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PROFESSIONAL BLACKJACK STANFORD WONG Pi Vee Press PROFESSIONAL BLACKJACK by Stanford Wong Pi Vee Press copyright Â©.This content was uploaded by our users and we assume good faith they have the permission to share this book.

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PROFESSIONAL BLACKJACK STANFORD WONG Pi Vee Press PROFESSIONAL BLACKJACK by Stanford Wong Pi Vee Press copyright Â© 1975, 1977, 1980, 1981, 1994 by Pi Yee Press All rights reserved No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.

Inquiries should be addressed to Pi Yee Press, 7910 Ivanhoe 34, La Jolla, CA 92037-4511.

ISBN 0-935926-21-6 Printed in the United States of America 4 5 6 7 8 9 10 4 PROFESSIONAL BLACKJACK PREFACE This 1994 edition of Professional Blackjack is a major update and expansion.

It covers many rules variations that the previous edition did not cover.

It contains simulation results to replace the "best guesses" in previous editions.

Appendixes C, D, and E are new material in this edition.

Parts of the previous edition were removed and expanded into another book, Black.

The main part involved was the chapter entitled "How to Win Without Getting Kicked Out.

Professional Blackjack is a reference book for card counters.

It contains a virtually complete set of strategy numbers for the high-low counting system.

The numbers in this book have been objectively derived on a computer.

This material has been thoroughly tested in casinos throughout the world.

Win rates herein are estimates based on simulations by Blackjack Count Analyzer.

The win rates con- 5 tained in this book are based on simulations totaling more than ten billion hands of blackjack.

You can buy this software and reproduce almost any strategy recommendation and almost any win rate in this book.

Small parts of this material first appeared in one or another of the newsletters: Stanford Wong s Blackjack Newsletter, Current Blackjack News, Blackjack World, and Nevada Blackjack.

Of those, only Current Blackjack News is still published.

Thanks to the people who read pre-publication copies and whose suggestions have made this a better book: Anthony Curtis, Michael Dalton, Dave Douglas, Frank Polo, Donald Schlesinger, and John Speer.

This book can be improved.

If you find passages that are wrong, or if your questions go unanswered, please put your comments in writing and email them to me at or mail them to me at Pi Vee Press, 7910 Ivanhoe 34, La Jolla, California 92037-4511.

I reserve the right to publish your questions and my answers.

When you travel, if there is any chance you will encounter a blackjack game, take this book with you.

Whatever rules you encounter at blackjack, you should be able to open this book and find the strategy numbers you need to attack the game.

Chapter 2 presents generic basic strategy.

Chapter 3 presents the high-low counting system for one commonly-found set of rules.

Following that are chapters presenting strategy indexes for insurance, double down, splitting, the no-hole-card game, surrender, multiplecard bonuses, and bonuses for particular hands such as 6-7-8 of the same suit.

You ought to be able to find the strategy index numbers you need for the rules you face in the casino of your choice.

You can bet that your first two cards will total over thirteen or under thirteen.

Aces count as one and not eleven, and thirteen loses.

Chapter 12 presents a more powerful counting system, the halves.

Chapter 13 is devoted to double exposure.

This is a form of blackjack in which both the dealer's cards are exposed before the players act, and ties go to the dealer.

Chapter 14 explains some of the finer points of blackjack, including expected win, risk, optimal betting, and counting in a casino.

Chapter 15 presents several studies of casino shuffles.

The appendix contains tables of strategy indexes, expectations, and frequencies of particular hands.

Dealer Action on Soft 17 Some of the tables in this book have been derived for dealer stands on soft seventeen, and some have been derived for dealer hits soft seventeen.

An abbreviation is used to distinguish between the two: s17 means dealer stands on soft seventeen, and hI7 means dealer hits soft seventeen.

The reason for this usage is to 17 INTRODUCTION make the advice on card counting unambiguous: "count per deck" means "count per 52 cards.

For example, sometimes with 4-4 it is correct to hit at very low counts, double down at very high counts, and split at intermediate counts.

Where this complication exists, adequate explanation is provided in the text.

Another example: With soft eighteen you may want to stand, you may want to hit, or you may want to double down.

This complication generally is handled by having two lines in the table, one for stand versus double and one for hit versus stand.

Win Rates This book contains numerous win rates.

They are based on computer simulations of at least 30 million hands of blackjack each.

Their purpose is to aid in understanding the importance of various rules options.

Benchmark Rules I have selected a benchmark, an arbitrary set of rules, playing conditions, and bets.

This benchmark is shown on the next page.

Use of a different benchmark would result in different numbers.

However, the relative importance of different rules variations would be approximately the same no matter what set of rules were used as the benchmark.

The benchmark betting scheme is not the best way to bet.

Much better is to leave the table on negative 18 PROFESSIONAL BLACKJACK Benchmark Rules Six decks.

One-deck cut, meaning five decks are dealt out.

Dealer stands on soft seventeen.

Double down is allowed on any first two cards but not after splitting.

Resplits are allowed to a total of four hands.

Split aces receive one card each, and cannot be resplit.

Insurance is offered, but no other side bets.

Naturals pay 3:2, and tie a natural by the dealer.

The high-low counting system is used, with strategy numbers from -10 to +10.

There is a total of two players at the table.

On positive counts, you must bet in a manner that does not attract unwanted attention to yourself.

See Blackjack Secrets for in-depth betting advice.

Simulating the base set of rules and betting scheme finds a win rate that is a benchmark for evaluating the cost or benefit of various deviations from the base rules.

In a casino you will be able to play 50 to 300 hands per hour, so 100 hands represents roughly an hour's play.

Throughout this book results are stated in dollars per hour, which really means dollars per 100 hands.

Though it is an estimate for 600 million hands of blackjack, it still contains what statisticians call "sampling error.

The term used to describe the precision of an estimate is the standard error.

It would be nice to have a smaller standard error of course, and the way to do that is to increase the sample size.

That is the reason why the sample size was run up to 600 million hands - to reduce the standard error to a reasonable number.

That is the average initial bet; it does not include additional amounts wagered on doubles, splits, and side bets such as insurance.

Risk varies with the rules.

Some rule changes mean more risk, and some mean less risk.

Being able to double down more often such as after splitting pairs means more risk, as represented by a higher standard devia- INTRODUCTION 21 tion.

Restrictions on doubling down mean a lower win rate but that is partially offset by lower risk.

Win Rates Reported A small box is used to indicate a win rate, and is the result of a simulation of at least 30 million hands.

The box explains the manner in which the rules of this simulation differ from the benchmark, the win rate per 100 hands using the benchmark betting scheme, and the standard deviation applicable to 100 hands.

The rest of the information pertaining to that simulation is the same as the benchmark as discussed above.

A summary of win rates is presented in table 82 page 185.

Double-exposure win rates are presented separately, as table 81 page 182.

If you already know basic strategy, skip this chapter.

Basic strategy is the best way to play a blackjack hand on the first round after a shuflle, assuming you see no cards other than your own and the dealer's upcard.

For a person who does not count cards, basic strategy is the best way to play every hand.

This chapter presents basic strategy for single exposure, which is blackjack where the dealer has one card face up for you to see as you are playing your hand.

Basic strategy is what plays you should make if you are not counting cards and you do not have any information about the dealer's hole card.

It is presumed that you GENERIC BASIC STRATEGY 23 know the total in your own hand and the dealer's upcard, but no other cards.

Chapter 13 contains basic strategy for double exposure, which is blackjack where the dealer has two cards face up.

Another book, Basic Blackjack, presents basic strategy for all the various rules that have been offered for blackjack, such as dealers take ties on seventeen, six-card hand pays double, surrender after doubling down, etc.

Calculating Basic Strategy Basic strategy can be either total-dependent or composition-dependent.

Total-dependent means the strategy numbers require only the dealer's card and the total points in your hand.

Composition-dependent means the strategy numbers require knowledge of the dealer's card and the precise cards that make up your hand.

For example, total-dependent strategy says stand on twelve against 4.

Composition-dependent strategy for twelve against 4 requires you to specify how you get to twelve: Do you have 7-5, 8-4, 3-2-2-5, or what?

If you get to twelve by 10-2 or 2-10 where 10 means any 10-count cardand two or fewer decks are being used or seven or fewer if the dealer stands on soft seventeenyou should hit.

If you get to twelve by any other route, or enough decks are being used, you should stand.

See Peter Griffin's The Theory of Blackjack for a good discussion of compositiondependent strategy.

The 10-2 versus 4 advice is from page 176 of Griffm's book.

