# 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

### Independent 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 or the next one. The probability of any particular number being hit remains the same in 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 to be a lucky number, whereas many people believe 13 brings bad luck.

In 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 has certain effects on the likelihood for the next one. In other words, when one card is dealt on the table, it changes the composition of the remaining cards in the shoe. Therefore, it changes the likelihood of certain hands being formed.

If this explanation sounds too vague, let us take this example – in single-deck blackjack, we have 4 Aces, which means that the likelihood of drawing an Ace is 4 in 52. The dealer deals 3-4 to the player and draws an Ace to himself. Suddenly, the likelihood that the player will receive an Ace gets slimmer (3 in 52) because there are only 3 Aces left in the deck. As we can see, the probabilities of winning or losing the game change with every card dealt by the dealer.

### Deck Penetration

Another expression players would probably encounter in blackjack guides is deck penetration. This term describes the percentage of cards that have been dealt before the dealer reshuffles the cards. Remember that not all cards in the shoe will be dealt – usually, the dealer leaves a small portion of them at the bottom of the shoe. For example, if 34 cards have already been used in a single-deck version of blackjack before the dealer decides to reshuffle, he has used 65 percent of all 52 cards. The deck penetration is, then, 65% and it is determined by the rules of the casino.

This principle has little use to players who rely on basic blackjack strategy but it is very important for card counters. For them, it is essential to count the cards that have been dealt so they can estimate what cards are likely to be drawn by the dealer next. Therefore, they should be looking at games with greater deck penetration. Usually, it is around 70% or 75% and if the blackjack variation offers even a greater one – let’s say 80%, the card count will be more accurate.

Of course, the number of decks is very important when we discuss penetration. While a 65% deck penetration is sufficient for a true count in a 6-deck game, one would want a penetration of at least 75%. If it is less than that, players would not be able to accurately count the remaining cards and would not know when to increase their bets.

## 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.

Probability
Odds

### Probability

Probability refers to the likelihood of an event occurring and ranges from 0 to 1, where 0 means that the event is impossible to happen and 1 means that 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 it 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.

Let us assume we play a single-deck blackjack variation where there is no hole card. The player receives 9-6 while the dealer draws a Queen. In this situation, the basic strategy calls for a Surrender but the game does not allow this option. So, the best solution here would be determined by the probabilities for busting and winning. To calculate those, we simply compare the number of favorable cards to all cards still left in the deck.

Since 3 cards have already been dealt, the shoe consists of 49 cards in total. The hand totaling 15 will bust with cards of 7 or higher – the 7s, 8s, and 10s and the three 9s still left in the deck, along with the four Jacks, four Kings, and three Queens. So, all cards with a value of up to 6 plus the Aces would be favorable – 23 in total. The probability of getting a good card will then be 23/49, which is 0.4693 or expressed as a percentage, 46.93%.

The probability for busting the hand 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 different variation of blackjack.

### Odds

In gambling, we can often see chance expressed with odds but this term is a bit different than 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 shown in reverse as expressing player’s chances against winning. The reason for this is very simple – casino patrons are (almost) always more likely to lose than to win.

If we use the same example, where 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).

## 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

### Player 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%

## Blackjack House Edge and Variance

### 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.

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