# Odds and Probabilities

Blackjack is a game of skills and as such, it allows players to predict to some extent the outcome of their choices and moves. In order to win in the long term, they need to acquire the needed knowledge and skills and do their best to improve their game every time they play.

However, it should be noted that regardless of all the strategies and useful tips players may know, after all gambling is gambling and there is no guarantee about the outcome of a particular hand.

No strategy promises that they will inevitably reach the desired result, however the right approach increases their chances of winning significantly and this is what makes Blackjack so interesting and exciting.

## Important Things to Consider About Blackjack Odds

One of the first things, new players should look for when deciding what to play, is the game’s odds and probabilities. Many people often choose to try their luck on Blackjack due to the fact that in most casinos the odds of winning are in favour of the players.

However, what needs to be taken into account is that the game’s outcome can be predicted to some extent only on the condition that they apply the basic strategy correctly and have the needed skills and discipline. Otherwise, the game becomes just like any other casino game where skills and knowledge make no difference.

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## History of the Odds

The prominent mathematicians Julian Braun and John Scarne were among the first people to observe and study the odds and probabilities of the game. They managed to discover how particular moves led the game to a whole new direction and that its odds can be determined to some extent due to the fact that it follows a particular pattern every time.

Their discoveries and strategies are famous all over the world and have contributed greatly to the blackjack games we know today. Other gaming experts and mathematicians that have driven players’ understanding of the game forward include Edward O. Thorp, Harvey Dubner, Peter Griffin, Donald Schlesinger, and Stanford Wong, among others.

## The Basic Concepts of Odds and Probability

Before we delve any deeper, we would like to provide readers with some brief explanations of what odds and probabilities are. Probability is a separate branch of mathematics that studies how likely or unlikely events are to occur. Blackjack players can use probability to determine the likelihood of winning or losing with certain hands against specific upcards of the dealer.

Probability is calculated by dividing the number of winning outcomes by the number of all possible outcomes. The probability is presented either as a percentage or as a decimal number.

Let’s assume a person wants to figure out the probability of drawing any given card out of a freshly reshuffled deck. With the jokers removed, the deck contains 52 cards, any of which has the same probability of being drawn as the rest.

Thus, the probability of drawing any given card, say the Queen of spades, is 1 in 52, or 1 / 52 = 0.0192, which is 1.92%. The likelihood of drawing any Queen from a fresh full deck is 4 in 52 (4 / 52 = 0.0769, or 7.369%). There are four Queens in a single 52-card deck, one for each of the four suits (spades, diamonds, clubs, hearts), hence the higher probability of pulling out one of them.

Odds are related to probability but the two are not the same thing as some players think. The term “odds for” denotes the ratio of winning to losing outcomes whereas “odds against” is the opposite, i.e. it shows you the ratio of losing to winning results.

Therefore, the odds of drawing any of the four Queens from one full deck are 4 to 48 (4:48) while the odds against pulling out a Queen are 48 to 4 (48:4) because there are 48 other cards and only four Queens. The odds format is handy since it furthers the comparison between true odds and casino odds (the payouts on winning bets).

A fair game pays out at true odds, i.e. the mathematical probability of winning coincides with the winning wagers’ payouts. Casino games, blackjack included, do not pay at true odds due to the built-in house edge that generates long-term profits for the casino.

## Calculating Probabilities in Blackjack

Before we proceed with the actual calculations, let’s first demonstrate how blackjack odds fluctuate as the composition of the deck or shoe changes. It would be easier to do this with an example where we have a jar containing ten red and ten blue hard candies, all diligently mixed together.

The probabilities of pulling out a red or a blue candy at random coincide (10/20, or 0.50%) since we start with an equal number for each colour. You reach into the jar without looking and pull out one red candy. This causes the probability of grabbing a red candy the next time you reach in to drop to 9/19, or a little over 47%.

Meanwhile, the likelihood of taking out a blue candy increases with each red one that leaves the jar. Assuming you place bets on the colour of these sweets, you would stand better chances of winning if you wager on blue on the third try.

