Probability Blackjack Math
Blackjack: the math behind the cards 2 and Ace cards worth 1 or 11 points used at the player’s discretion. The objective of the game is receiving cards with a total point value of 21, hence giving Blackjack its nickname 21'. Tunica gambling Probability Of Blackjack Math 888 tiger casino mobile little wind river casino. How we use cookies. Cookies help us provide, protect and improve our services. We also use cookies to verify your financial information and identity and for fraud prevention purposes. Third parties use cookies on our site to tailor adverts to. The probability of a blackjack in a single deck game is 4.16/combin(52,2) = 64/1326. So the probability of four in a row is (64/1326) 4 = 16777216/ 176 = 1 in 184270. However the actual probability is much less, because as the player gets each blackjack the ratio of aces to cards left in the deck decreases. Learn how to beat the house! Or, if that's not possible, at least learn how much of an edge they have. In this course, you'll analyze three specific casino games: blackjack, craps, and poker. And, while you're learning how to master these games, you'll also strengthen your skills working with many foundational probability topics including probability trees and conditional probability. For example, the odds of a player receiving blackjack are 4.8%. That’s almost five natural blackjacks for every 100 hands played. Meanwhile, a player will be dealt cards totaling between 1 and 16 38.7% of the time. This means that you’ll have to make a decision on whether to stay or hit on approximately 39 of 100 hands dealt.
We first present the probabilities attached to card dealing and initial predictions. In making this calculus, circumstantial information such as fraudulent dealing is not taken into account (as in all situations corresponding to card games). All probabilities are calculated for cases using one or two decks of cards. Let us look at the probabilities for a favorable initial hand (the first two cards dealt) to be achieved. The total number of possible combinations for each of the two cards is C(52, 2) = 1326, for the 1-deck game and C(104, 2)=5356for the 2-deck game.
Probability of obtaining a natural blackjack isP= 8/663 = 1.20663% in the case of a 1-deck game andP = 16/1339 = 1.19492%in the case of a 2-deck game.
Probability of obtaining a blackjack from the first two cards isP= 32/663 = 4.82654%in the case of a 1-deck game andP = 64/1339= 4.77968%in the case of a 2-deck game.
Similarly, we can calculate the following probabilities:
Probability of obtaining 20 points from the first two cards isP = 68/663 = 10.25641% in the case of a 1-deck game andP= 140/1339 = 10.45556%in the case of a 2-deck game.
Probability of obtaining 19 points from the first two cards isP = 40/663 = 6.03318%in the case of a 1-deck game andP = 80/1339 = 5.97460%in the case of a 2-deck game.
Probability of obtaining 18 points from the first two cards isP= 43/663 = 6.48567%in the case of a 1-deck game andP= 87/1339 = 6.4973% in the case of a 2-deck game.
Probability of getting 17 points from the first two cards isP= 16/221 = 7.23981%in the case of a 1-deck game andP= 96/1339 = 7.16952%in the case of a 2-deck game.
A good initial hand (which you can stay with) could be a blackjack or a hand of 20, 19 or 18 points. The probability of obtaining such a hand is calculated by totaling the corresponding probabilities calculated above: P = 32/663 + 68/663 + 40/663 + 43/663 = 183/663, in the case of a 1-deck game and P = 64/1339 + 140/1339 + 80/1339 + 87/1339 = 371/1339, in the case of a 2-deck game.
Probability Blackjack Math Definition
Probability of obtaining a good initial hand isP= 183/663 = 27.60180%in the case of a 1-deck game andP= 371/1339 = 27.70724%in the case of a 2-deck game.
The probabilities of events predicted during the game are calculated on the basis of the played cards (the cards showing) from a certain moment. This requires counting certain favorable cards showing for the dealer and for the other players, as well as in your own hand. Any blackjack strategy is based on counting the cards played. Unlike a baccarat game, where a maximum of three cards are played for each player, at blackjack many cards could be played at a certain moment, especially when many players are at the table. Thus, both following and memorizing certain cards require some ability and prior training on the player’s part. Card counting techniques cannot however be applied in online blackjack.
