Blackjack Odds and House Edge Explained: A Mathematical Perspective

Among the casino games, blackjack has the reputation of offering the best odds for the house to be beaten.

It is what game theorists call an almost fair game, given it is actually symmetric for the players and the house (dealer). 

Moreover, some asymmetric elements of the game tend to favor the player:

• the player knows the dealers first card before starting the play;
• the dealer is not allowed to split or double;
• the player is paid one-half times the original bet if they achieve a blackjack, in the classical version of the game.

Like in any game of chance, the gaming events of blackjack have odds and probabilities attached. 

It is this mathematical information that is used to evaluate the players advantage in various moments of the game and to devise objective strategies. 

The apparent advantage of the player in blackjack could not be materialized without the use of this mathematical information. 

In this guide you will learn what are blackjack odds, probabilities, and house edge, and how they vary with the game progression, but also with the game rules or versions. 

You will also find which situations are favorable for the player and which are not relative to the blackjack odds and what are the roles of the odds in evaluating your advantage and taking strategic actions.

Here are the main topics covered in this guide:

• Expected Value and House Edge
• House Edge in Blackjack
• Defining Odds and Probabilities
• Why Knowing the Blackjack Odds is Important
• Blackjack Payout Odds

Without further ado, let's dive right in!

Expected Value and House Edge

Probability and payout rate give what is called the expected value (or mathematical expectation or, simply, expectation) of a bet (EV), by the general formula:

EV = (probability of winning) × (payoff if you win) + (probability of losing) × (loss if you lose), where the loss is inputted as a negative number. 

The EV of that bet is 0.2284 x $1 + 0 - 0.6516 x $1 =  - 0.4232 $. (The probabilities showing here are in the odds tables in the next sections.)

This result can be interpreted as:

By keeping playing this bet when that situation occurs, you may expect to lose on average 42.32 cents for every dollar bet. In this wording, average is used in a statistical sense and not as an arithmetic average or mean. 

The house edge or house advantage (HE) is defined as the opposite (as the sign +/-) of the expected value: HE = - EV. 

HE is expressed as a percentage and is calculated as the ratio of the average loss to the initial bet. In our above example, The HE of that bet is 42.32%. HE is also associated with a game, not only with a bet. The HE of a game is the average of the HEs of all possible bets.

It is the opposite in sign of the overall expectation of all the possible bets, calculated as a statistical average.

The HE is interpreted as the rate of the profit that the house makes over the long run from the players bets. 

A positive HE is the mathematical guarantee that the house wont get bankrupt with that game, whatever bets are placed, strategies are applied, or however many players are at the table. 

It is important to note that the house edge of a game is not a fixed parameter and the style of playing/betting changes the HE relative to that play. 

Optimal and near-optimal strategies (for the games that allow them, such as blackjack) reduce the house edge.

House Edge in Blackjack

We saw in the introductory section that house edge (HE) is the opposite sign of the overall expected value (HE = - EV). This is the statistical average or aggregate of the expectations of all possible bets of the game. 

In blackjack, like in any other game, the HE represents a statistical indicator that players should be aware of for being informed on the average loss they may expect over the long run relative to their wagers.

As a raw range for blackjack house edge, it could be anywhere between 0.1% and 2.7%.

As for the strategic play, it can lower any HE down to its lower limits with basic strategy. 

The highest reduction of the HE is obtained with optimal strategy based on card counting.

Traditional versions of blackjack have a HE under 0.5% for basic strategy.

But why does HE vary with rules or strategies?

The answer is just in the definition of HE (or EV), where products between probabilities and payout rates are added together: 

• Once a probability changes, the HE changes;
• Once a payout rate changes, the HE changes; 
• Once a new product appears or one is removed from the formula, the HE changes.

A blackjack rule impacts on the probabilities of winning and losing, whether we talk about the number of decks used, the mode of shuffling and dealing, doubling down allowing, number of hands to split up into, late surrender, resplitting aces, and so on. 

Blackjack paying 6 to 5 changes the payout rate for this event from 3/2 to 6/5 and consequently changes the HE. 

Bonus-type versions of blackjack (Bonus blackjack, Hi/Lo 13, high streak, perfect pairs, and so on) also assume new payout rates coming into the equation and thus new house edges.

