• Blackjack Probabilities
• Math Behind Blackjack

Blackjack (also known as twenty-one or sometimes pontoon) is one of the most popular casino card games in the world. The name blackjack comes from the fact that when blackjack was first introduced in the U.S. it wasn't very popular, so casinos and gambling houses tried offering different bonus payoffs. One of those was a 10-to-1 payoff for a hand consisting of the ace of spades and a black jack (that is, the jack of spades or the jack of clubs). With the current rules, a blackjack hand doesn't even need to contain a jack.

Rules

• The basic blackjack strategy is a mathematical approach to blackjack that tells you the best moves to make based on the cards you’ve been dealt. The best way to learn this approach is to use a basic blackjack strategy chart. This shows you when you should.
• A blackjack game has a dealer and one or more players. Each player plays against the dealer. All players are initially dealt two cards and the dealer is dealt one card face down and one face up (these are called the hole card and up card respectively).
• Blackjack math and card counting have a strong connection, because the latter would not make sense without the former. Card counting is an advanced technique that demonstrates truly the usage of math in blackjack. Using this technique and a computer, experts including computers have been able to devise a card counting technique that works.

Simple Blackjack Math Using Cards. Blackjack uses a regular deck of cards, which helps you make.

A blackjack game has a dealer and one or more players. Each player plays against the dealer. All players are initially dealt two cards and the dealer is dealt one card face down and one face up (these are called the hole card and up card respectively). Each player can then hit (ask for an additional card) until her total exceeds 21 (this is called busting) or she decides to stand (stop taking cards for the rest of the hand). Face cards count as 10 and an ace may be counted as 1 or 11. After all of the players have finished, the dealer reveals the hole card and plays the hand with a fixed strategy: hit on 16 or less and stand on 17 or more.

The player loses if she busts and wins if she does not bust and the dealer does (observe that if both the player and the dealer bust, the player loses). Otherwise, the player wins if her total is closer to 21 than the dealer's. If the player wins, she gets twice her bet; if she loses, she loses her money. If the dealer and player tie it is called a 'push;' the player keeps her bet but does not earn any additional money. If the player's first two cards total 21, this is a blackjack and she wins 1.5 times her bet (unless the dealer also has a blackjack, in which case a tie results), so she gets back 2.5 times her bet.

Soft Hand. A hand that contains an ace that can be counted as 11 is called a soft hand, since one cannotbust by taking a card. With soft hands, the basic strategy is to always hit 17 or less and even hit 18 if the dealer's up card is 9 or 10 (where the 10 refers to a 10, J, Q, or K).

Doubling down. After the player is dealt her initial two cards she has the option of doubling her bet and asking for one additional card (which is dealt face down). The player may not hit beyond this single required card. With the basic strategy, you should always double with a total of 11, double with 10 unless the dealer's up card is 10 or A, and double with 9 only against a dealer's 2 to 6. (Some casinos only allow doubling down on 11).

Splitting pairs. At the beginning of a hand, if the player has two cards with the same number (that is, a pair) she has the option of splitting the pair and playing two hands. In principle, a pair of aces should of course be split, but in this case blackjack rules allow you to get only one card on each hand, and getting a 10 does not make a blackjack. With the basic strategy, you should never split 10's, 5's or 4's, always split 8's, and, in the other cases, split against an up card of 2 to 7, but not otherwise.

Strategies for the Player

Blackjack is almost always disadvantageous for the player, meaning that no strategy yields a positive expected payoff for the player. In the long run, whatever you do, you will on average lose money. Exceptions exist: some casinos offer special rules that allow a player using the right strategy to have a positive expected payoff; such casinos are counting on the players making mistakes.

The so called basic strategy is based on the player's point total and the dealer's visible card. It consists of a table that describes what you should do in any situation in the game (you can find an example of this table at Wikipedia). Under the most favorable set of rules, the house advantage against a player using the basic strategy can be as low as 0.16%.

Many people assume that the best strategy for the player is to mimic the dealer. A second conservative strategy is called never bust: hit 11 or less, stand on 12 or more. Each of these strategies leads to a player disadvantage of about 6%.

Edward Thorp, in his 1962 book Beat the Dealer, describes a simple strategy that makes blackjack an almost even game: if the dealer's up card is 2 to 6, play never bust; if it is 7 to ace, mimic the dealer. The exception to this simple rule is that one should hit a 12 if the dealer's up card is 2 or 3. More advanced strategies include features such as taking into account the player's hand composition (as opposed to just considering the point total) and the other players' hands, specially card counting (that consists of keeping track of the cards that have been dealt so as to know the composition of the remaining cards in the deck), and shuffle tracking (which is far more complicated than card counting, and consists in roughly following groups of cards as they are shuffled). These two last strategies are usually forbidden in casinos.

What does it mean to have a 0.16% disadvantage?

When discussing casino games, one usually finds statements such as the ones above saying something like: 'the house advantage in this game is about 0.16%'. A first explanation is the following: betting ten dollars each hand, you will in the long run lose an average of 1.6 cents per hand. It would be nice to have an idea of the probability of winning any particular bet when playing some specific strategy. Indeed, we can infer this from the player's disadvantage. Let's take, as an example, the potential 0.16% disadvantage when playing the basic strategy.

Suppose you bet $1 at each of 10,000 bets playing the basic strategy. Let's call p the total probability of winning a pass line bet (so p is the number we are trying to calculate). If p was, for example, 0.5, it would mean that, on average, half the times you should win the bet, so you would win 0.5 · 10,0000 = 5,000 times. Since each time you win a bet you get twice what you bet and each time you lose the bet you lose all the money, you would end up with 5,000 · $2 = $10,000, that is, the same total amountyou bet (10,000 times $1). In this case, the house advantage is 0%, as is the player advantage.

The same idea applies for any p: if you bet 10,000, you should, on average, win the bet 10,000p times, so your average payoff is $20,000p. In our case, the house advantage is 0.16%, so if you play $10,000, on average you end up with $10,000 - $10,000 · 0.0016 = $10,000 - $16 = $9,984. So we only have to solve the equation $20,000p = $9,984 to get p = 0.4992.

Links

You can find more information on blackjack's rules, strategies, and history on the Internet. For instance, you can try Wikipedia.

A very interesting free on-line blackjack trainer can be found here.

1. If you are dealt a point total of 16, what is the probability of busting if you hit, assuming that a whole deck will be used to choose among when you are dealt your next card?
2. If you are dealt a 3 and an ace, what is the probability of not busting if you hit, assuming that a whole deck will be used to choose among when you are dealt your next card?
3. Suppose you are the only player against the dealer, and you are in the first hand of a game played with one deck. You are dealt an 8 and a 6, while the dealer is showing a queen. What is the probability that you bust if you decide to hit?

Solutions

Return to Lesson Index

Top of Page

Math isn’t something that most gamblers want to think about when they’re gambling, but it can help in so many ways. Blackjack is a casino game that shows exactly how much knowing a little bit of math can improve results.

In fact, if you embrace the arithmetic behind the game, you can actually win more than you lose when you play blackjack.

Here are seven mathematical facts that every blackjack player needs to know.

1 – Odds Based on a Specific Set of Cards

The best thing about playing blackjack for real money is that everything that happens when you play is controlled by a specific set of cards. Most blackjack players don’t think about the game this way, but there are only 52 cards in a deck.

Even if the game is using six or eight decks of cards, there are still only 52 possible cards. And the base game of blackjack really only uses 13 sets of cards. This is because the jack of diamonds and jack of spades are the same. And because there are 13 ranks of cards, there are only 13 sets of cards that matter, no matter how many decks are being used.

This might not seem like it matters much, but once you understand why this is important and how you can use it, it helps you make smart decisions at the blackjack table.

If you have a hard 16 and the dealer is showing a six, do you know what the correct play is? You’re going to learn more about strategy cards in a different section. But if you don’t have a strategy card, you can get a good idea of the correct play using the fact that there are only 13 possible card ranks in the deck.

If you hit and get any card of six through king, you bust. If you bust, it doesn’t matter what happens to the dealer hand. This means that eight cards will make you bust, and only five help your hand.