There click the following article few differences between compositiondependent and total-dependent strategies for single deck, and none that are important for multiple decks.

This book uses total-dependent strategy.

The details of basic strategy depend on the particulars of the rules.

However, you must start someplace.

This chapter presents a version of basic strategy that is approximately correct for the most common sets of rules - a generic basic strategy.

For modifications of this real online money australia blackjack basic strategy that are appropriate for different sets of rules, see Basic Blackjack.

Blackjack is most commonly played with the dealer's hand showing one card face up.

If you do tens when you blackjack split in playing blackjack in a game where you get to see two cards face up in front of the dealer before you play your hand, go to chapter 13 for playing-strategy advice.

Table 1 presents generic basic strategy.

It contains advice for every decision the blackjack player commonly makes.

Each column is a different dealer upcard.

Each row is a different player hand.

Technically, table 1 is basic strategy for multiple decks and dealer stands on soft seventeen.

Pairs The order of decisions presented in table 1 is the order in which you evaluate your hand.

Do you have a pair?

At most casinos, any two 10count cards, e.

J-K, are a pair and may be split.

If you have a pair, the first part of table 1 tells you how to play your hand.

Use this part of the table to decide whether to split your pair.

To split means to make another bet equal in size to your fIrst bet, and play each card as the start of a separate hand.

GENERIC BASIC STRATEGY 25 If you split a pair and catch another card of the same value, resplit if you can.

If it is correct to split a pair, it is correct to resplit.

You mayor may not be allowed to double down after splitting a pair.

For example, if you split 8-8 and catch a 3 for eleven, you mayor may not be allowed to double down on that eleven.

If doubling down after splitting is allowed, then splitting is more attractive and you should split more often.

The first part of table 1 assumes you are not allowed to double down after splitting.

If you are allowed to double down after splitting, then use the last part of table 1 - the part on the facing page.

The decisions that are hits in the first part of table 1 and splits on the facing page are: 6-6 against 2, 4-4 against 5 or 6, 3-3 against 2 or 3, and 2-2 against 2 or 3.

Soft Hands Do you have an ace?

Aces count your choice of either eleven or one.

A hand in which an ace counts eleven is called a soft hand, and the total points in it are called a soft total.

The second part of table 1 explains how to play soft hands.

The double-down advice is broken down into db and dbs.

The reason is you need to know what to do with a particular total if you cannot double down.

For example, suppose you have soft eighteen and the dealer shows 3.

Your best play is to double down, so that is what you do if you can.

But if your soft eighteen is a three-card hand, say ace-2-5, then you probably will not be allowed to double down.

Table 1 lists "dbs" for that hand, which means if you are not allowed to double down then you should stand.

Double down; if you cannot double, then hit.

If you have soft eighteen and the dealer shows 9, 10, or ace, then hitting your soft eighteen is better than standing on it.

One decision that is very close is soft thirteen against 5.

It does not matter whether you hit or double down.

Hard Hands The lower two parts of table 1 explain how to play the rest of your hands~ Hands labeled "hard" might contain aces, but under the circumstances all such aces are counted as one.

Hands tabulated as from 5 to 11 do not contain an ace; if a hand totaling eleven or less has an ace it is a click hand and is visit web page according to the "soft" part of the table.

Surrender Surrender means losing half a bet for the privilege of not playing out the hand.

Late surrender means you cannot surrender if the dealer has a natural.

The table of generic basic strategy includes strategy for late surrender.

Most of the' value of late surrender comes from surrendering sixteen against 10.

Late surrender is worth 0.

If you are playing blackjack at a casino that does not offer surrender, or if you are not allowed to surrender due to having more than two cards, then hit those hands for which table 1 advises surrender.

GENERIC BASIC STRATEGY 29 Insurance Table 1 does not show insurance.

Basic strategy says never take insurance.

Even if you have a natural, you are better off not insuring it.

You are better off winning 3:2 most of the time than winning even money for sure.

If you see two, go to chapter 12, which discusses double exposure.

This chapter discusses singleexposure blackjack.

Winning at blackjack requires two things: You must bet more when you have the advantage and less when the dealer has the advantage; and you must make correct decisions on insurance, surrender, splitting pairs, doubling down, and hitting or standing.

Parts of this chapter are similar to chapter 3 of Blackjack Secrets.

One difference is that this book gives "surrender" its own chapter.

Another difference is that this book presents high-low read article numbers for a 31 HIGH-LOW SYSTEM wider range of counts per deck, -10 to +10, instead of -1 to +6.

High-Low Count You need a counting system to tell whether you have the advantage and to aid in making decisions.

Aces and lOs favor you because naturals are worth half again more to you than they are to the dealer.

Small cards favor the dealer by decreasing the dealer's chance of busting.

The high-low system, first introduced in 1963 by Harvey Dubner, is both simple and powerful.

Thorp in the revised edition of Beat the Dealer, Lawrence Revere in Playing Blackjack as a Business, and Julian Braun in How to Play Winning Blackjack, present strategy numbers for it.

This chapter contains my independent calculations for it.

Peter Griffin in The Theory Of Blackjack says the betting correlation of the high-low system is 0.

Counting cards in the high-low system is relatively simple.

Start with a count of zero after the cards are shufiled.

Add one for every small card 2, 3, 4, 5, 6 that Table 2 High-Low Count Card Count 2,3,4,5,6 7,8,9 1O,J,Q,K,Ace +1 0 -1 32 PROFESSIONAL BLACKJACK you see.

Subtract one for every ace or 10-count card that you see.

Do not change your count for 7, 8, or 9.

This is shown in table 2.

Keep a running count.

Accumulate the total is pokerstars good the last shufile.

A full deck contains the same number of +1 cards as -1 cards.

Therefore, at the end of the deck the running count should come back to zero.

This is called a "balanced count.

The high-low count tells you when the best cards, the 1Os and aces, outnumber the small cards.

The richer the pack, that is, the more 1Os and aces relative to small cards, the better for you.

You need to know how rich the pack is in order to make appropriate bets and decisions in play.

Richness of the pack depends on the proportion of excess 1Os and aces.

For decision purposes, you must relate the running count to the number of decks you have not seen.

For example, twelve lOs and aces remaining to be used when the dealer is halfway through a single deck is two 1Os and aces more than average; that is as favorable to the player as if there were four excess 1Os and aces in 52 cards, and is described as a count per deck of +4.

For betting and playing decisions, a running count of + 1 with half a deck remaining is equivalent to a running count of +2 with one deck remaining, to a running count of +4 with two decks remaining, and to a running count of +8 with four go here remaining.

Therefore, you must convert the running count into count per 33 HIGH-LOW SYSTEM deck for making decisions.

Simply divide the running count by the number of decks or fraction of please click for source deck that you have not seen.

If less than one deck remains, your count per deck will be greater than your running count.

If more than one deck remains, your count per deck will be less than your running count.

For example, suppose you approach a blackjack table with a single-deck game in progress and see the following cards.

First Player Second Player 2-3-2-4-7 10-10 Dealer 3-8-9 You glance at the cards and see that the running count is +3.

Since what you have not seen is almost one deck, the count per deck is slightly above +3.

The dealer picks up the cards, and you place a bet.

You receive 2-10, the dealer's upcard is 2, the first other player has a natural, and you do not see the second other player's cards.

You must decide whether to hit or to stand.

The running count is now +2.

It gives a count per deck of +3.

So you stand instead of hitting.

You need only approximate the count per deck.

You do not need an exact count of the number of cards remaining.

If you had seen roughly half a deck in a single-deck game, a running count of +2 would translate to a count per deck of about +4.

If you had seen about half a deck 26 cards in a double-deck game, a running count of +2 would translate to a count per deck slightly greater than +1 2 divided by 1.

A rough estimate of the count per deck suffices for decisions because you 34 PROFESSIONAL BLACKJACK need to know only whether it exceeds an integer in a table.

Estimating Your Edge Your advantage or disadvantage if you play basic strategy varies with the rules and number of decks used.

Current Blackjack News is a good source for keeping up to date with the rules at various casinos and your advantage or disadvantage.

Commonly you are at a disadvantage of about 0.

Table 3 contains more detail on estimating the casino's edge right after a shuffle.

The starting point depends on the number of decks used.

Simulation shows that a basic-strategy player playing with the benchmark rules has an expectation of 0.

The negative signs mean the casino has an edge.

The rest of table 3 contains numbers that https://partysutra.com/blackjack/online-blackjack-basics.html can add to one of those starting points, depending on how the rules you face vary from those benchmark rules.