As for optimal strategies, they increase the probability of winning for some special configurations in the progress of the game, which impacts expectations and the HE.

The next table notes the house edges of the most known versions of blackjack, for basic strategy: 

Game	House Edge
Atlantic City Blackjack	0.36%
American Blackjack	0.35%
Big 5 Blackjack	0.47%
Classic Blackjack	0.13%
Double Exposure	0.69%
European Blackjack	0.42%
Pontoon	0.41%
Premier Blackjack	0.42%
Spanish Blackjack	0.38%
Super Fun 21	0.94%
Vegas Downtown Blackjack	0.39%
Vegas Single Deck	0.35%
Vegas Strip Blackjack	0.35%

The next table notes the increase or decrease of the house edge depending on the rule variation, relative to the house edge of classical blackjack:

Rule variation	House edge effect
Blackjack pays 6 to 5	+1.39%
Dealer hits soft 17	+0.22%
Double down on 10 or 11 only	+0.18%
Cannot double after split	+0.14%
European No Hole Card Rule	+0.11%
Split up to two hands	+0.10%
Double down on 9, 10 or 11 only	+0.09%
No late surrender	+0.08%
Split up to three hands	+0.01%
Can resplit Aces	-0.10%
Double down rescue (surrender after double)	-0.10%
Hit or double down on split Aces	-0.19%
Double down on any number of cards	-0.23%
Player 21 wins straight away	-0.54%
Blackjack pays 2 to 1	-2.27%

Separately, the following table notes the house edge increase relative to one-deck game, depending on the number of decks used:

No. of decks	House edge
2	+0.412%
4	+0.556%
5	+0.584%
6	+0.603%
8	+0.627%

The most important thing to note is that players who master optimal strategies can benefit from the lowest house edges for the same version of blackjack.

Just looking into the tables above one may see that certain combinations of rules give the lowest house edges in the overall range. 

Obviously, 3 : 2 blackjack is far more advantageous than 6 : 5 blackjack and the 2 : 1 blackjack offers the highest decrease in HE (- 2.27%).

A 3 : 2 one-deck game with any of the rules; Hit or double down on split Aces, Double down on any number of cards and Player 21 wins straight away is a combination that reduces the HE of a classical blackjack game with one of the highest rates of reduction with respect to rules.

They say blackjack offers the best odds among the casino games. 

This statement is true in terms of players advantage or house edge. Blackjack standard 0.5% house edge is not an advantage that you could easily find among the games of chance. 

Take for instance European Roulette, whose HE is 2.7%, 5.4 times bigger than blackjack standard HE or 24.5 times bigger that the HE of blackjack played with optimal strategy.

The essential difference is that the HE of roulette cannot be reduced with strategies, while blackjack HE can.

However, the best odds statement is not true if you think of odds as probability of winning. With this meaning for odds, roulette offers the highest odds of winning in casino games - with some combined bets, one may reach over 90% of winning, although this high probability is counterbalanced by a low profit rate. 

This balance between winning probability and profit is also a reflection of the house edge, and blackjack obviously is in the lead of the casino games with the lowest HE.

Defining Odds and Probabilities

In games of chance and in general, probability of an event is defined as the ratio between the number of the equally-possible outcomes favorable for that event to occur and the number of all equally-possible outcomes. 

The set of these possible outcomes is called the sample space. No matter whether the outcomes are produced with dice, cards, symbols or numbers on reels, wheels, or screens; the probability of a certain outcome or result is computed as a ratio, as described. 

This probability is called the classical discrete probability and is applicable in all games of chance. Probability of an event is always a non-negative number less than or equal to 1. 

The event for which we compute the probability is uncertain, while the information we use for this computation is certain.

Examples from blackjack   

As you see, probability depends on the available information. 

Therefore, the same event may come to have different probabilities at various moments of a game, depending on the progression of the game and the change in that information. 

Every change in the configuration of the game changes the sample space and the associated probabilities for the events.

Probability is an objective measure for the possibility of an event to occur. It reflects the likelihood of this occurrence, expressed in mathematical terms. 

In the above examples simple events were measured in probability, however, the games of chance and gambling involve much more complex events, of various kinds. For such events as the latter, probability may be computed by using the properties of probability and various mathematical techniques. 

