Player Busting with a Hit on First 2 Cards
| Player Busting with a Hit on First 2 Cards | |
|---|---|
| Hand Total | Chance for Busting (%) |
| 21 | 100.00% |
| 20 | 92.00% |
| 19 | 85.00% |
| 18 | 77.00% |
| 17 | 69.00% |
| 16 | 62.00% |
| 15 | 58.00% |
| 14 | 56.00% |
| 13 | 39.00% |
| 12 | 31.00% |
| 11 or less | 0.00% |
Dealer 2 Cards Probability
| Dealer 2 Cards Probability | |
|---|---|
| Hand Total | Probability (%) |
| Blackjack | 4.82% |
| 21 (3 or more cards) | 7.36% |
| 20 | 17.58% |
| 19 | 13.48% |
| 18 | 13.81% |
| 17 | 14.58% |
| 16 | 28.36% |
Dealer Bust Probability with Up Card
| Dealer Bust Probability with Up Card, Player Advantage with Basic Strategy | ||
|---|---|---|
| Dealer Up Card | Bust Probability (%) | Player Advantage (%) |
| 2 | 35.30% | 9.80% |
| 3 | 37.56% | 13.40% |
| 4 | 40.28% | 18.00% |
| 5 | 42.89% | 23.20% |
| 6 | 42.08% | 23.90% |
| 7 | 25.99% | 14.30% |
| 8 | 23.86% | 5.40% |
| 9 | 23.34% | -4.30% |
| 10, J, Q, K | 21.43% | -16.90% |
| A | 11.65% | -16.00% |