Early surrender gives the player 0.

Table 3 allows you to find your expectation for the first hand after a shuffle.

These are simulation results, and thus contain a small amount of random error.

To a small extent, the effect of a given rule also depends on the number of decks used.

The best presentation of how your edge after a shuftle depends on the rules is in Peter Griffin's The Theory of Blackjack.

You also can generate such numbers yourself with Blackjack Count Analyzer if you ask for a shuffle after every hand.

Table 3 applies to the first hand after a shuflle.

For subsequent hands, your edge might be higher or lower.

HIGH-LOW SYSTEM 35 Table 3 Player Edge benchmark: benchmark: benchmark: benchmark: benchmark: rule one deck two decks fOUf decks six decks eight decks 0.

It goes up by increasing amounts at high counts, presumably due to insurance becoming more valuable the higher the count.

Generally, in a multiple-deck game, when you have a count per deck of +1 you are playing even with the casino - no advantage or disadvantage.

At a count per deck of +3 you have an advantage of about 1%.

At a count per deck of -1 you are at a disadvantage of about 1% - if you play perfectly.

If the count per 36 PROFESSIONAL BLACKJACK deck is -4, you can expect to lose at a rate of about 2.

Winning with card counting requires betting more money when you have an edge than when you are at a disadvantage, and playing your hands correctly.

Jumping into a Game in Progress When the best available table is a game in progress and you want to jump in without waiting for a shuflle, just start counting from zero and play according to your count.

Treat the unseen discards the same as cards yet to be dealt.

An unseen card is an unseen card whether it has already been used or remains to be dealt.

Example: You jump into a four-deck shoe after about one deck has been used.

You play and count as another deck is used.

If your running count is +6, what is your count per deck?

Two of the four decks have been used, but you have counted the cards in only one deck.

Since about three decks remain unseen by you, divide your running count by three to get a count per deck of +2.

Effect of Ru les on Strategy Some casinos use one deck, some use six, and some use other 333 washburn blackjack />Strategy numbers vary only slightly with the number of decks used.

The tables in this chapter should be thought of as based on multiple decks.

They actually are based on four decks, but are almost exactly the same as strategy numbers based on other multiples such as two, six, etc.

One-deck strategy numbers are slightly different from multiple-deck strategy numbers.

You can use Blackjack Count Analyzer soft- HIGH-LOW SYSTEM 37 ware to find strategy numbers tailored to any number of decks and any set of rules.

You should truncate, and not round, when using these tables.

Examples: If the count per deck required to double down is +4, then double down only if you have a count per deck of +4 or more, and do not double down if your count per deck is say 3.

If a count per deck of -1 is required to stand, then stand if your count per deck, truncated, is -lor higher.

Zero is click at this page than -1, and a count per deck of -1.

At some casinos dealers stand on soft seventeen, whereas at other casinos dealers hit it.

The discussion of this chapter initially assumes that the dealer stands on soft seventeen, and then also covers what to do when the dealer hits soft seventeen.

The decisions are discussed in the order in which you make them at a casino: insurance, pair splits, double down, and hit or stand.

The strategy numbers in this book are derived for multiple decks, and are also very close for one deck.

You must memorize them; when you are playing blackjack in a casino, you can refer to these strategy numbers only in your head.

Insurance Advice of well-meaning but ill-informed gamblers that you should insure only a natural is worth its cost - nothing; you should buy insurance if more than onethird of the unseen cards are lOs.

As soon as you see the dealer's ace, begin considering whether you should buy insurance.

If the count per deck is greater than or equal to +3, buy insurance.

If you can double down after splitting, then use table 5 instead.

Count per deck also determines whether you should split pairs.

The higher the count per deck, the more pairs you should split.

The exception is 8-8 against 10, which you should split when the count per deck is less than +6 and not split when the count per deck is +6 or more.

Dealer stands on soft seventeen.

Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number.

Split only if the count per deck is less than the number in the table.

Examples: A You get 3-3 against 3.

The number from table 4 is +3.

If the count per deck equals or exceeds +3, you should split the pair.

If the count per deck is less than +3, you should hit.

B You get 3-3 against 9.

That spot in table 4 is blank, so do not split, no matter what the count per deck.

C You get 8-8 against 9; table 4 says to split no matter what the count per deck.

Split only if the count per deck is less than the number in the table.

You should be more likely to double down 'if the count per deck is higher.

If you decide to double down, always put up additional money equal to the full amount of your original bet.

The soft-doubling parts of all double-down tables in this book assume that the ace in your hand can be valued at either 1 or 11 after you split.

If the ace cannot swing, then do not double down on soft hands of nineteen or less.

Examples: A You get 6-5 against 4.

Your hand totals eleven, and table 6 says to double down no matter what the count per deck.

B You get 4-5 against 9.

Your hand totals nine and table 6 says do not double down, so you take a hit.

You may call your total nine and double down if you wish, and table 6 advises doing so if the count per deck equals or exceeds +1.

If the count per deck is less than +1, or if doubling and catching a 2 would give you eleven instead of twenty-one, call your total nineteen and stand.

Dealer stands on soft seventeen.

Soft doubling as in table 6 takes precedence, so use table 7 only if you are not doubling down.

If you do not double down you should hit a soft total of seventeen or less and stand with a soft total of nineteen or more.

With soft eighteen if you do not double down you should stand against 2 through 8, and hit against 9 or 10.

Against ace you should hit soft eighteen when the count per deck is less than +1, but stand when the count per deck is +1 or more.

Table 7 Soft Standing, S17 Player's Hand soft 19 soft 18 soft 17 KEY blank h s 17 2 3 4 h h h Dealer's Upcard 5 6 7 pokerstars fpp h h h h 9 10 A h h h h 1 h Stand.

Dealer stands on soft seventeen.

With a total of eleven or less, never stand.

With a hard total of twelve or more, you should be more likely to stand when the count per deck is higher.

Examples: A You have 9-8 for a total of seventeen against 2; table 8 says to stand.

B You have 10-4 for fourteen against 8; table 8 says to hit.

Suppose you receive a 7 to bring your total to fifteen.

If the count per deck is less than +4, you should ask for another hit; if the count per deck equals or exceeds +4, you should stand.

Dealer stands on soft seventeen.

Stand if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

If you are playing blackjack at a casino where the dealer hits soft seventeen, you should play slightly differently, primarily against an ace.

The differences against an ace are: You will be more likely to hit soft eighteen, more likely to stand on hard sixteen or hard fifteen, more likely to double down on eleven, and more likely to split 9-9.

Hitting soft seventeen has these effects on strategy because the dealer is less likely to finish the hand with exactly seventeen and is more likely to bust.

The major changes to make to tables 4-8 if the dealer hits soft seventeen are as follows: Table 7: Always hit soft 18 against ace.

Table 8: With hard 16 against ace, stand if the count per deck is +3 or more.

With hard 15 against ace, stand if the count per deck is +5 or more.

With hard 12 against 6, stand if the count per deck is -3 or more.

So the dealer hitting soft seventeen is a rule variation that is costly to the card counter.

Table 9 applies if the dealer stands on soft seventeen, and table 10 applies if the dealer hits soft seventeen.

Table 10 includes the major changes listed above plus minor changes.

The line for doubling takes precedence.

Note that in table 9, the 1 for ace-7 against ace is for choosing between hit and stand, and does not say to double down.

Importance of Strategy Tables This chapter has presented multiple-deck playing strategy, but at some casinos blackjack is dealt with one deck.

Multiple-deck strategy numbers are only slightly different from single-deck strategy numbers.

You will win almost as much by using multiple-deck strategy numbers with one deck as you will win by using onedeck strategy numbers with one deck.

With freshly shuftled cards, the casino has a larger advantage as the number of decks increases.

You might be wondering how important bet variation is relative to strategy variation.

See the hI- row for the number saying whether to hit or to stand.

Split only if the count per deck is less than the number in the table.

With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

Playing only positive counts and leaving the table on negative counts, as popularized by earlier editions of Professional Blackjack, is a form of bet variation.

So varying strategy with the count per deck improves on basic strategy by only a small amount when flat bets are used in a six-deck game.

Most of that improvement is due to taking insurance, standing on hard sixteen against 10, and standing on hard fifteen against 10, all at the appropriate times of course.

After those three decisions, the amounts a card counter gains by correctly deviating from basic strategy drop off dramatically.

See Donald Schlesinger's "Attacking the Shoe" in the September 1986 issue of Blackjack Forum for more detail on this point.

Using a smaller range of strategy numbers should result in a lower win rate, but I got a higher rate.