We saw that probability is a ratio, but it can be expressed not only as a fraction or percentage, but also as odds. 

The odds are a way of expressing the probability of an event reported to the probability of the contrary event. 

Even though the terms odds and probability have different definitions, they express the same measurable aspect of an event and using them interchangeably is not quite an error. 

When asking What are the odds of that?relative to an uncertain event we may either respond with a fraction/percentage as numerical probability or odds (against).

However, in gambling jargon the term odds is also used with another meaning, as related to the payoff or payout of a bet. 

It expresses the multiplier by which the stake of a bet is multiplied and given as payoff in case the bet is won. 

Blackjack Odds Explained

In the card game of blackjack, the events that we can measure in probability are expressed as totals of points: the odds of achieving 18, 19, 20, or 21 points,  achieving more than 18 points, or more than 21 points (bust). 

In card games, the outcomes are combinations of cards and the probability computations use combinatorial calculus. 

This is why computing odds for these games is more difficult than for other games of chance. Not all probabilities can be computed manually or by explicit formulas. When the measured events are too complex, mathematicians use specific methods or even simulations-based software for such computations.

In blackjack, the events expressed as totals of points can be unfolded into combinations of cards. 

For computing the probability of that event in a given situation, we have to remove from these final combinations those containing any seen card (the players cards already drawn and dealers card), count the remaining combinations and divide their number to the number of all possible two-card combinations for the two cards.

Still, blackjack odds are more difficult to be computed than the odds in poker, for instance. 

This happens because the combinatorial effort extends to the decomposition of any total in sums corresponding to the values of the cards (partitioning) for some events.  The number of these partitions are in the order of hundreds.

Immediate odds in blackjack

The easiest to compute blackjack odds are the so-called immediate odds. These are the probabilities of the events expressing an outcome with the next card to come. They can be computed by the player simply by a minimal count.

If you formulate your event as a range for the total (implying a range for the values to come with the next card), like in our example above, you have to apply the formula for each of those values in the range, then add the results together.

Of course, the ability to perform such calculations during the game, including counting the values remaining in the deck is not at everybodys hand, but is trainable. 

The above formulas and results are applicable in the case the game is played with a complete deck or decks of cards and classical shuffling and dealing. 

Modern blackjack is very often played with several decks (usually six), shuffled and divided off by a blank card at about one-fifth of the pack. 

This rule prevents either computation of immediate odds or card counting since the values are not evenly distributed in the composition of the stack in play.

The immediate odds have a role in evaluating your position or advantage in every moment of the game, but they do not reflect all the relevant information about who is more likely to win the hand in the end. 

There are 3 reasons for that: 

• If at the moment of evaluation your total is low (say, under 10), there is not necessarily the next card drawn that will make the difference, but the next two or more; 
• The odds for certain total or totals change dramatically with every new card dealt; 
• Even though you might achieve your target total, there comes the dealers turn and dealers odds become as important as yours.

Long-shot odds in blackjack

The long-shot odds are probabilities of events expressing an outcome with more than one card to come.

Usually final events are considered when estimating long-shot odds, like the totals to achieve or who is going to win. 

Such odds are computed or estimated using the information available at an intermediary stage of the game for events reflecting the end of the game. 

The more information is available for the long-shot odds, the more relevant is their measurement. This condition reverts of course to the number of viewed cards.

However, most of the long-shot odds are actually impossible to be computed in a combinatorial mode, either manually or by explicit formulas. 

The reason is the hugeness of possible combinations of cards that a certain total may be unfolded in as partitions and the non-linear progression of the totals.

Mathematicians have struggled ever since the 1950 to work out such odds when dealing with the optimal strategies in blackjack. 

The big idea was to assign a constant probability of 1/13 for the individual card values, except for the cards of value 10, which are assigned a probability of 4/13 (as being fourth times more numerous than the cards with a certain value less than 10). 

Of course, such an assumption would only be valid for an abstract stack of cards with infinite size, since the probabilities change with every new card dealt out. 

Still, this supposition allowed mathematicians to model the progression of totals with the drawn cards as a Markov chain. 

By this model, mathematicians were able to approximate overall long-shot probabilities by using recursive and iterative methods. 

The big prize of such a mathematical approach was the conception of the optimal strategy of blackjack based on maximizing expectation, between 1956 and 1964.

