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Blackjack Math

 Nicholas Colon, Author

A Primer to Understanding Blackjack Probabilities and Statistics

Blackjack math is important, whether that’s how to count cards, learning about bet variation or otherwise. Blackjack is the most notable game when advantage play is discussed, although it is certainly not the only game susceptible to exploitation by a skilled player. This article is intended to serve as a blackjack math starting point for the inclined student. For a more complete derivation of the subject, Peter Griffin’s “Theory of Blackjack,” covers the subject in its entirety.


Blackjack Mathematics and Blackjack Statistics

Blackjack probabilities change: at its core, Blackjack is the study of shifting percentages. Each card that is played has a specific value that it adds to, or subtracts from the initial advantage that the casino has over the player. We will discuss the source of the initial houses advantage shortly. The percent value that each card holds varies from card to card and is carried out to several decimal places. There several counting systems that have been developed to keep track of the shifting percentages, each having varying degree efficiencies. The Hi-Lo count is the simplest while the Uston APC count is the most effective as well as the most complex.


The Math behind Blackjack

In a standard deck of cards there are 13 cards in each of the 4 suit. From simple deduction we can see that the probability that any one card is dealt is 1 in 52.


1/(4*13)=  1/52
Equation 1


To get the percentage of any one card in N number of decks, simply multiply the fraction by N.


N/((4*13)*N)=  N/(52*N)
Equation 2


This reduces to equation 1.

Further exploring the deck composition we can see that there are only 13 different values in a deck (2-Ace). And because there are 4 suits in each deck, each value can occur 4 times in each deck. This is stated in the following equation:


Probaility Per Value=4/52
Equation 3


Adjusting for N number of decks we get:


Probaility Per Value in N decks=(4*N)/(52*N)
Equation 4


In each suit 4 of the 13 Cards have a 10 value. They are 10, J, Q & K. Carrying this through the a single deck we find that 16 out of 52 cards have a 10 value.


Probaility of a 10 card=(16 )/52
Equation 5


Multiplying this through N number of decks we get:


Probaility of a 10 card for N number of decks=(16*N )/(52*N)
Equation 6


The Probability of Blackjack

The probability of any particular card being dealt off the top of the shoe is straight forward to calculate. As each card is played, the chance of that card appearing again gets smaller and smaller. Keeping track of the cards played and percentage shift is the function of the count system. For the full derivation of the high-low count system see my previous three-part article titled “The M.I.T. Guided to Card Counting,” found at the following links:

Determining the initial house advantage is a little more complex then determining the straight forward probabilities of a single card being dealt. The casino’s advantage comes from a combination of the Double Bust Property and the rules selected by the casino. First, let us consider the Double Bust Property. The Double Bust Property is summed up in the order of who draws first. In Blackjack the player decides whether to draw or stand prior to the dealer knowing their total, and subsequently whether or not they need to take a card or not. In the case where the player bust, that is attains a total of 22 or more and the dealer also busts (goes over the value of 21) the player loses their bet and the dealer does not have to pay out. This is the source of the casino’s advantage.

Every other rule that the casino offers benefits the player. Whether or not the dealer hits or stands on a soft 17 (where the dealer hits or stands with the two initial cards A,6), both benefit the player. It merely benefits the player more when dealer is bounded by rule to stand on a soft 17. Additional rules, like Doubling Down and Splitting like Pairs, Doubling after Splits and Early or Late Surrender adds value the player as well. Knowing when to apply these options optimally is determined by computer simulation and is found in basic strategy chart in my three-part article titled “The M.I.T. Guided to Card Counting.” It is worth noting that many casinos sell basic strategy cards in their gift shops. The player should be cautious that there is usually one or two mistakes on those cards that increase the houses’ advantage over the player. A common one is stating that a player should stand with a 12 against a dealers 3 up card, when the correct play is take a hit. The other error is usually found somewhere in the doubling down options.


Blackjack Probabilities

There are two common variations of Blackjack found in most casinos. The first is a 6 deck shoe game, with the rules dealer Hits on Soft 17, Pair Splitting, Double after Split, with Blackjacks paying 3:2. This game has an initial house advantage of approximately 0.53%, assuming the player plays perfect basic strategy. The second is a 6 deck shoe game, with the rules dealer Stands on Soft 17, Pair Splitting, Double after Split and Late Surrender. This game has an initial house advantage of approximately 0.26%, this also assumes the player plays perfect basic strategy. To determine the house advantage on of any rule set combination a player simply do a web search on the key term “blackjack calculator,” and the player will be able to select the rules they are playing and the initial house advantage will be calculated.

The information provided here gives the aspiring player a starting point on the map of beating Blackjack. It is by no means a full analysis of the subject, however not much else is required before learning to attack the game for profit. One is able to take these fundamental ideas and derive dozens of ways to attack the game for profit.


For even more blackjack tips – head to our online blackjack page!


DISCLAIMER: This article is intended for educational use only, and cannot promise any winnings from using its tips and blackjack math.


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