I increased the sample sizes to see whether the gap would disappear or the difference would become statistically significant.

This is a puzzle, and I am continuing to work on it.

My suggestion for beginners, and for anyone else who wants to cut down on the volume of numbers to memorize, is ignore all strategy numbers smaller than -1 and larger than +6.

For winning money at multiple-deck blackjack, bet variation is much more important than strategy variation.

In fact, all of these tables on deviating from basic strategy are of minor value.

More important is to avoid being barred and to find good rules and good penetration.

Appendix A contains more details on strategy rules for the high-low system.

Table 11 gives the minimum counts per deck for which insurance is a good buy, as a function of number of decks shuffled together.

For example, if six decks are used, insurance is likely to be profitable if the count per deck, after counting the dealer's ace and as many other cards as you can see, exceeds 3.

My suggestion to simplify the strategy numbers: Use 1.

Table 11 also includes advice for the halves counting system, which is explained in chapter 12.

Take insurance at a count per deck equal to or greater than the strategy number shown in table 11.

The insurance number does not depend on whether or not the dealer takes a hole card.

For single-deck games, insurance is quite valuable.

The following boxes report the results of two simulations, one with and one without insurance.

The shuflle point is 26 cards yet unseen.

The IO-count is the best count for insurance.

The only way to make better insurance decisions is to get a glimpse of the dealer's hole card.

The simplest way to do the 10-count is to keep the following as a second count as shown in table 12-: assign the value +1 to all cards except 10 and assign the value -2 to every 10.

If this running IO-count gets read more four times the number of decks shuftled together +4 with a single deck, +8 with a double deck, +16 with four decks, or +24 with six decksbuying insurance is profitable.

Note that the 10-count insurance decision is based learn more here the running count and not on the count per deck.

For example, to buy insurance on a six-deck game you need a IO-count of +24 or more no matter how many decks have yet to be dealt.

Also note that the pluses and minuses do not balance out - when all the cards are gone, your running count will be four times the number of decks used.

Table 12 to-Count for Insurance Card ace, 2, 3, 4, 5, 6, 7, 8, 9 IO,J,Q,K Count +1 -2 INSURANCE 55 Suppose the dealer shuflles a single deck and deals the following cards: ace-9 6-6-8 7-10 After that round the high-low running count is zero.

Suppose you place a bet, are dealt 8-9, and the dealer shows an ace.

Would you buy insurance?

Though the high-low count of -1 says no, you should.

The reason is that an excess of 7s, 8s, 9s, and aces has appeared.

Of the 42 cards you have not seen, 15 are lOs; when you buy insurance you will be paid 2 to 1 for something that will happen 15 out of 42 times, giving you an expected-win rate of roughly 7%.

If you are keeping a IO-count for insurance decisions, the IO-count would be +7 and thus indicate that you should buy insurance.

A single running count that relates 1Os to non-lOs for accuracy on insurance could be used for all decisions, but such a count is less accurate for estimating advantage; gains resulting from accuracy on insurance are more than offset by losses on accuracy on estimating advantage.

IO-count strategies are obsolete for this reason.

Bustout Bet Some casinos offer a bustout bet.

Winning bustout bets pay 2:1.

If the dealer's first two cards total twelve through sixteen and you still have a hand in play, you can bet that the next card will be a 10 or face card.

If you bust, or if you have a natural for which you have already been paid, you do not have the option of making the bustout bet.

If you split or double down, the amount you can bet on the bustout is not doubled.

If you use the high-low or halves see chapter 12 counting system with the bustout bet, use 3.

The reason the insurance number does not apply to the bustout bet is the dealer does not show an ace.

With basic strategy, you will have the option of making the bustout bet 32% of the time, or four times as often as the dealer shows an ace.

The max on the bustout bet is your original bet.

So the bustout bet is approximately eight times as profitable as insurance.

With insurance the dealer always shows an ace, and insurance is a better bet the more non-lOs have been removed 57 INSURANCE from the pack.

But with the bustout bet, you get to see more cards before deciding whether to make the bet.

My suggestion is to play bust-out blackjack with a partner, keeping two counts and helping each other make decisions.

One of you uses a ten-count to make correct bust-out and insurance bets, and the other uses a conventional counting system such as high-low for betting and making decisions other than bustout and insurance.

Table 14 is doubling down when the dealer hits soft seventeen.

As you can see by comparing those two tables, how the dealer plays soft seventeen has almost no effect on whether you should double down.

If you can double down on certain totals only, then use only the appropriate part of the tables.

For example, if you can double down on ten or eleven only, then ignore the parts of the tables that give advice on doubling down on nine or less.

Double on Any Number of Cards Tables 13 and 14 are for use when this is your only chance to double, and if you hit instead you no longer DOUBLE DOWN 59 have the option of doubling.

This is the most common form of doubling down.

At a few casinos, you can double down on any number of cards.

This rule makes doubling on small twocard hands less attractive.

The strategy numbers for doubling down when you can double on any number of cards are shown in tables 15 and 16.

With ace-2 or ace3 when the count is high enough, the tables say to double down against 4, 5, or 6; however, you might as well just hit all hands of A-2 and A-3 because the value of doubling is approximately equal to the value of hitting no matter how high the count.

That is true no matter which action the dealer takes on soft seventeen.

Double on Two or Three Cards At Las Vegas Club in Las Vegas, starting in 1993, you can double down on two or three cards, except after splitting.

This requires two double-down strategies.

One strategy table 16 is for use if you will still have the option of doubling down after hitting once, and the other strategy table 14 is for use if this is your last chance to double on this hand.

For example, with eight against 5 and a high count, you ought to hit if you will have the option of doubling after that hit, but double down now if you will not have the option of doubling after hitting.

With Multiple-Card Surrender In some Asian casinos you can or could surrender on any number of cards.

This makes hitting more attractive relative to doubling can blackjack free slots seems />The appropriate doubling strategy is shown in table 17.

The only change from table 13 is the strategy number for ten against 10 is +6.

Dealer stands on soft seventeen.

Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number.

With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

Dealer hits soft seventeen.

Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number.

With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

Dealer please click for source on soft seventeen.

Double down if the count per deck equals or exceeds the number; do not double down if the count per deck is less than the number.

With Surrender After Doubling If you have multiple-card surrender plus surrender after doubling down, use the strategy numbers from table 18.

Surrender strategy can be found in the surrender chapter.

If the dealer hits instead of stands on soft seventeen, the only change to the ace column of table 18 is the strategy number for doubling down on ten against ace; it should be o.

Table 5 page 39 covers pair splits if the dealer stands on soft seventeen and you can double down after splitting; it is reproduced to beat noraut poker as table 20.

Tables 21 and 22 are parallel, and apply if the dealer hits soft seventeen.

Table 21 contains the strategy numbers for splits if the dealer hits soft seventeen and you cannot double down after splitting.

Table 22 contains the strategy numbers for splits if the dealer hits soft seventeen and you can double down after splitting.

Note that the asterisk reverses the meaning of the strategy number.

Normally you split if the count per deck equals or exceeds the number.

But if there is an 68 PROFESSIONAL BLACKJACK asterisk, you split if the count per deck is less than the number.

Comparison of the four tables with each other shows that being able to double down on any two cards after splitting has large effects on playing strategy, but whether the dealer hits or stands on soft seventeen makes a difference only for splitting 8-8 against ace.

Resplits Resplits make no difference in the sense that the strategy for splits is the same whether or not resplits are allowed.

Resplits follow split strategy: The strategy number for resplits is the same as the strategy number for splits.

Not being able to resplit costs you about a dollar per 100 hands.

Double Any Number of Cards Suppose you can double down on any number of cards.

This rule makes splitting small pairs such as 3-3 less attractive, because you might hit the total 6 with a small card, such as a 4 or 5, bringing your hand to a total that would be an attractive three-card double.

Tables 23 and 24 contain strategy numbers for splitting if you can double down on any number of cards, but cannot double down after splitting.

Table 23 is for dealer stands on soft seventeen; and table 24 is for dealer hits soft seventeen.

Being able to double down on any number of cards increases the value of hitting small totals, including SPLIT 69 pairs.

But being able to double down on any number of cards after splitting increases the value of splitting.

The stronger effect is the increased value of being able to double down after a hit.

Therefore if you can double down on any number of cards, you are less likely to split small pairs.

Tables 25 and 26 contain strategy numbers for splitting if you can double down on any number of cards after splitting; table 25 is for dealer stands on soft seventeen; and table 26 is for dealer hits soft seventeen.