With two red candies already removed, the blue ones become the better bet since their probability increases to 10/18, or 55.55%. If you are pressing for bigger wins, you will bet more on blue in this case, would you not?

Notice that the probabilities change only on condition you do not return the candies to the jar after each result. Had you been doing this, the likelihood of taking out either colour would have been the same each time, or 10/20.

This sugary game we just made up is the same as blackjack where outcomes are also dependent and interact with each other. What has happened in the past influences what will happen in the future. Here are a few more examples to see how this applies to the game of 21.

### The Three Laws of Probability

Probability is governed by three rules, or laws, to be precise, those of multiplication, addition, and complementation. Under the law of complementation, the likelihood of something not happening equals 1 minus the likelihood of the same thing happening.

There are 13 cards of each suit in a single deck. Therefore, the likelihood of you pulling out any card of the clubs suit is equal to 13 / 52 x 100 = 0.25, or 25%. Respectively, the probability against pulling clubs is 1 – 0.25 = 0.75, or 75%.

The law of addition states the likelihood of one out of two, three, four, etc. outcomes occurring is a result of the total of the events’ individual probabilities minus the likelihood of all of them happening. Therefore, the likelihood of getting a deuce or a card of the clubs suit is equal to [(4 / 52) + (13 / 52)] – 1 / 52 = 17 / 52 – 1 / 52 = 16 / 52 = 0.3076, or 30.76%.

Finally, there is the law of multiplication. For games of dependent trials like blackjack, it dictates that the likelihood of all possible events from a fixed set of outcomes is the result of their conditional probabilities. This is the more complex way of saying the likelihood of an outcome occurring is influenced by the outcomes that occurred before it.

To show you how this works, let’s check out the probability of drawing three ten-value cards from a single 52-card deck without replacing the discards as is the case in blackjack. There are 16 ten-value cards, so the probability of pulling one of them out is 16 / 52. The likelihood of drawing another ten next decreases to 15 / 51 since now we have 15 tens out of a total of 51 cards.

The same applies to getting a third ten in a row as now we have 14 tens out of 50 cards. The likelihood drops to 14 / 50. It follows that the probability of getting three tens out of a freshly opened pack is (16 / 52) x (15 / 51) x (14 / 50) = 0.0253, or 2.53%. If all 16 tens leave the deck, this percentage will drop to zero.

## How Card Removal Influences Blackjack Odds and Probabilities

Unlike casino games based on sole chance, blackjack revolves around dependent trials. Instead of remaining static like in chance games, the odds in blackjack change each time a card is removed from play. The precise value of the discard also influences the odds and the players’ expectation.

Having a flexible value, the ace is the single most important card in the entire game. On one hand, it helps you obtain blackjacks and receive a bonus payout of one-and-a-half times your wager. On the other, aces give you soft hands you cannot bust with on the next hit. It makes sense the removal of aces has the most dramatic effect on your expected value.

The probability of hitting blackjacks drops with each ace that leaves the deck. If all four aces are dealt out a fresh full deck, players’ chances of receiving the coveted 3 to 2 payout would be nil. But what about cards of other denominations? Is their removal as important as that of the aces?

Gaming specialists have tackled this question via computer simulations to determine that each card denomination influences players’ expected value differently. The simulations were run under a given ruleset. All possible hands that could be dealt were then played by the simulator in line with perfect basic strategy to determine the expected value for players when a certain card was removed from the deck.

This procedure was then repeated under the same set of blackjack rules, only different cards were removed each time to recalculate the expectation. It was determined that removing certain cards improves players’ expectation while that of others decreases it.

This makes perfect sense because, in blackjack, some cards work to the advantage of the dealer while others are more helpful to the player. Small cards with a pip value of 4, 5, and 6 help the dealer who is required under house rules to draw to hard totals of 12 through 16.

It would be more difficult for the dealer to bust if there is an excess of low cards in the shoe. High cards, aces and tens, in particular, are better for the player because they can result in blackjacks. Ten-value cards can also help you with more successful double downs.