A distribution for the apriori overall probabilities for the dealer achieving various totals is in the following table:

Total	Probability
17	0.1451
18	0.1395
19	0.1335
20	0.1803
21	0.0727
Blackjack	0.0473
Bust	0.2816

Table: Apriori long-shot odds for the dealer to achieve 17 through 21 points, blackjack, or bust.

The odds in the table above are determined with no additional information available, that is, before any card is dealt. 

Obviously, such odds gets more relevant if computed conditionally on the dealers first card. 

This can be done iteratively, by calculating the final-value probabilities conditioned on arbitrary intermediate states of the dealer:

1. Begin with dealer value 16. Final values 17, 18, 19, 20, and 21 are reached with probability 1/13 each. The dealer exceeds the value 21 with probability 8/13.
2. Next, consider the dealers value 15. Again, values 17 through 21 are reached with probability 1/13 each. The case that an ace is drawn next reverts to the already considered case of a deales value of 16. The final results 17 through 21 are reached with probability 1/13 + 1/169 = 14/169.
3. Move now to dealers value 14. We have again probability 1/13 for each of the final values 17 to 21. The two already considered values 15 and 16 are reached with the same 1/13 probability. Altogether, the final values 17 through 21 are reached with probability 1/13 + 14/2197 + 1/169 = 196/2197.

Continuing this computation in the reverse order of the chronology of the game, one can get the probabilities for all 13 situations relative to the dealers first card.

These odds are in the following table:

These odds do not take into account players cards, as they reflect the information available before the player starts their play.

We can see in this table on the last row that the highest odds for the dealer to bust are for their first card 5 or 6.

You might be more interested in what are the players winning odds when they decide to stay with the total they reached. These odds can be obtained from the dealers odds in the previous table. 

 The next table notes all the odds for the player to win if they decide not to draw, conditioned on dealer’s first card.

Players Total Value	Probability to Bust
21	1
20	0.92
19	0.85
18	0.77
17	0.69
16	0.62
15	0.58
14	0.56
13	0.39
12	0.31
11 or less	0

Table: Odds for the player to bust with one hit.

As I said, odds are more relevant as they rely on more information (dealt cards), but not all the odds can be computed in blackjack.

The rawest estimation is the apriori overall probability of winning the game of blackjack: You have a probability of 42.43% to win the game, 8.48% to go into a tie, and 49.09% to lose. 

This estimation stands if no additional information is available, that is, no cards have been dealt yet. 

Of course, these odds are not of much help in the course of the game and just the first card of the dealer or the first card of the player may change them dramatically. They are just illustrative for the small advantage that the house has over the player in terms of chance.

Why Knowing the Blackjack Odds is Important

Blackjack odds are an objective measure for their chances and advantage against the dealer in the course of the game. 

Just looking carefully into the odds tables of this section, one may find general and particular situations reflecting advantage or disadvantage for which a certain strategic action may be recommended.

From the dealer;s long-shot odds table no. 4 we get that the highest chances for the dealer to bust (between 39.4 - 42.3%) are for the dealers first card 4, 5, or 6.

In a larger range, the highest chances for dealers bust are for their first card 2 through 6 (35.3 - 42.3%), then the chances drop for their first card 7 through ace. 

Other figures jumping out of the monotonicity are that the dealer has the highest odds of reaching 19 points (about 35%) for their first card 9 and the highest odds of reaching 18 points (about 36%) for their first card 8. 

Now, moving to the players odds table no. 5 with an eye on what we found from table no.4, lets observe that:

• Staying at less than 17 points wins with the highest probability for dealers first card 5 or 6 (41.62 - 42.32%), which have associated the highest odds for the dealer to bust.
• Probability for the player to bust with the next card is higher (56 - 69%) for a total between 14 and 17 points than for a total between 12 and 13 (31 - 39%), from Table 6.

• Staying at 18 points wins with a probability over 50% for the dealers first card 3, 4, 5, 6, or 7, being maximal for 6 or 7 (about 59 - 63%). 
• Dealers probability to bust with their first card 6 is 42.32%, but with 7 is only 26.23%.