In 1993 Las Vegas Club in Las Vegas introduced a hybrid: you can double down on any two or three cards before you split, but you can double down only on any two cards after you split.

You cannot double down on four or more cards, and after splitting you cannot double down on three cards.

Tables 27 and 28 cover Las Vegas Club splitting.

Table 27 is for dealer stands on soft seventeen, and table 28 is for dealer hits soft seventeen.

Split only if the count per deck is less than the number in the table.

Dealer stands on soft seventeen.

Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number.

Split only if the count per deck is less than the number in the table.

Dealer hits soft seventeen.

Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number.

Split only if the count per deck is less than the number in the table.

Split only if the count per deck is less than the number in the table.

Dealer hits soft seventeen.

Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number.

Split only if the count per deck is less than the number in the table.

Split only if the count per deck is less than the number in the table.

Split only if the count per deck is less than the number in the table.

Split only if the count per deck is less than the number in the table.

Dealer hits soft seventeen.

Split if the count per deck equals or exceeds the number; do not split if the count per deck is less than the number.

Split only if the count per deck is less than the number in the table.

The players get two cards each to start, and finish their hands in the normal mamier.

Then the dealer takes a second card, and if necessary, more cards.

Suppose the dealer gets a natural, a two-card twentyone.

At most casinos you lose unless you also have a natural; if you have gotten to twenty-one in three or more cards you lose if the dealer gets a natural.

At some casinos you lose only your original bet and not any extra amounts you have wagered on doubles or splits.

At other casinos if you have doubled down or split and the dealer gets a natural you lose your whole doubled bet.

There also are casinos that allow you to keep your money i.

Lose Only Original Bet to a Natural Whether the dealer's second card is face down under the upcard or is in the pack yet to be dealt should not matter to you.

This is true even if you are the last player to act.

If you not taking a card means it will be the dealer's second card, the proper play of your hand is the same as if the dealer already had a second card.

If you have sixteen and the proper play is to hit, then you should hit whether the dealer has a hole card or not.

What does make a difference in strategy is what happens when the dealer gets a natural.

Lose All to a Natural In Europe and a few other places, the dealer does not take a hole card, and if the dealer ends up with a natural you lose your whole bet if you have doubled down or split a pair.

This means you should be less willing to wager extra money on your hand when the dealer shows a 10 or ace.

If you cannot lose more than your original bet in the event of a natural by the dealer, this section does not apply.

This section applies only for the situation where if you put out an extra bet and the dealer has a natural, you lose your original bet and the extra bet too.

If it is correct to hit your hand when the dealer takes a hole card, it is also correct to hit your hand if the dealer does not take a hole card.

See elsewhere in the table for hit or stand.

Dealer stands on soft seventeen.

Stand or double or split at a count per deck equal to or greater than the number; hit or do not split at a count per deck less than the number.

With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

The dealer winning all on naturals also does not affect the play of your hand if the dealer shows other than a 10 or ace.

The tables in this chapter have colwnns for dealer showing 2-9, but those columns are identical to those in table 9 page 46.

Basic strategy, if you lose all to a natural by the dealer, is to split ace-ace against 10 but otherwise do not double down or split against 10 or ace.

Advice for card counters if the dealer might have a natural to beat your doubled bet is contained in table 29.

Note that the only doubling down to consider against a 10 or ace is with eleven against 10.

The only splitting to consider against a 10 or ace is ace-ace against 10.

The strategy numbers for playing against 10 and ace do not depend on whether you can double down after splitting.

The advice to never double or split against an ace holds whether the dealer hits or stands on soft seventeen.

Table 29 assumes the dealer stands on soft seventeen.

If you are playing against the lose-all rule in a casino where the dealer hits soft seventeen, then use the 10 and ace columns of table 29 with columns for other upcards from table 10 page 48.

Table 30 summarizes the amendments to those parts of the appendix that deal with rules for doubling down and pair splitting against 10 or ace when you lose all to a dealer's natural.

The appendix contains strategy indexes for the high-low and halves counts for one and multiple decks and both dealer actions on soft seventeen.

No other decision is affected by whether or not the dealer takes a hole card.

Any 21 Ties Dealer Natural This section applies if the dealer's two-card twentyone counts as an ordinary twenty-one and not as a natural winner.

You should be slightly more likely to hit.

See elsewhere in the table for hit or stand.

Dealer stands on soft seventeen.

Table 31 is the same as table 9 page 46 except for the 10 and ace columns.

Table 31 assumes you can tie all the dealer's twocard twenty-ones by reaching twenty-one yourself in any number of cards.

If your three-or-more-card twentyone ties the dealer's two-card twenty-one for only some upcards and not for others, then use part of table 31 and part of table 9.

For example, you might find a casino where the dealers do not check hole cards under 10s but do check hole cards under aces.

Ace-up naturals are turned up immediately, so you never get a chance to get a three-card twenty-one if the dealer shows an ace.

In that case, use the 10 column from table 31 and the ace column from table 9.

The other columns are the same on both tables.

The ace column of table 31 applies whether the dealer stands or hits with soft seventeen.

Thus for a casino in which the dealer hits soft seventeen, the 10 and ace columns of table 31 can be combined with the other columns from table 10 page 48.

So this rule is quite valuable for the card counter.

At the time of this writing, this rule is offered at several casinos in the Reno area.

Surrender is a good play only if there is no other way to play your hand that has an expected win of better than -0.

In other words, since surrender means losing half a bet, any other play that on average loses less than half a bet is superior to surrender.

The tables in this chapter have two columns for dealer ace.

The column labeled "A-s17" is for use when the dealer stands on soft seventeen, and the column labeled "A-hI7" applies when the dealer hits soft seventeen.

Whether the dealer hits or stands on soft seventeen does not matter when the dealer starts with 7 through 10.

Surrender can be worthwhile if the dealer shows 8, 9, 10, or ace.

When the dealer starts with 2 through 7, you should not surrender.

For a complete set of strategy 90 PROFESSIONAL BLACKJACK numbers when surrender is offered, combine the appropriate table of this chapter with the 2-7 columns of the appropriate table of another chapter, such as chapter 3.

The tables of this chapter break 8-8 from other player hands of sixteen because splitting 8-8 is an attractive alternative.

There is no reason to separate 7-7 from other fourteens, because for upcards you might surrender against you should not split 7-7.

Early Surrender At some casinos, you have the option of early surrender; that is, you may surrender before the dealer checks the hole card.

The higher the count per deck, the more likely you should be to surrender.

Table 32 summarizes early-surrender decisions.

Do not surrender against 2- 7.

Examples: A You get 9-7 for sixteen against ace; table 32 says always surrender.

B You get 8-8 against 8; table 32 says do not surrender.

C You get 10-4 for fourteen against 9; surrender if the count per deck is +6 or more, and do not surrender if the count per deck is less than +6.

If you lose all to a natural by the dealer and thus will not split 8-8 against 10, then for 8-8 against lOuse -5, the same number blackjack california casinos for other sixteens against 10.

Unfortunately, early surrender is also rare; late surrender is the common form of surrender.

Dealer hits soft seventeen.

Dealer stands on soft seventeen.

The higher the count per deck, the more likely you should be to surrender.

Table 33 summarizes latesurrender decisions.

Examples: A You get hard thirteen against ace; table 33 says do not surrender.

B You get 10-4 for fourteen against 9; surrender if the count per deck is +6 or more, and do not surrender if the count per deck is less than +6.

If you lose all to a natural by the dealer and thus will not split 8-8 against 10 or ace, then for 8-8 against 10 or ace use the same numbers as for other sixteens against 10 or ace.

Late surrender is not as profitable as early surrender, of course, since you cannot surrender against a natural.

Dealer hits soft seventeen.

Dealer stands on soft seventeen.

Surrender if the count per deck equals or exceeds the number; do not surrender if the count per deck is less than the number.

Dealer hits soft seventeen.

Dealer stands on soft seventeen.

Surrender if the count per deck equals or exceeds the number; do not surrender if the count per deck is less than the number.

Multiple-Card Surrender At a few casinos, primarily in Asia, you can surrender any number of cards.

Dealer hits soft seventeen.

You are less likely to surrender a hand with a low total if you will still have the option of surrendering after one more card.

The strategy numbers for multiple-card early surrender are shown in table 34.

The strategy numbers for multiple-card late surrender are shown in table 35.

No strategy table is provided for this rule because except for seventeen against ace, the best play is simple: After doubling down, surrender all hands of sixteen or less if the dealer shows 8, 9, 10, or ace.