Mathematician and Blackjack Hall of Fame inductee Peter A. Griffin calculated the impact on players’ expectation each card’s removal had and published the results in his The Theory of Blackjack*. His calculations were made for a blackjack game that plays with one full deck of cards. Mr. Griffin determined that the removal of a single ace leads to a drop in players’ expectation of 0.61%.

Assuming the house set of rules yields a casino advantage of 0.50%, the absence of a single ace would increase this percentage to 1.11%. Meanwhile, the removal of any of the 4s leads to the opposite effect, reducing the casino’s advantage by 0.55%. The odds tilt in the player’s favour, giving them an advantage of 0.05%.

As you can see from the table below, the removal of 8s does not impact the odds of the game. This is because 8s are neutral cards and as such, have no effect on the advantage of the player or the dealer.

The Effect of Card Removal (EOR) on Players’ Expectation in a Single-Deck Blackjack Game
Aces -0.61%
Ten-value cards (10, K, Q, J) -0.51%
9s -0.18%
8s No effect, 8s are neutral cards
7s 0.28%
6s 0.46%
5s 0.69%
4s 0.55%
3s 0.44%
2s 0.38%

*Griffin, P. A. (1996). The Theory of Blackjack: The Compleat Card Counter’s Guide to the Casino Game of 21. Huntington Press, p. 44

## How Are Blackjack Odds Determined

Two terms are used in gambling to indicate the odds for the players – a positive expected win rate and negative expected win rate. As their names hint, these are the odds gamblers have throughout the game. They work the following way – if players’ expected win rate is 2%, the same percentage will be their loss rate.

The odds and probabilities in Blackjack depend on the players’ strategy, knowledge, skills, number of decks, casino rules and others. As these factors have been studied very carefully over the years, now it is possible to make the needed calculations and find out what odds players have whenever they make a specific move. They are usually measured in percentage for their convenience and for a better understanding.

## Odds and Probabilities Depending On the Decks

The odds and probabilities’ estimations are based on the number of decks. Every deck contains 52 cards and in order to find out the players’ chances of getting one of the aces for instance, we need to consider the probability regardless of their current position in the game. Every card type has four of each which means that the chances of getting one of them are 4/52 which is 7.7 %. Using this information, the chance of going bust can be also determined which is the reason why it is really important to get familiar with the odds and probabilities of the game in advance.

In order to explain how they are calculated in a particular situation, there are a few examples that are very straightforward. Let’s substitute the ace with a ten card and see what will be the odds in this case. The possibility that it will be the hole card is 4/13 or 30% and in this case players’ chances of going bust are 8/13 or 61%. If dealer’s card is five or less, the possibility that this will be the hole card is 4/13 or 38.5% and the chances players can bust are 0%. This is how the game odds can be calculated according to the dealer’s possibility of the hole card.

## The Effect of Card Removal Weakens in Multi-Deck Blackjack

One interesting thing about card removal is that its impact on the odds differs, depending on the number of decks a given blackjack variation uses. The particular set of rules of the game also matters and should be taken into account, of course. But why does deck number impact the EOR?

Some players wrongfully assume the number of decks is irrelevant because the ratio of high to low cards increases proportionately to the number of decks one introduces into the game. While this is true, the more cards are introduced into play, the less pronounced their absence from the deck becomes.

It is one thing to discard one of four aces from a single deck, it’s another thing to remove it from an eight-deck shoe where there are 31 more aces left. The effect on the expected value for the player is not as dramatic.

## How Deck Number Changes Blackjack Probabilities

With that in mind, more decks translate into a higher advantage for the house, which, in turn, leads to a lower expectation for basic strategy players. This is because the instances where players hit blackjacks, split, or double down successfully also get diluted by the greater number of cards in play. This sounds counterintuitive to many people but the probabilities speak clearly enough.

Here is how this works. The probability of obtaining a blackjack from a freshly shuffled deck is equal to the probability of drawing an ace multiplied by that of pulling out a ten-value card. If you remember, this is what the law of multiplication dictates for dependent events. The result is then multiplied by two to account for the possible permutations, i.e. the specific order in which the cards leave the deck (ace-ten and ten-ace in this instance).