You also know that the dealer has one of the 13 cards as his or her face down card. If the dealer has a 10 through king, they have a hard 16 and have the same chance to bust as you. You also know that if they have a six through nine, they have to hit and have a decent chance to bust. This means that eight out of 13 cards are bad for the dealer.

If these are the only things you know, it’s clear that the best play is to stand. This is why a specific set of cards is valuable to you when you play blackjack. You can use this information every time you play.

2 – Splitting Eights

Blackjack Probabilities

Now, you can see some specific ways to use the information you learned about in the first section. Consider your hand if you start with a pair of eights.

You have to place another wager, but it’s worth it to avoid starting with a hard 16. You receive another card on each of your eights.

Here’s a list of cards that help you a great deal on an eight. A two, three, nine, 10, jack, queen, king, and ace all put you in a good spot. This is eight out of 13 cards. Even a four through seven is better than a hard 16. This means that, by splitting, you can’t possibly be in a worse spot than playing it as a hard hand.

If you get another eight, you can split again.

3 – Splitting Aces

When you start with a pair of aces, you have either two or 12. Neither of these is a good starting total. But when you split aces, you have an opportunity to improve by a large amount.

Any of the cards 10 through king give you a total of 21. A six through nine all give you a soft 17 or better. The only cards that don’t help you much are two through five, and you still have a soft hand with each of these. You still have a chance to win. This is another situation where eight out of the 13 possibilities improve your hand quite a bit.

You can see how the mathematics of blackjack help you improve your chances to win by the examples in these first three sections.

4 – Doubling Down

The usefulness of mathematics in blackjack doesn’t stop with the examples above. It’s also useful in situations where you need to determine how to get more money in play when you have a good chance to win.

When you have a hard 11, 10, or nine, it’s often more profitable to double down than to hit. When you double down, you get twice as much money in play on the hand, and you receive one more card to complete your hand.

With an 11, you should double down every time except when the dealer has an ace showing. With a hard 10, doubling down is the most profitable play except when the dealer has an ace or 10 point card showing. And with a nine, you should double down when the dealer shows a three through six.

A few soft hands also are more profitable to double down with, and you can learn more about these hands by using a strategy card. You can learn more about strategy cards in the next section.

5 – Casino Strategy Cards

Now that you see how valuable it is to use math when gambling, you might be wondering if you need to do all of the math yourself. I’m going to put your mind at ease. Someone else has already done all the blackjack math for you and put it in an easy-to-use strategy card.

You can get a strategy card and use it every time you play blackjack. The card has all of the best possible plays, so you simply follow the card on every hand. It covers hard hands, pairs, and soft hands.

6 – Counting Cards

Even if you use a strategy card, you still are fighting a house edge. This means that you’re losing money playing blackjack, even though you’re playing with a small house edge.

But you can use math to help you turn this around and get a small edge against the casino. The way to do this is to use card counting.

If you play blackjack, you should learn more about counting cards. It’s not as hard as you probably think, and once you learn how to use proper card counting systems, you can benefit every time you play.

7 – Sizing Your Bets

Math is also helpful when it comes to figuring out how much you need to bet when you play blackjack. And it’s actually pretty simple to figure this out.

When you play blackjack and the casino has an edge, you need to bet the lowest amount possible. This means that every time you play blackjack and aren’t counting cards, you need to make the table’s minimum wager.

Even when you’re using a strategy card, the casino has an edge. It isn’t a big edge, but it’s still an edge. This means that the bigger your bets are, the more you lose.

Depending on the size of your bankroll, you should bet as much as possible when you have an edge. In real life, it’s hard to always bet the least and the most possible depending on the situation. But you’re going to learn how to do this as you learn more about counting cards.

Conclusion

The power of blackjack math starts with the 13 ranks of cards in the deck. Once you understand how you can use this knowledge, your results are going to improve right away.

Math Behind Blackjack

Use the math behind a blackjack strategy card to get started, then learn how to count cards. If you practice long enough, you can learn how to get an edge every time you play.