If you have doubled down and ended up with seventeen and the dealer shows an ace, you might want to surrender.

Follow the advice for seventeen against ace in the appropriate table from earlier in this chapter.

When to double, if you can surrender after doubling down, is explained in the chapter on doubling down.

The difference between this second-chance option and ordinary surrender is you must play a new hand for the remaining half of your bet rather than simply putting 97 SURRENDER those chips back on your stack.

If the dealer shows a card that is powerful enough to make surrender worth considering, then the dealer is showing a card you would rather not play against.

Therefore, basic strategy says ignore the second-chance option.

If you are counting cards and the count gets high enough, you sometimes are better off giving up half your bet to get a new hand.

The situations arise when you have hard thirteen to seventeen and the dealer shows 7 to 9.

How high the count per deck must be to justify giving up half your bet to get a new hand is shown in table 36.

This table applies to both dealer actions on soft seventeen.

Table 36 Second-Chance 21 Player's Hand hard 17 hard 16 hard 15 hard 14 hard 13 2 3 4 Dealer's Upcard 5 6 7 8 9 8 4 4 8 5 5 9 3 4 5 9 10 A KEY blank Ignore the second-chance option.

The strategy for playing them depends on the size of the bonus and the number of additional cards required to be eligible for the bonus.

The tables in this chapter tell blackbelt in blackjack ebook when you should hit, given the number of additional number of cards you need to qualify for the bonus.

The more cards already in your hand the closer you are to the bonus, and thus the more likely you are to hit.

Of course you stop hitting and smile when you have enough cards to qualify for the bonus.

Every live-game bonus I have seen has been optional, and if you accept it the hand is over and the dealer picks up your cards.

Some blackjack slot machines have a bonus that is paid automatically because no more cards will fit on the screen.

For example, Macao has a bonus of half a bet for a fivecard hand.

With a bonus that small, some five-card hands are worth more when played out to maybe win, maybe push, maybe lose.

Tables 37-44 are for use when the bonus is half a bet.

If you happen to be offered a half-win bonus for six cards, then use tables 38-40 and 42-44 for hands of one more card; for example use table 40 for six-card hands instead of five-card hands.

Tables 37-44 assume the dealer stands on soft seventeen.

They also assume you can double down on any first two cards, but cannot double down on more than two cards.

They also assume you lose only one bet if the dealer has a natural.

The casinos that I know have offered this rule all are located in Asia.

At all of them, dealers stand on soft seventeen.

If you need five-card half-win tables for dealers hit soft seventeen, you can generate them yourself with Blackjack Count Analyzer.

Tables 37-40 assume you cannot collect a half win if the dealer has a natural.

Double down; if you cannot double, then hit.

Dealer stands on soft seventeen.

Stand or double or split at a count per deck equal to or greater than the number; hit or do not split at a count per deck below the number.

Hit or do not split at a count per deck equal to or greater than the number; stand or double or split at a count per deck less than the number.

Dealer stands on soft seventeen.

Hit at a count per deck equal to or greater than the number; stand at a count per deck less than the number.

Dealer stands on soft seventeen.

IIit at a count per deck equal to or greater than the number; stand at a count per deck less than the number.

Reject bonus of half a bet, and stand.

Dealer stands on soft seventeen.

Reject bonus and stand at a count per deck equal to or greater than the number; accept bonus at a count per deck less than the number.

Accept bonus at a count per deck equal to or greater than the number; reject bonus and stand at a count per deck less than the number.

FIVE CARDS 105 At some casinos that offer the five-card half win, dealers do not take hole cards and you lose allan doubles or splits if the dealer has a natural.

At those casinos, if you get a five-card hand you can collect your half-win before you fmd out if the dealer has a natural.

This book calls this "early half-win.

The columns for 2-9 are identical to tables strategies advanced since the dealer cannot get a natural except with a 10 or ace up.

If you are playing the five-card half win at a casino where the dealer checks under the ace but not under the 10, then use the ace columns from tables 37-40 and the 10 columns from tables 41-44.

Double down; if you cannot double, then hit.

See elsewhere in the table for hit or stand.

Dealer stands on soft seventeen.

Stand or double or split at a count per deck equal to or greater than the number; hit or do not split at a count per deck less than the number.

Reject bonus of half a bet, and stand.

system blackjack betting oscars stands on soft seventeen.

Dealer stands on soft seventeen.

A h h h h number Stand at a count per deck equal to or greater than the number; hit at a count per deck less than the number.

Double down; if you cannot double, then hit.

Dealer hits soft seventeen.

Hit or do not split at a count per deck equal to or greater than the number; stand or double or split at a count per deck less than the number.

With 4-4, splitting is better than doubling down if doubling is allowed after splitting.

See the h1- row for the number saying whether to hit or to stand.

In this case it is always worthwhile accepting the bonus - you should never turn it down.

Note that it is worthwhile hitting soft twenty-one if one more card will give you an automatic winner.

A sure winner is worth more than a hand that will probably win but might push.

Tables 45-51 assume the automatic winner applies to six-card hands.

If you happen to be offered an automatic win for five cards, then use the tables 45-51 for hands of one less card; for example use table 47 for fourcard hands instead of five-card hands.

Likewise if you happen to be offered an automatic winner for seven-card hands, then use tables 45-51 for hands of one more card.

Tables 45-51 also assume you lose only one bet if the dealer has a natural.

Tables 45-47 assume the dealer stands on soft seventeen, and they assume you can double down on any first two cards.

Tables 48-51 assume the dealer hits soft seventeen and also assume you can double down click to see more two or three cards.

Las Vegas Club in Las Vegas at this writing has these rules, and allows double down on two cards after splitting.

The numbers for soft eighteen in table 48 are for stand versus double.

If the count per deck is below the number in the table, then stand; if the count per deck is equal to or greater than the number in the table, then double.

Soft eighteen is more complicated if: you have it in three cards, you can double down on three cards, and a six-card hand is an automatic winner.

Hit or do not split at a count per deck equal to or greater than the number; stand or double or split at a count per deck less than the number.

Dealer stands on soft seventeen.

Double Pay on Five-Card 21 Tables 52-54 cover the situation where five cards is an automatic 2: 1 winner only if you get exactly twentyone and the dealer does not also get twenty-one.

This bonus is less valuable than if you were to get the bonus no matter what the total points in your hand, and no matter whether that total was a winner.

This bonus has been offered at a few tables in Atlantic City since 1991.

Each table in this chapter is divided into several sections.

The reason is there are several ways that the bonus can interact with surrender.

You might be paid the bonus even if the dealer turns out to have a natural on the hand, or the bonus might be paid only when the dealer does not click to see more a natural.

And there are both early and late surrender.

All of the tables in this chapter assume you are paid the bonus if the dealer achieves twenty-one in more than two cards.

If the bonus applies only to one suit, such as spades, the numbers in the sections for 6-7-8 suited apply.

You simply use these numbers when your first two cards are of the appropriate suit.

Tables 55-58 are for use when the dealer stands on soft seventeen; tables 59-62 are parallel and apply when the dealer hits soft seventeen.

Double Pay for 6-7-8 Double for Mixed Suits In 1990 The Red Lion in Elko paid double for any 67-8; the cards did not have to be of the same suit.

This made hitting 6-7, 6-8, and 7-8 more attractive.

The appropriate strategy numbers defining where hitting becomes more attractive than standing are shown in table 55.

Red Lion did not offer surrender.

However, table 55 also indicates when it is better to hit than to earlysurrender, and when it is better to hit than to latesurrender.

Note that hi-means do not surrender; instead use the hit-stand part of the table.

Dealer stands on soft seventeen.

That is not nearly as valuable as double pay for blackbelt in blackjack ebook 6-7-8.

Table 56 indicates when it is better to hit than to stand, when it is better to hit than to early-surrender, and when it is better to hit than to late-surrender.

However, should you encounter this rule, table 57 indicates when it is better to hit than to stand, when it is better to hit than to early-surrender, and when it is better to hit than to late-surrender.

The value of this rule to a card counter is minimal.

Table 58 indicates when it is better to hit than to stand, when it is better to hit than to early-surrender, and when it is better to hit than to late-surrender.

Dealer stands on soft seventeen.

Stand against 2 through 6 or surrender against 8, 9, 10, or ace if the count per deck equals or exceeds the number; hit if the count per deck is less than the number.

See the fIrst three lines of the table for hit or stand.

Dealer hits soft seventeen.

See the first three lines of the tabIe for hit or stand.

See the fIrst three lines of the table for hit or stand.

See the fIrst three lines of the table for hit or stand.