Note that the overall number of cards for the ten’s probability decreases to 51 since one card has been dealt. It follows the likelihood of a person pulling out an ace and a ten from one full deck is (4 / 52) x (16 / 51) x 2 = 0.04826.

If you perform these calculations for eight decks, you will arrive at the following result: (32 / 416) x (128 / 415) x 2 = 0.0474. This makes for a difference of 0.00086, which seems minuscule but it can still grind you down in the long run.

The same tendencies are observed for successful double downs, with the probabilities again decreasing along with the introduction of more decks. Interestingly enough, the opposite applies when it comes to being dealt pat 20 with two ten-value cards. These strong hands tend to occur less frequently the fewer decks are in play. However, this is nothing to cry about since it affects the player and the house equally.

## Calculating Expected Value of Blackjack Games

All casino games are designed to favour the house, yielding different percentages in terms of casino advantage. The same applies to blackjack, which has one of the lowest house edges (around 0.50%) with basic strategy.

It is possible for players to overcome this edge and play at an advantage on condition they learn to count cards. This approach to the game is profitable in the long term as it enables you to gain more information about the composition of the remaining deck or shoe. This information is obtained by keeping track of the discards. Casino Guardian shall cover this topic in more depth later on in this guide.

The above-mentioned house edge is closely related to the expected value of players’ bets. The term refers to the amount players will lose or win in the long term. The vast majority of casino games have a negative expected value since they favour the house. This is not the case for card counters who get to enjoy a positive expected value over time when the advantage swings in their favour.

Basic strategy players imminently face a negative expectation. They can calculate how much playing consistently will cost them over the long haul by multiplying their chosen blackjack game’s house edge by their average wager and the average number of hands they play per hour.

Assuming that one plays a six-deck game with a peeking dealer who hits soft 17, no surrender, doubling on any two cards, and doubling after a split, they will face a house edge of 0.006 (0.66%) with basic strategy.

Let’s suppose the person is a flat bettor and wagers \$20 per hand on average. One such player would generate roughly sixty hands per hour if joining full tables consistently. Consequently, the expectation they would produce for the house would be 0.006 x \$20 x 60, or -\$7.2 per hour.

Of course, our hypothetical player will not lose precisely this amount during a single hour of betting. In the short term, they will win or lose more at times. In the long run, however, they are doomed to arrive at those average hourly losses.

## How to Keep the Odds Favourable

Many professional players know that in order to have an overall positive outcome of every game, they need to concentrate on winning in the long term. This means that whether they made the right decision on one isolated hand is not of a great importance. What is really crucial for them is to make sure they do their best and make their choices according to their chosen strategy every time.

Imagine a single drop of wine is put in a cup filled with water whoever drinks from it won’t be able to taste anything apart from the water. In the same sense, one incorrect move doesn’t doom players to a certain loss, however a sequence of particular actions does. In order to make a profit, they need to use the right strategy and stick to it throughout the game as this is the only way to keep the odds in their favour. Many players lose faith and get discouraged when they make a few incorrect moves and this is one of the most common mistakes.

## Conclusion

Every game’s odds can be predicted to some extent which gives players respectively better or worse chances of winning. In Blackjack, they have the opportunity to gain advantage over the casino and achieve the desired goal. It is possible to win with the right strategy and knowledge but in order to do so players need to master every aspect of the game. The odds and probabilities allow them to understand better the concept that lies behind the game and give them the chance to make better decisions when they are on the table.

The only way to know what their actions might result in is if they are aware of their possible outcomes. Many players don’t look at the odds of the games as they believe that all that it takes to win is to have luck. This is simply not true, firstly, because the fact that Blackjack is a game of skills indicates that it does matter whether players have a strategy and knowledge and secondly, because this excuse serves them well as they can put the blame on it.

No matter what kind of specialised literature and books about Blackjack players might read, it is highly unlikely to find a chapter dedicated to being lucky. The reason behind this is simple – there are more important things than luck, such as strategy, knowledge, money management skills, discipline and attitude. Prior to entering the game, it is an important part of the players’ preparation to get familiar with the possibilities and odds of Blackjack in order to know which moves place them in an advantageous position and which are to be avoided.