Blackjack Mathematical Strategy

Mar 6, 2014

• The mathematical way of playing Blackjack is known as the Blackjack basic strategy chart. This basic strategy has the ability for cutting the casino's edge to 0.5%. The game's chart was derived from a.
• What is Basic Blackjack Strategy? Quite simply, basic strategy is the optimal way to play blackjack. It has been devised using statistical probabilities to calculate the best possible move in any particular circumstance according to your cards and the dealer's starting card.
• 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.
• 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.

A strategy is a collection of deterministic decisions the player will make during the hand regarding hitting, standing, doubling, and splitting, where each decision depends only on the player's total and the dealer's up-card.2 While the strategy S may be chosen according to the player's knowledge of the distribution D at.

Card counting is a science and while a number of books and sites list the ways in which to beat the dealer at his game, what is the exact science behind smart strategies in blackjack and card counting?

To some blackjack is a game of chance, but to others it's one of the few casino games where the outcome can be calculated using mathematical and observational skills.

In the 1980s and 90s, a team from MIT set out to beat the system using a tried and tested card counting system.

21 blackjack greek subs. You might have heard of the Hi-Lo method as a strategy to win at blackjack, but what is the scientific basis behind card counting and why is it an effective way to beat the house?

The basics of card counting

Card counting is a blackjack strategy used by keen players. In principle it's not too difficult to implement, but requires hours of training and practice.

There are different methods, with the Hi-Lo method being the most popular for its effectiveness and ease. Here, cards are assigned values based on where they're located in the set.

Other card counting strategies:

• The KO method follows the same strategy as the Hi-Lo method, but the 7s are worth +1.

• Omega II assigns a worth of +2 to cards 4, 5 and 6, and the 10 and face cards a value of -2, where aces are worth 0.

• In Halves, the 2 and 7 cards are worth +5, the 5 is worth +1.5 and the 9 is worth -5.

Low cards ranging from 2-6 are assigned a value of +1, 10s, face cards and aces are -1, and all other cards are set to zero.

The card counter keeps a running tally of the cards that have been dealt, but the true count also depends on the decks used, where this is obtained from the run divided by the number of decks used.

The idea is that once the count goes above +2, then the game is in the player's favor and bigger bets can be made.

Other methods include the High Opt 2 and Zen versions, which yield improved results but are harder and easier to miscount.

Also, the High Card Count is another possible strategy to use, since this is much easier where only the high cards are counted, but as a result, it's still not as effective as the other techniques.

Playing for variance

In the gambling world, luck is a word that often gets tossed about, but the word 'variance' can give us a mathematical difference of the expected advantage and the results produced.

Say you're playing a game and applying your card counting technique of choice, and your hourly expected value is $25 per hour, then playing for 100 hours means your expectation is $2500.

However, in reality if you're only $2000 ahead, you fall $500 below the expectation value, which is the variance of what you would expect to win, and can also work in reverse by having $3000 after 100 hours.

Variance does mean good news for players with effective strategies, since the longer you play, the closer you'll get to balancing out the variance for winning.

Deviating the standard

Blackjack strategies might seem simple on the surface, but the game really does boil down to its mathematical elements.

It's impossible to discuss statistics without referencing standard deviation. The standard deviation refers to how often an outcome will deviate from the mean or the average.

You'll see this in many areas of statistical analysis, and it's quite easy to determine. The standard deviation is essentially the square root of the variance.

How does the standard deviation affect your game? Well, there is no one answer, but if you can figure out the variance and the standard deviation, then you can assess to see if you're winning a game or how much money you should play with.

The standard deviation can give you an idea of what to expect in terms of variance and help you predict your own game play.

Nothing but zero

The N-Zero is a good number to place your bets on, but is often forgotten by novice players.

This N0 gives you the theoretical number of hands that you should reach before hitting your statistical goal of being ahead by one order of standard deviation.

To explain this in dummies' terms, 4xN0 gives the value needed to overcome the 2SD in your game play.

But the N0 is dependent on playing with the same set of rules and strategies. If this is met, then the $25 expected value per house gives N0 BY N0 = Variance/25².

To play to win in the long run, then play games and bet spreads at values lower than the N-Zero.

The risk adjusted return of Certainty Equivalence

When you take the product of taking the expected win rate and you adjust it to your level of risk in proportion to the inputted money and level of risk tolerance, you get the Certainty Equivalence.

This number will give you a mathematical litmus test on whether it is worth playing the game or not when compared to your bank size.

Even though you might assume a return expected value of $100 per hour, the CE might only show that the game is only worth $50 since your bankroll is low and you haven't got the money to burn.

If the CE value goes negative, it means that you are over betting what you have in your account and it's not worth the risk.

The number essentially gives you an idea on whether it's worth taking a risk or not and can help you to make the smart call of when to play and when not to play.

The truth about blackjack is it's a game of statistics and probability that can be dominated by mathematics. The question is, do you have what it takes to master the mathematical side of blackjack?

Tags: blackjack strategies, card counting system, Certainty Equivalence, Hi-Low, High Opt 2, N-Zero, smart strategies, Standard deviation, Variance, win at blackjack, Zen

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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

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%.

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.

Blackjack Strategy Chart

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

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%.

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.

Blackjack Strategy Chart

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?

Basic Strategy Blackjack