It assumes that you are not allowed to double after splits.

Table 63 gives the strategy appropriate for splitting a pair of 7s when 7-7-7 any mixture of suits pays double, and table 64 applies when 7-7-7 pays triple.

Tables 63-64 are for use when the dealer stands on soft seventeen; tables 65-66 are parallel and apply when the dealer hits soft seventeen.

See the fIrst join blackjack video strip game congratulate of the table for hit or split.

See the fIrst line of the table for hit or split.

See the fIrst line of the table for hit or split.

Dealer hits soft seventeen.

You can bet that your first two cards will total over thirteen, or that they will total under thirteen.

Aces count one, and thirteen loses.

This innovation started in 1988 at Caesars Tahoe and hopefully will continue to spread.

You can make money on them only by counting cards.

Both the over and under are horrible bets right after a shuflle.

The over loses at the rate of more than 6% and the under loses at a rate of more than 10%.

Here is an example showing how strong the over bet is.

With a freshly-shuffled single deck, there are 1326 two-card combinations, of which 618 exceed thirteen and win while 708 lose, for a casino edge of 6.

Removing two small cards leaves 50 cards from which can be dealt 1225 two-card combinations, of which 618 win and 607 lose, for a player edge of 0.

Removing a third small card leaves 49 cards from which can be dealt 1176 two-card combinations, of which 618 win and blackbelt in blackjack ebook lose, for a player edge of 5.

Removing a fourth small card leaves 48 cards from which can be dealt 1128 twocard combinations, of which 618 win and 510 lose, for a player edge of 9.

Removal of one card per deck can make a difference of about 4% in expected value of the over bet.

The most effect any card has in regular blackjack is about 0.

You do not fmd profitable over or under bets very often, but some of the bets you make have very high expected values.

While waiting, I calculated my expected win on the coming hand.

I had a running crush count of +9 with about half a deck remaining to be dealt.

At 2% per count per deck, my edge was 30% on my over bets!

Mentally I shouted "Please don't shuffle!

This increased risk means you may want to decrease your bet size somewhat or take a larger bankroll to handle the larger fluctuations you will encounter.

For the high-low, bet the under at counts below -6 and bet the over at counts above 5.

Those numbers apply to all numbers of decks.

Counting systems that are good for the over are also good for the under.

I have checked the numbers in this paper, including calculating all the correlation coefficients for one to eight decks, and I found no mistakes.

I am not going to reproduce Snyder's work here.

The crush count has a correlation coefficient of 0.

It is a level 2 count, meaning that the values you assign to the cards are integers from click to +2.

Snyder's best level 2 count is slightly more efficient at 0.

I prefer the crush COWlt because it assigns the value of zero to more cards, making it easier to use.

The crush count is shown in table 67.

It counts +2 for A and 2, +1 for 3, and -1 for 9-10.

Bet the over on crush counts per deck of 3 or more, and bet the under on crush mackie onyx blackjack per deck of -5 or lower.

Those numbers apply to all numbers of decks.

A reminder: -6 is lower than -5; -4 is not lower than -5.

Each count per deck of the crush count changes your expectation on over and under bets by about 2.

The break-even count per deck on the over is 3.

The break-even count per deck on the under is -4.

The simulation results reported in the rectangles are for the benchmark betting scheme, including a matching bet on the over or under if justified.

Better penetration means a higher expected win rate, probably more so than with any other blackjack rule.

The counting system introduced in this section will be referred to as counting halves.

Both counting systems have the same efficiency with respect to insurance and each requires the same amount of memorizing from tables.

Counting halves is more difficult to do, but will result in a slightly higher expected-win rate.

Counting halves means assigning values to cards as shown in table 68.

If you prefer to work with integers you can double all these card values, and also double the strategy numbers in tables 69 and 70.

A full deck contains 20 plus and 20 minus points in the high-low system, and 22 plus and 22 minus points in the halves system.

The halves system finds more situations of advantage than the high-low system does.

The 149 HALVES COUNT halves count is a very close approximation of the precise value of each card as given on page 162 link the 1962 edition of Thorp's Beat the Dealer.

With typical rules, the dealer's advantage disappears when either high-low or halves count per deck rises to +1.

According to Thorp's book, each additional count per deck gives you an extra 0.

Thus the value of a high-low count per deck plus or minus one standard deviation is 0.

Chapter 14 will explain standard deviation.

In the halves count, each additional count per deck also gives you an extra 0.

These numbers come from comparisons of the counting systems with Thorp's card values.

The low standard deviation means that the halves count makes almost no errors in estimating advantage.

Peter Griffin in The Theory Of Blackjack says the betting correlation of the halves counting system is 0.

Simulation shows the halves count results in higher w.

The sample sizes were increased to the point where that difference is statistically significant.

The high-low count overestimates the values of 2 and 9 and underestimates the values of 5 and 7.

The halves count is a very close approximation of the true value of each card.

As an example, suppose you are watching two tables simultaneously, both tables are single deck, and both dealers are shuffling.

One dealer is a bit careless and shows you the bottom card, a 5.

The other dealer is even more careless and shows you both the burned card and the bottom blackbelt in blackjack ebook, and they are both 2s.

You should immediately make a bet at one table because you have an advantage, but which table?

According to Thorp's data, one 5 gone gives you a greater advantage than two 2s gone.

The halves count tells you accurately that the table with the 5 gone is better, since it gives you counts of +1.

The high-low count gives you the wrong answer in this case, since it gives you counts of +1 and +2.

Griffin claims that a 5 is worth almost exactly the same as two 2s.

An attractive feature of counting halves is that the strategy tables are similar to those of the high-low system.

Once you are expert with the high-low system, HALVES COUNT 151 you can switch to counting halves in easy stages.

For example, you can count halves but make your playing strategy decisions by using the high-low tables that you have already memorized; and, if counting halves is too difficult at fIrst you can always switch back to the highlow system.

When you feel so confident counting halves that you know you do not want to go back to the high-low system, then learn the halves strategy numbers and forget the high-low strategy numbers.

Table 69 is the halves strategy numbers for dealer stands on soft seventeen; it is the counterpart of table 9 page 46.

Table 70 is the halves strategy numbers for dealer hits soft seventeen; it is the counterpart to table 10 page 48.

Insurance Insurance is a good bet if the halves count per deck exceeds +1.

The under is a good bet at counts per deck of -6 and lower.

Appendix B Appendix B contains strategy rules for the halves system.

Table B1 is for a single deck when the dealer hits soft seventeen.

Table B2 is for four decks when the dealer hits soft seventeen.

Table B3 is for a single deck when the dealer stands on soft seventeen.

Table B4 is for four decks when the dealer stands on soft seventeen.

See the hI- row for the number saying whether to hit or to stand.

Dealer stands on soft seventeen.

See the hI- row for advice on whether to hit or to stand.

This sounds great, but there is another rule that is to your disadvantage: The dealer wins pushes unless you both get naturals.

At most casinos with double exposure, naturals win even money instead of three to two but your natural wins even if the dealer also has a natural.

Blackjack Secrets has a chapter on double exposure.

That book has basic strategy and https://partysutra.com/blackjack/blackjack-ballroom-spam.html strategy numbers from -1 to +6.

A few paragraphs of this book are word-for-word copies from that chapter of Blackjack Secrets.

This book has three things the Blackjack Secrets chapter does not have: high-low strategy numbers for a DOUBLE EXPOSURE 157 broader range -10 to +10strategy numbers for the halves counting system, and simulation results.

Estimating Your Edge Simulation shows a basic-strategy player's approximate expectation for double exposure for four or more decks, dealer stands on soft seventeen, no resplits, and naturals pay even money, is -0.

The negative sign means the casino has the edge.

This is shown in table 71.

The rest of table 71 contains numbers that you can add to that -0.

For example, if you are playing a double-exposure game in which the dealer Table 71 Player Edge for Dbl Exposure benchmark both dealer cards face up, pushes lose except natural against natural, dealer stands on soft seventeen, double any first two cards, no resplits -0.

If you get paid though your natural is tied by the dealer's natural, table 71 says you gain 0.

If naturals pay 3:2, the value of getting paid on a tied natural is 0.

Table 71 shows that allowing double after splits gives the player 0.

So if you can double after splits but only on totals of nine or more, you gain 0.

Basic Strategy Tables 72-74 contain generic basic strategy for double exposure.

Table 72 contains the part of basic strategy for double exposure that applies no matter what the dealer does on soft seventeen.

In this table, split strategy is the same whether or not you can double down after splitting.