The formula of probability for obtaining a certain favorable value is similar to that for baccarat and depends on the number of decks of cards used. If we denote by x a favorable value, by nx the number of cards showing with the value x (from your hand, the hands of the other players and the face up card in the dealer’s hand) and by nv the total number of cards showing, then the probability of the next card from the deck (the one you receive if you ask for an additional card) having the value x is:
This formula holds for the case of a 1-deck game. In the case of a 2-deck game, the probability is:
Generally speaking, if playing with m decks, the probability of obtaining a card with the value x is:
Example of application of the formula: Assume play with one deck, you are the only player at table, you hold Q, 2, 4, A (total value 17) and the face up card of the dealer is a 4. Let us calculate the probability of achieving 21 points (receiving a 4).
We havenx = 2, nv = 5, so:
.
For the probability of achieving 20 points (receiving a 3), we havenx = 0, nv = 5, so:
.
For the probability of achieving 19 points (receiving a 2), we havenx = 1, nv = 5, so:
.
If we want to calculate the probability of achieving 19, 20 or 21 points, all we must do is total the three probabilities just calculated. We obtainP = 9/47 = 19.14893%.
Unlike in baccarat, where fewer cards are played, the number of players is constant (two), and the number of gaming situations is very limited, in blackjack, the number of possible playing configurations is in the thousands and, as a practical matter, cannot be entirely covered by tables of values.
Sources |
A big part of the gaming situations that require a decision, where the total value held is 15, 16, 17, 18, 19 or 20 points, is comprised in tables in the section titled Blackjack of the book PROBABILITY GUIDE TO GAMBLING: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets.You will also find there other issues of probability-based blackjack strategy . See the Books section for details. |
Card counting
Card counting is the practice of keeping track of the cards dealt in previous blackjack hands in order to calculate the possibility of specific cards being dealt in the current hand. Players good at this method can gain an advantage over the house. Card counting is much easier in one-deck games of blackjack, but can become quite difficult in multi-deck games, and in games with frequent reshuffles. While card counting mentally is completely legal, using external devices to enable this tracking is illegal. Players visibly seen as counting the cards at the blackjack tables are unwelcome guests in casinos and are frequently asked to leave the premises.
Card counters typically assign a point score to each rank of card that is dealt at the table, keeping a running score in their heads. As each card is exposed, the player adds the score of that card to the running total, which is considered the 'count'. After each reshuffle of the deck, the count starts again at 0.
The films “Rain Man” and “21” and some other blackjack movies included scenes of players counting cards, giving them an advantage over the casinos where they played blackjack.
Hi-Lo strategy
Hi-Lo strategy is a card counting system, and it is actually the easiest one to learn. Using the Hi-Lo card counting calculation system, all cards are assigned values of +1, 0 and -1 depending on their value in the game itself. Cards numbered 2–6 are called low cards and assigned a value of +1. Aces, tens, jacks, queens, and kings, the high cards, are assigned a value of -1. The remaining cards (7, 8, 9) are assigned a value of 0.
As each card is dealt at the blackjack table it is easy to add up the point values in your head. The greater that number is, the more high cards remaining in the deck to be dealt. This is a simple method of card counting that will make it easier to decide whether to hit, stand, split or double down on the cards you are dealt.
Shuffle tracking
Shuffle tracking is an advanced form of card counting. Instead of tracking the count of individual cards, the player keeps track of one or more subsections of cards as they are played. These sections are referred to as tracking zones. After the cards are played, and then shuffled, there are zones of cards in the upcoming deals which contain most of the cards in the original tracking zones. This method requires keen eye observation and memory skills
Probability in blackjack
Card counters and all avid blackjack players who rely on skill in order to achieve an advantage at the table employ some element of probability in their calculations as to what card will next be dealt. Players know the number of two-card combinations, the possibility of getting a blackjack, and the total number of possible outcomes in each hand. All of these skills are applied blackjack math.
When you employ a card counting technique, you can calculate the probability of specific cards being dealt to you, and, what the dealer’s face down card may be. Keeping track of the score will allow you to calculate the blackjack odds of being dealt a 3, or alternatively a face card. The more you keep track, the more you will be able to apply blackjack mathematics and know how to play.