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Blackjack Math: A Primer to Understanding Blackjack Probabilities and Statistics
by Nicholas G. Colon
Nicholas is an 18 year veteran of the Casino Gaming Industry and Managing Director of Carnivore Investment Strategies LLC. He is an accomplished intellectual engineer with profound contributions in Analytic's, Finite Mathematics and is leading analytical consultant . He can be contacted at
nicholas.g.colon@gmail.comKnowing the mathematics behind any casino game is the first step that must be taken towards attacking it for monetary gain. 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 starting point for the inclined student. For a more complete derivation of the subject, Peter Griffins "Theory of Blackjack," covers the subject in its entirety.
Blackjack at its core 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. (I 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 are several counting systems that have been developed to keep track of the shifting percentages, each having varying degree of efficiencies. The Hi-Lo count is the simplest, while the Uston APC count is the most effective but also the most complex.
In a standard deck of cards, there are 13 cards in each of the four suits. From simple deduction, we can see that the probability of any one card being dealt is 1 in 52.
Equation 1
To get the percentage of any one card in N number of decks, simply multiply the fraction by N.
Equation 2
(Note: Equation 2 reduces to equation 1.)
Further exploring the deck composition, we can see that there are only 13 different values in a deck of cards (2 through Ace). Moreover, there are four suits in each deck (
), each value can occur four times in each deck. This is stated in the following equation:
Equation 3
Adjusting for N number of decks we get:
Equation 4
In each suit, 4 of the 13 Cards have a 10 value, which are 10, J, Q & K. Carrying this through a single deck, we find that 16 out of 52 cards have a 10 value.
Equation 5
Multiplying this through N number of decks we get:
Equation 6
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.
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 busts, (i.e., attains a total of 22 or more) and the dealer also busts (goes over the value of 21), the player loses his bet and the dealer does not have to pay out. This Double Bust Property is the source of the casinos advantage.
Every other rule that the casino offers benefits the player. Whether or not the dealer hits or stands on a soft 17 (i.e., two initial cards being A-6), both benefit the player. It merely benefits the player more when the dealer is bounded by rule to stand on a soft 17. Additional rules, like Doubling Down, Splitting like Pairs, Doubling after Splits and Early or Late Surrender adds value to the player as well. Knowing when to apply these options optimally is determined by computer simulation and is found in the basic strategy chart in my three-part article mentioned above. It is worth noting that many casinos sell basic strategy cards in their gift shops. The player should be cautious because there are usually one or two mistakes on those strategy 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 in the doubling down options.
There are two common variations of Blackjack found in most casinos. The first is a six- deck shoe game, with these rules: dealer Hits on Soft 17, Pair Splitting, Double after Split, and 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 six-deck shoe game, with these 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% (assuming the player plays perfect basic strategy). To determine the house advantage for any mix of rules and number of decks, a player can 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 basic ideas and extrapolate dozens of ways to attack the game for profit.
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