It's important that you know whether the rules on a particular blackjack table require that the dealer stand on soft 17 or must hit soft 17; whether you can double down on any two initially dealt cards or if there are restrictions on what hands you can double; whether surrender is allowed; whether re-splitting of aces is allowed; and, most importantly, what the payoff is for an untied blackjack (is it 3-2, 6-5, 7-5, or even. Blackjack is a difficult game to analyze, and differences in methodology can play a minor effect on the results. Both sets of figures are more than adequate. The 'cut' column shows the number of decks yet unseen when the dealer usually shuffles.
With the odds on a blackjack table for instance, there is a little more variance used that has an effect on the amount of the house edge. The hold is different on the blackjack table. Out of the total cash dropped on the tables, the hold is what the casino keeps. Blackjack should need no introduction. It is the most popular table game in the United States, and is easily found in casinos throughout the world. The object of the game of Blackjack is simply to get more points than the dealer without going over 21. Rules Hand Signals Wizard's Simple Strategy Basic Strategy Blackjack.
Blackjack is a slightly deceptive game. Its simple rules of play may fool you into believing it is easy to master but if you delve deeper, you will quickly find this is a purely mathematical game that is all about odds and probabilities. Blackjack Hall of Fame inductees Edward Thorp and Julian Braun were among the first people to come to this realization in the 1960s. They ran millions of simulations on old IBM computers to refine the basic blackjack strategy Ed Thorp published in his Beat the Dealer book, which is now a classic in the blackjack canon.
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If you truly want to win at this game, you need to gain a good understanding of what the odds and probabilities for every possible scenario at the table are and base your playing decisions on these odds. The following article aims at introducing you to the foundations of blackjack odds and probabilities. Toward the end, we have also included several charts that may prove to be useful.
Understanding Blackjack Probabilities
Many people use 'probability' and 'odds' as two interchangeable terms but in fact, there is a pronounced difference between the two. While inherent in gambling, probability is, first and foremost, a separate branch of mathematics that deals with the likelihood of different events occurring. Probability permeates all aspects of our lives, from weather reports to statistics and playing at your local casino.
Probability is calculated on the basis of known data but cannot be used to predict exact outcomes, like the outcome of a hand in the game of blackjack, for example. It simply shows you the likelihood of an event happening, based on the knowledge of the number of desired outcomes and the number of all possible outcomes. You can use this knowledge to execute the best play at the blackjack table but it alone does not show you with absolute certainty which card the dealer will pull out next.
Statisticians use what is known as a 'probability line' to represent the probabilities of events, which can be classified as certain, likely, unlikely, and altogether impossible. The further to the left an event is positioned on the probability line, the more unlikely it is to take place. Conversely, when an event is positioned further to the right of the line's center, it has a higher likelihood of occurrence.
The probability of a given outcome happening is quite simple to calculate. All you have to do is divide the number of desired outcomes by the number of all possible outcomes. In the context of gambling, this translates into dividing the number of ways to win by the number of all possible outcomes.
Independent vs. Dependent Trials
Before we proceed with concrete examples, we would like to make a distinction between independent and dependent events (or trials in statistics). An independent event has no impact on another event's probability of occurring (or not occurring). Such is the case with the dice tosses in the game of craps and roulette spins, where previous outcomes have no influence on the results of the trials that are to follow.
Here is an example of determining the probability of rolling a 2 with a six-sided dice. There is only one possible way for you to roll a 2 out of 6 possible outcomes. It follows that the likelihood of a 2 being rolled is 1/6 = 1.166 * 100 = 16.66%.
Casino Odds Blackjack
The probability of rolling a deuce with two dices is even slimmer because there are more permutations (36 to be precise) but there is only one two-dice combination that results in a 2. Respectively, the likelihood of this independent event occurring is 1/36 = 0.027 * 100 = 2.77%. No matter how many times you toss the dice, the probabilities of the tosses, or trials, will always remain the same.
With dependent events, on the other hand, previous trials influence the probabilities of the trials that are to follow. Unlike roulette and dice games, blackjack is a game of dependent trials where each card dealt changes the composition of the remaining deck and therefore, influences the likelihood of forming specific hands on the next rounds of play.