• Staying at 18 points wins with the lowest probability for the dealers first card ace (24.60%), followed by 10, 9, and 8 (32.36 - 37.33%). 
• The dealer having an ace gives them about 59% odds of achieving more than 19 points without busting and reduces the chances for you to improve the hand if hitting, since ace is a favorable value for your hand. 
• The probability for you to bust is 77%. Dealers odds for a total exceeding 18 are 53.16% when their first card is 9.

Odds may also indicate recommendations on how to deal with bad hands. One such hand is one in which you reached a hard 16. 

It is so because the total of 16 is just below the threshold of 17 and the odds to bust with the next card are relatively high, since no ace is in your hand (62%). 

One recommendation for that hand is to surrender, if this option is allowed, to minimize your loss. 

For each split blackjack hand, the immediate odds of drawing an ace, 10, or face card are - according to the formula of p(x) in the immediate odds section - about 41% for the one-deck game and about 39% for the six-deck game, which are relatively high odds and get even higher in aggregate with the two hands to play. 

Depending on the configuration of the game at the moment of analysis, such recommendations can be made on the basis of immediate odds, if these latter are computable. 

Looking at the blackjack odds only, we may point out such particular situations as those above where the player or the dealer has a relative advantage or disadvantage over each other that may be turned into a win or a minimization of loss. This advantage or disadvantage is just a reflection of the probabilities involved and hence we must be precautious when interpreting them in terms of winning, because:

1. Probability does not say anything about whether the measured event will actually happen in that game. Not even a 90% probability guarantees winning.
2. All relevant probabilities in blackjack are conditional. That is, we may estimate a probability of achieving a total, but the probability of winning with that total depends on what the dealer will achieve and it is this latter probability that counts toward the former. Moreover, each probability changes with every new card dealt so it is worth relying on them only at the moment of the evaluation.
3. Probability is a measure for an uncertain event, however as a blackjack player you wont play only one game and lose or win it according to or against the odds, but many games. When drawing the line, it is not about how many wins compared with losses you achieved, but what money you have won or lost overall. This financial outcome is not expressed in probability, but in (mathematical) expectation.

It is this third reason for which strategic recommendations in blackjack should not be made on the basis of odds solely, but rather on expectations. 

We saw that the definition of expectation involves both probabilities and the payout odds of blackjack. 

Expectation (EV) reflects a statistical average of the profit or loss over the long run and optimal strategies are based on expectations and not on odds alone.

It is this feature of the optimal strategies that allows such a strategy to cover every possible gaming situation. 

That is, an optimal-strategic recommendation is made in terms of actions:

• Stand
• Hit
• Split
• Double
• Surrender

For every moment of the game or configuration of the players and dealers hands. Odds alone cannot provide such coverage. 

Odds inform, but cannot rule alone over the optimally strategic actions in blackjack.

Within an optimal strategy, every recommendation associated with a configuration is based on comparing players expectations for each possible action and choosing that action that maximizes players EV. 

As such, an optimal strategy only works over the long run.

The first optimal strategy of blackjack (called the fixed optimal strategy) was devised in 1956 by a group of American mathematicians. 

It is called fixed because it does not consider the information available to the player about the cards already played. 

This changed with the monumental work of Edward Thorp (1961-1962) who discovered the High-Low count optimal strategy.

Blackjack Payout Odds

Payout odds (payout rates) are the other factor that counts for the computation of EV along with probabilities. 

They express the rate at which the player is paid their wager after the hand is played. 

There are 4 values for the blackjack payout odds: -1, 0, 1, and 3/2 (or 6/5, for a non-classical version of the game), applied as explained below:

Payout Odds	Rules of application
-1	When the dealer achieves a better hand than you or a blackjack they take your wager.
0	No money will exchange if you and the dealer achieve the same total value (a push); they return your wager.
1 (or 1 1)	If you beat the dealer without a blackjack, you are returned your wager and paid the same amount.
3/2 (or 3:2)	When you achieve a blackjack and beat the dealer, you are returned your wager and paid 1.5 times your wager.
6/5 (or 6:6)	When you achieve a blackjack and beat the dealer, you are returned your wager and paid 1.2 times your wager.

There is also a rare 2 : 1 version of the game, with a similar rule of application for the payout odds 2/1 (or 2 : 1).

Taking into account the payout rates and the associated probabilities, one can compute the expectations for every possible blackjack bet. 

For example, the table above is used to obtain the following table of expectations: