Blackjack Deviations

• Blackjack Deviations 18
• Blackjack Deviations From Basic Strategy
• Blackjack Deviations 18

Deviations If this is your first visit to the Blackjack Forum, be sure to check out the FAQ by clicking the link above. You will have to r e g i s t e r (free) before you can post: click the r e g i s t e r link to proceed. These drills are the first step to being an effective card counter. There's still deviations, betting, bankroll management, avoiding heat, and a lot more. Not only have I taken over $600,000 from casinos and co-managed a blackjack team that took nearly $4Million from casinos, but through Blackjack Apprenticeship, we've trained more successful card counters than anyone.

• Blackjack pros resort to using indices for their playing deviations. An index is a number that tells you at what true count you must diverge from the basic strategy. Let's use hard 12 against a dealer with a deuce as an example just to give you a better idea of how playing deviations for hitting and standing work.
• 15.6k members in the blackjack community. A subreddit dedicated to the card game Blackjack for counters and casual players alike!

Mar 26, 2014

While poker requires people reading skills to win and roulette just requires a hell of a lot of luck, blackjack on the other hand boasts one of the lowest houses edges in the casino, making it the game to place your bets on, but how can you use mathematics to your advantage?

It doesn't take long to learn perfect basic blackjack strategy, and if you know what you're doing you can even get as far as lowering the house edge to 0.5%.

The house edge alone is not a determining factor in whether you win or lose, since the unexpected always happens in cards, which is why it's called gambling in the first place, but there is a mathematical area for deviation that can give you an insight into your winning and losses.

The secret behind blackjack lies in the mathematics behind statistics, in the variance and its standard deviation.

Deviating from the standard

When you use strategy to lower the house edge down to 0.5%, theoretically this means that after playing a certain amount of time, you'll lose 0.5% of your money, which goes back to the casino.

Some basic blackjack strategy tips:

• Always hit a hard 4 to 8

• Always stand on a hard 17 and on any combination above 18

• Surrender a 16 versus 10

• Double a hard 10 or 11 if more than the dealer

However, real life is not as simple as that, since there is always room for a variation, which mathematically is deemed as the 'variance', which is calculated by measuring the standard deviation.

In statistics, probability is not an act of chaos with no pattern, but rather follows a bell curve, where the middle corresponds with the average, and the distribution shows all the possible outcomes and their probabilities.

The standard deviation is essentially the number marking the number of units falling to each side of the average, which means that 68% of the outcomes will fall within the standard deviation of the average.

Mathematically, this also means that 95% and 99.7% of the outcomes will fall within 2 and 3 standard deviations, respectively.

Standard deviation and its effect on blackjack

If you account for all the blackjack rules and basic strategy of blackjack, the standard deviation of the game falls at the value of 1.14, in general.

This means that in a game with a 0.5% house edge, the standard deviation marks the odds to win and lose on both sides of the bell curve. This means that 68% of the time you'd either win or lose 1.14 units, and 95% of the time win or lose 2.28 units.

But again, statistics also shifts depending on how many hands you play, and the more hands you do play mean that you get closer to the average. Thereby by including the variable of your dealt hands, you can predict your winning and losing likelihood by a fixed number.

To calculate the amount you win or lose, you basically take the square root of the number of hands played and multiply by 1.14.

So that means for 100 hands of blackjack, yields a standard deviation of 11.4 in this case. So betting, say, $1 per hand would yield an expected loss of 50cents, since the house edge is 0.5% (for 200 hands, this would go up to $1).

If you play 100 hands, then there is a 68% chance you could win or lose $11.4, for example. While there is room for manoeuvre, you can use standard deviation to budget your losses.

Show me the money

Let's take a more concrete example. Let's see how often you would lose $50 in a game of blackjack using the above examples.

To calculate this you need to factor in the expected loss (50cents for 100 hands) and measure the difference from the actual loss ($50 in this case). So the difference applied here is $49.5.

You then need to apply the standard deviation for your set of hands, so for 100 hands this is 11.4, as seen before, and divide this difference so 49.5/11.4 = 4.34.

This means losing $50 is over 3 standard deviations, so the chances of winning or losing the amount is in the margin of around 0.3% of time.

Using standard deviation can help you to optimize and minimize your risk. Playing 100 hands of blackjack and betting $1, with setting your maximum loss at $50, means that you're only likely to lose that amount 0.3% of the time.

To maximize your blackjack game, you simply need a smart strategy and a bit of statistics know-how.

Playing with standard deviation means you can leave your card counting system at home for once, and not get yourself banned from a casino.

Tags: blackjack strategy, card counting system, smart strategy, Standard deviation, Variance

• The Mathematics of Card Counting and Advantage Play in Blackjack
  A look into mathematical strategies behind card counting and advantage play....
• Banned for Good: Blacklisted Blackjack Counters
  Blacklisted blackjack counters are always polarizing, with some seeing them as heroes and others as...
• How the Wolf of Wall Street Can Help You Become a Professional Blackjack Player
  If you want to be a professional blackjack player watch The Wolf of Wall Street;...
• How to Get Caught Using a Card Counting System
  Getting caught using a card counting system will get you the boot from any casino,...
• How You Can Knockout the Casino Using the K-O Card Counting System
  If you're looking for an alternative to the usual Hi-Lo card counting system, then try...