This phenomenon is best explained through examples, so let's calculate the probability of drawing an Ace from a single deck of cards. Single-deck blackjack utilizes a full deck of 52 cards without any jokers where we have 13 card denominations of 4 different suits each, which is to say there are only 4 Aces out of 52 cards. Therefore, the likelihood of you pulling an Ace at random is P(Ace) = 4/52 = 0.0769 * 100 = 7.69%.
Independent vs. Dependent Trials Additional TipsProvided that the Ace you have already drawn is discarded instead of being reintroduced back into the pack, the probability of pulling an Ace on the next trial will be slimmer. We have three Aces left and the number of cards in the deck has now dropped to 51. The calculations are as follows: P(Ace) = 3/51 = 0.0588 * 100 = 5.88%. The more cards of a given denomination have left the deck, the slimmer the likelihood of drawing a card of said denomination on your next try.
The only unknown factor in the game of blackjack is which card we will pull out next. We can determine the probability of drawing a specific card but cannot say with absolute certainty which card will leave the pack on the next draw.
The only random factor that impacts the draws is the reshuffle. If we place the Ace back into the deck and reshuffle after each trial, the probability of pulling it will remain the same as now you will be dealing with independent trials.
Another Example – Hard 16 vs. Dealer 10 (No Surrender)
Let's demonstrate how probabilities in blackjack work when more cards have left the deck. We assume you start a fresh round in a no-hole-card game after the single deck has been reshuffled. You are dealt Q-6 against a dealer K but surrender is unavailable, in which case you are forced to hit your hard 16. What is the likelihood of improving your total on the next hit?
We are left with 49 cards since 3 cards have been removed from the deck already. The following cards can help you improve your 16: an Ace for a total of 17 (your Ace will be counted as 1 in this case since otherwise it will bust your hand), a deuce for a total of 18, a 3 for a total of 19, a 4 for a total of 20, and a 5 for the best possible outcome of 21.
Thus, you have 20 cards that can help you out of 49. The probability of drawing a 'good' card is 20/49 = 0.408 * 100 = 40.8%. Conversely, the likelihood of you busting by pulling one of the 29 'bad' cardsis 29/49 = 0.591 * 100 = 59.1%.
The Probability of Obtaining a Blackjack
Naturals are the strongest hands you can obtain in the game of 21. Not only it is impossible to lose with a natural (in the worst-case scenario you will push with the dealer) but you get a little extra in terms of profits since blackjacks return 1.5 times your original bet (on condition you are not foolish enough to play 6-to-5 games). Because of this, it is important for you to gain a proper understanding of the probabilities of getting blackjacks.
Knowing the number of decks in play, you can easily determine the likelihood of receiving a natural after the reshuffle. For the purpose, you must multiply the probability of pulling an Ace by the probability of pulling ten-valued cards like 10, J, Q, and K (there are four of each in a single pack for a total of 16). It is also necessary to multiply the result by 2 because there are two possible permutations of cards in a hand of blackjack, for example A-Q and Q-A, K-A and A-K, and so on.
The probability of drawing an Ace is 4/52 while that of pulling one of the ten-value cards is 16/51. Is huuuge casino real money. The number of cards has dropped to 51 in the second case to account for the Ace that has been removed from the deck. Therefore, we calculate the probability of getting dealt a blackjack in the following way: P (Ace) * P (Ten-Value Card) * 2 = (4/52) * (16/51) * 2 = 0.0482 * 100 = 4.82%.
The likelihood of getting naturals decreases as more decks are introduced into the game, which, inturn, slightly increases the advantage the casino has over you. This often sounds antithetical to inexperienced players who reason it should be the other way around because there are more Aces and ten-value cards when multiple decks are used.
The Probability of Obtaining a Blackjack Additional TipsThis line of reasoning is erroneous because the effect of the individual cards' removal is not as pronounced in multiple deck games as it is in single or double deck variations. The probability formula we use is the same, however, no matter how many decks are in play.
Below are the probabilities of drawing a blackjack at the start of a fresh shoe with two, four and six decks. You will notice the difference becomes less pronounced the more decks are introduced into play. The difference in blackjack probabilities between six and eight decks is so minuscule, we did not bother including the calculations here.
- The probability of a blackjack with two decks is (8/104) * (32/103) * 2 = 0.0479 * 100 = 4.77%
- The probability of a blackjack with four decks is (16/208) * (64/207) * 2 = 0.0475 * 100 = 4.75%
- The probability of a blackjack with six decks is (24/312) * (96/311) * 2 = 0.0474 * 100 = 4.74%
Converting Probability into Odds
Odds Of Winning Blackjack At Casino
Odds are different than probability in that they show us the ratio between the number of desired outcomes occurring and the number of ways in which the desired outcome will not occur. In the context of gambling, this corresponds to the ratio between winning and losing outcomes. Unlike probability, the odds are normally expressed as fractions instead of as percentages.
Here are a couple of examples so you can get a firmer grasp on how odds work. Let's suppose you are interested to know the odds of you hitting number 9 in single-zero roulette where there are 37 numbers on the wheel in total. There is only one number that wins, so it follows there are 36 ways for you to lose. Respectively, the odds for you succeeding are 1 to 36, or 1/36. This corresponds to implied probability of 2.70% which weirdly enough coincides with the advantage of the casino in this game.
Let's use another example with a single-deck blackjack game. What are the odds of you pulling the Queen of Spades from the 52-card pack? There is only one Queen of Spades in the deck opposed to 51 cards of different suits and denominations, so the odds of you drawing this card are 1 to 51 or 1/51.
In gambling, odds are normally expressed in reverse showing you the chances 'against' an outcome occurring, like so: 51 to 1 and 36 to 1. You can convert the implied probability into odds with the following formula: (100/P) – 1 where P stands for probability.
Converting Probability into Odds Additional TipsIn the example with you hitting number 9 on roulette, the calculations run as follows: (100/2.70) – 1 = 37.03 – 1 = 36.03, or roughly 36 to 1. In the one with the Queen of Spades, the implied probability of 7.69%, when converted into 'odds against', corresponds to (100/7.69) – 1 = 13 – 1 = 12, or 12 to 1.
The above calculations show us the actual, true odds of hitting a 9 and of drawing the Queen of Spades from a full 52-card deck on the first trial. The casino extracts its advantage (and profits) by ensuring it always retains a percentage of all players' cumulative wagers.
In games like roulette and craps, this is achieved solely through payout reduction. The true odds of hitting an individual number on a single-zero wheel are roughly 36 to 1 whereas the casino pays you only 35 to 1.In blackjack, the house extracts its edge in a variety of ways including payout reduction for naturals (6 to 5 instead of 3 to 2), unfavorable rules, and increasing the number of decks in play.
The Probability of the Dealer Busting and the Effect of Cards' Removal
In blackjack, the odds and probabilities fluctuate with each card that leaves the deck or shoe. This is so because small cards 2 through 6 favor the dealer, whereas high cards 10, J, Q, K, and A favor the player. Cards 7 through 9 are neutral because they favor neither the player nor the dealer.
The dealer has higher chances of exceeding 21 when they start their hand with small cards like 4, 5, and 6. The player's advantage increases when the dealer exposes one of these cards. Respectively, the player's advantage begins to drop when high cards are removed from the deck. Examine the table below for more information on the dealer's probability of busting with individual upcards.
| The Dealer's Exposed Card | The Probability of the Dealer Busting with This Card in S17 Games | The Player Advantage against a Dealer Showing the Card |
|---|---|---|
| Ace | 11.65% | -16.00% |
| 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% |
Blackjack is the only casino game where cards 'have a memory' since your chances of winning change each time a card is removed from the deck. In fact, this is the basic premise of card counting which we discuss further on in this guide.
When the composition of the deck or shoe is such that ten-value cards and Aces outnumber small card, the player holds an advantage over the dealer. It is the other way around when there are more small cards left to be played. The table below shows you how the cards of different ranks impact your chances of winning:
| Card That Leaves the Deck | Impact of the Card's Removal on Players' Chances of Winning |
|---|---|
| A | -0.59% |
| 2 | 0.40% |
| 3 | 0.43% |
| 4 | 0.52% |
| 5 | 0.67% |
| 6 | 0.45% |
| 7 | 0.30% |
| 8 | 0.01% |
| 9 | -0.15 |
| 10, J, Q, K | -0.51 |
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 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.
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. Lost bets games tube. 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:
Blackjack Odds Table Chart
.
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. |