• Appendices
• Miscellaneous
• External Links

Introduction

This appendix presents information pertinent to the standard deviation in blackjack. It assumes the player is following basic strategy in a cut card game. Each table is the product of a separate simulation of about ten billion hands played. As a reminder, the total variance playing x hands at once is the variance plus covariance × (x-1).

The following table is the product of many simulations and a lot of programming work. It shows the variance and covariance for various sets of rules.

Summary Table

By way of comparison, Stanford Wong, in his book Professional Blackjack (page 203) says the variance is 1.28 and the covariance 0.47 for his Benchmark Rules, which are six decks, dealer stands on soft 17, no double after split, no re-splitting aces, no surrender. The second row of my table shows that for the same rules I get 1.295 and 0.478 respectively, which is close enough for me.

Effect on Variance of Rule Changes

The next table shows the effect on the expected value, variance and covariance of various rule changes compared to the Wong Benchmark Rules.

Effect of Rule Variation

What follows are tables showing the probability of the net win for one to three hands under the Liberal Strip Rules, defined above.

Liberal Strip Rules — Playing One Hand at a Time

The first table shows the probability of each net outcome playing a single hand under what I call 'liberal strip rules,' which are as follows:

• Six decks
• Dealer stands on soft 17 (S17)
• Double on any first two cards (DA2)
• Double after split allowed (DAS)
• Late surrender allowed (LS)
• Re-split aces allowed (RSA)
• Player may re-split up to three times (P3X)

6 Decks S17 DA2 DAS LS RSA P3X — One Hand

Deviations If this is your first visit to the Blackjack Forum, be sure to check out the FAQ by clicking the link above. You will have to r e g i s t e r (free) before you can post: click the r e g i s t e r link to proceed. These drills are the first step to being an effective card counter. There's still deviations, betting, bankroll management, avoiding heat, and a lot more. Not only have I taken over $600,000 from casinos and co-managed a blackjack team that took nearly $4Million from casinos, but through Blackjack Apprenticeship, we've trained more successful card counters than anyone.

• Blackjack pros resort to using indices for their playing deviations. An index is a number that tells you at what true count you must diverge from the basic strategy. Let's use hard 12 against a dealer with a deuce as an example just to give you a better idea of how playing deviations for hitting and standing work.
• 15.6k members in the blackjack community. A subreddit dedicated to the card game Blackjack for counters and casual players alike!

Mar 26, 2014

While poker requires people reading skills to win and roulette just requires a hell of a lot of luck, blackjack on the other hand boasts one of the lowest houses edges in the casino, making it the game to place your bets on, but how can you use mathematics to your advantage?

It doesn't take long to learn perfect basic blackjack strategy, and if you know what you're doing you can even get as far as lowering the house edge to 0.5%.

The house edge alone is not a determining factor in whether you win or lose, since the unexpected always happens in cards, which is why it's called gambling in the first place, but there is a mathematical area for deviation that can give you an insight into your winning and losses.

The secret behind blackjack lies in the mathematics behind statistics, in the variance and its standard deviation.

Deviating from the standard

When you use strategy to lower the house edge down to 0.5%, theoretically this means that after playing a certain amount of time, you'll lose 0.5% of your money, which goes back to the casino.

Some basic blackjack strategy tips:

• Always hit a hard 4 to 8

• Always stand on a hard 17 and on any combination above 18

• Surrender a 16 versus 10

• Double a hard 10 or 11 if more than the dealer

However, real life is not as simple as that, since there is always room for a variation, which mathematically is deemed as the 'variance', which is calculated by measuring the standard deviation.

In statistics, probability is not an act of chaos with no pattern, but rather follows a bell curve, where the middle corresponds with the average, and the distribution shows all the possible outcomes and their probabilities.

The standard deviation is essentially the number marking the number of units falling to each side of the average, which means that 68% of the outcomes will fall within the standard deviation of the average.

Mathematically, this also means that 95% and 99.7% of the outcomes will fall within 2 and 3 standard deviations, respectively.

Standard deviation and its effect on blackjack

If you account for all the blackjack rules and basic strategy of blackjack, the standard deviation of the game falls at the value of 1.14, in general.

This means that in a game with a 0.5% house edge, the standard deviation marks the odds to win and lose on both sides of the bell curve. This means that 68% of the time you'd either win or lose 1.14 units, and 95% of the time win or lose 2.28 units.

But again, statistics also shifts depending on how many hands you play, and the more hands you do play mean that you get closer to the average. Thereby by including the variable of your dealt hands, you can predict your winning and losing likelihood by a fixed number.

To calculate the amount you win or lose, you basically take the square root of the number of hands played and multiply by 1.14.

So that means for 100 hands of blackjack, yields a standard deviation of 11.4 in this case. So betting, say, $1 per hand would yield an expected loss of 50cents, since the house edge is 0.5% (for 200 hands, this would go up to $1).

If you play 100 hands, then there is a 68% chance you could win or lose $11.4, for example. While there is room for manoeuvre, you can use standard deviation to budget your losses.

Show me the money

Let's take a more concrete example. Let's see how often you would lose $50 in a game of blackjack using the above examples.

To calculate this you need to factor in the expected loss (50cents for 100 hands) and measure the difference from the actual loss ($50 in this case). So the difference applied here is $49.5.

You then need to apply the standard deviation for your set of hands, so for 100 hands this is 11.4, as seen before, and divide this difference so 49.5/11.4 = 4.34.

This means losing $50 is over 3 standard deviations, so the chances of winning or losing the amount is in the margin of around 0.3% of time.

Using standard deviation can help you to optimize and minimize your risk. Playing 100 hands of blackjack and betting $1, with setting your maximum loss at $50, means that you're only likely to lose that amount 0.3% of the time.

To maximize your blackjack game, you simply need a smart strategy and a bit of statistics know-how.

Playing with standard deviation means you can leave your card counting system at home for once, and not get yourself banned from a casino.

Tags: blackjack strategy, card counting system, smart strategy, Standard deviation, Variance

• The Mathematics of Card Counting and Advantage Play in Blackjack
  A look into mathematical strategies behind card counting and advantage play....
• Banned for Good: Blacklisted Blackjack Counters
  Blacklisted blackjack counters are always polarizing, with some seeing them as heroes and others as...
• How the Wolf of Wall Street Can Help You Become a Professional Blackjack Player
  If you want to be a professional blackjack player watch The Wolf of Wall Street;...
• How to Get Caught Using a Card Counting System
  Getting caught using a card counting system will get you the boot from any casino,...
• How You Can Knockout the Casino Using the K-O Card Counting System
  If you're looking for an alternative to the usual Hi-Lo card counting system, then try...

• Appendices
• Miscellaneous
• External Links

Introduction

This appendix presents information pertinent to the standard deviation in blackjack. It assumes the player is following basic strategy in a cut card game. Each table is the product of a separate simulation of about ten billion hands played. As a reminder, the total variance playing x hands at once is the variance plus covariance × (x-1).

The following table is the product of many simulations and a lot of programming work. It shows the variance and covariance for various sets of rules.

Summary Table

By way of comparison, Stanford Wong, in his book Professional Blackjack (page 203) says the variance is 1.28 and the covariance 0.47 for his Benchmark Rules, which are six decks, dealer stands on soft 17, no double after split, no re-splitting aces, no surrender. The second row of my table shows that for the same rules I get 1.295 and 0.478 respectively, which is close enough for me.

Effect on Variance of Rule Changes

The next table shows the effect on the expected value, variance and covariance of various rule changes compared to the Wong Benchmark Rules.

Effect of Rule Variation

What follows are tables showing the probability of the net win for one to three hands under the Liberal Strip Rules, defined above.

Liberal Strip Rules — Playing One Hand at a Time

The first table shows the probability of each net outcome playing a single hand under what I call 'liberal strip rules,' which are as follows:

• Six decks
• Dealer stands on soft 17 (S17)
• Double on any first two cards (DA2)
• Double after split allowed (DAS)
• Late surrender allowed (LS)
• Re-split aces allowed (RSA)
• Player may re-split up to three times (P3X)

6 Decks S17 DA2 DAS LS RSA P3X — One Hand

The table above reflects the following:

• House edge = 0.28%
• Variance = 1.303
• Standard deviation = 1.142

Probability of Net Win

I'm frequently asked about the probability of a net win in blackjack. The following table answers that question.

The next three tables break down the possible events by whether the first action was to hit, stand, or surrender; double; or split.

Net Win when Hitting, Standing, or Surrendering First Action

Net Win when Doubling First Action

Net Win when Splitting First Action

Liberal Strip Rules — Playing Two Hands at a Time

The following table shows the net result playing two hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the two hands.

Blackjack Deviations 18

6 Decks S17 DA2 DAS LS RSA P3X — Two Hands

The table above reflects the following:

• House edge = 0.28%
• Variance per round = 3.565
• Variance per hand = 1.782
• Standard deviation per hand= 1.335

Liberal Strip Rules — Playing Three Hands at a Time

The following table shows the net result playing three hands at a time under the Liberal Strip Rules, explained above. The Return column shows the net win between the three hands.

6 Decks S17 DA2 DAS LS RSA P3X — Three Hands

Blackjack Deviations From Basic Strategy

The table above reflects the following:

• House edge = 0.285%
• Variance per round = 6.785
• Variance per hand = 2.262
• Standard deviation per hand= 1.504

Internal Links

Blackjack Deviations 18