Table 73 is the rest of basic strategy for use if the dealer stands on soft seventeen.

Table 74 is the rest of basic strategy for use if the dealer hits soft seventeen.

Thus a complete basic strategy for double exposure is table 72 plus either table 73 or table 74.

There are important differences in strategy that depend on whether the dealer hits or stands on soft seventeen.

One important difference is if the dealer stands on soft seventeen, then A-6 is played according to the column labeled 17 in table 72.

If the dealer hits soft seventeen, there is a separate column for that hand in table 74.

With soft eighteen against small dealer totals, it does not matter much whether you hit or stand if you 159 DOUBLE EXPOSURE cannot double.

Eighteen is not a strong hand when the dealer wins ties.

You might be wondering why basic strategy says stand on sixteen against eight at double exposure, but hit at single exposure.

Standing on sixteen has equal value at either variety of blackjack - you win if the dealer busts, and you cannot think, samsung blackjack smartphone entertaining />Hitting sixteen is more valuable at single exposure because you might end up with a push.

The dealer wins ties at double exposure.

Seventeen is the worst hand you can get at double exposure - you cannot push, and the only way you can win with seventeen is if the dealer busts.

Counting Cards Each count per deck is worth about 0.

This is in contrast to the 0.

Your advantage at double exposure is more volatile than your advantage at single exposure - at higher counts your advantage is higher.

Therefore, for equivalent bet sizes you can make more money at double exposure than at single exposure.

Counting Cards: High-Low The high-low counting system, used in this book for single-exposure blackjack, is also the best simple counting system for double exposure.

This chapter presents double-exposure strategy numbers for the high-low count.

Table 75 provides the strategy numbers that apply no matter what the dealer does on soft seventeen.

Table 76 is for use if the dealer stands on soft seventeen.

Table 77 is for use if the dealer hits soft seventeen.

All the calculations in tables 75-77 are based on six decks.

Double down; if you cannot double, then hit.

Double down; if you cannot double, then hit.

Double down; if you cannot double, then stand.

Dealer stands on soft seventeen.

Dealer hits soft seventeen.

For decision purposes you need a count per deck.

The method for finding it is the same as was discussed in chapter 3.

Start with zero after a shuffle.

Keep a running count, counting all cards you have seen since the shuffle.

Divide this running count by the number of decks you have not seen to get the count per deck.

You should stand, hit, or double down only if your count per deck truncated, and not rounded is equal to or greater than the number in the table.

Insurance at double exposure is the same as is explained in chapter 4, and the strategy numbers of table 11 page 53 are applicable to double exposure too.

Generally double exposure uses six decks, which means an insurance strategy number of 3.

What the numbers mean.

You might want to double down on a natural against fourteen, fifteen, or sixteen if the count is high enough.

This is true only if naturals pay only even money.

Do not double down on a natural if it pays 3:2.

Suppose you have 4-4 and have a choice of hitting, splitting, or doubling down.

Regardless of the dealer's action on soft seventeen, meaning you are using either table 76 or 77, doubling down on your total of eight is more profitable than splitting 4-4 if the dealer has six or less, or soft seventeen or less.

If the dealer has hard twelve to sixteen, then basic strategy says splitting 4-4 is more profitable than doubling down on eight.

When the count per deck gets above the numbers shown in table 75, then doubling on eight is more profitable than splitting 4-4.

DOUBLE EXPOSURE 167 Against twelve or thirteen, hit 4-4 if the count per deck is bad enough.

The count per deck must be at least -2 to justify splitting 4-4 against twelve, and it must be at least -7 to justify splitting 4-4 against thirteen.

With 6-6 against seventeen, split if the count per deck is less than +2 and hit if the count per deck is +2 or more.

With 3-3 and 2-2, splitting is always more profitable than doubling down.

If the count per deck is equal to or greater than the number in the table, then double down.

With soft eighteen against twelve through sixteen, double if you can.

For example, if you have soft eighteen and the dealer shows thirteen, double if you can.

If you cannot double, then hit if the count per deck is less than zero or stand if the count per deck is zero or more.

The numbers for soft eighteen against dealer A-A through A-5 are for when to double.

When the dealer stands on soft seventeen, you have soft eighteen, and the dealer shows A-A through A-5, if you do not double it does not matter whether you hit or stand.

So take a card on a negative count to hasten the shuftle, and stand on a positive count.

With 7-7, if you do not split then either hit or stand according to the strategy number for fourteen.

Likewise with 6-6: If you do not split, then play according to the strategy number for twelve.

Double down; if you cannot double, then hit.

Dealer hits soft seventeen.

Stand or double or split at a count per deck equal to or greater than link number; hit or do not split at a count per deck less than the number.

In it, 3, 4, 5, 6 count +1, lOis -1, and theÂ· other cards count zero.

Earlier editions of Professional Blackjack presented strategy numbers for double exposure for hiopt.

That feature has been removed from this book because simulation of it as a simple count i.

Counting Cards: Halves Simulations show that the halves counting system handily outperforms the high-low at double exposure.

Moving up to the halves count for deciding when to bet big means a bigger increase in expected win rate than learning strategy numbers.

For decision purposes you need a count per deck.

The method for finding it is the same as was discussed in chapter 3.

Start with zero after a shuffle.

Keep a running count, counting all cards you have seen since the shuft1e.

Divide this running count by the number of decks you have not seen to get the count per deck.

Insurance for the halves count at double exposure is the same as is explained in chapter 4, and the strategy numbers of table 11 page 53 are applicable to double exposure too.

Generally double exposure uses six decks, which means an insurance strategy number of 3.

For the halves count, use table 78 with either 79 or 80.

Use tables 78 and 79 if the dealer stands on soft seventeen.

Use tables 78 and 80 if the dealer hits soft seventeen.

The calculations for tables 78-80 are based on six decks.

Hit or stand decisions with soft eighteen against four to six or twelve to sixteen are insensitive to the DOUBLE EXPOSURE 175 halves count.

For all of these dealer hands basic strategy is neutral - it does not matter whether you hit or stand whatever the count.

Win Rates at Double Exposure Table 81 contains simulation results for double exposure.

The format is the same as for the rest of the win rates in this book.

The first row is benchmark win rates, and the rest of the rows are win rates that differ from the benchmark in the identified manner.

The benchmark is six decks, five dealt out; ties lose; dealers stand on soft seventeen; player can double any first two cards but not double after splitting; no resplits; naturals pay even money, and tied naturals push.

The benchmark rules are used in simulations of the highlow and the halves counts.

For simulations of deviations from the benchmarks, results are presented both as raw expected-win rates and as deviations from the appropriate benchmark win rates.

Note that varying bets with the count wins almost as fast when you use basic strategy as when you vary your strategy with the count.

That means all the strategy numbers of this chapter put together have only a modest effect on your win rate.

Double down; if you cannot double, then hit.

Double down; if you cannot double, then stand.

Hit or double; never split.

Stand or double or split at a count per deck equal to or greater than the number; hit or do not split at a count per deck less than the number.

Split if the count per deck is less than the number; else hit.

Dealer stands on soft seventeen.

These are initial bets only, and do not reflect additional amounts wagered on doubles, splits, insurance, etc.

Your expected win is proportional to your bet size and to the amount of your table time.

The more that you bet the more you will win on lucky streaks and the more you will lose on unlucky streaks.

Since you must be financially as well as emotionally prepared to handle an unlucky streak, the more that you bet the more you must take with you to the casinos and the more you must have as total capital.

He asked which book of mine he should buy to accomplish his goal.

If you want to win at a faster rate, you must: a Make bigger bets in those situations where you have an edge, or b Find a game with better rules, or c Find a game with fewer decks, or d Find a game with better penetration, or e Play more hands per hour.

Table 82 summarizes the information in most of the rectangles throughout this book.

It does not contain double-exposure win rates, and it does not contain win rates for some of the rarer rules.

It also contains many win rates that do not appear in rectangles elsewhere in this book.

Every number in table 82 was generated by simulation of at least 30 million hands of blackjack.

In many cases the sample sizes are more than 100 million.

For example, if you are playing blackjack at a casino that allows double after split and late surrender and also has its dealers hit soft seventeen, simply combine the appropriate win rates.

These are initial bets only, and do not reflect additional amounts wagered on doubles, splits, insurance, etc.

Risk The discussion of risk should not be separated from the discussion of expected win.

If you want to double your expected win you have to bet twice as much, and that means having ups and downs that are twice as large.

Those little rectangles sprinkled throughout this book have numbers for standard deviation as well as win rate.

These numbers are for playing one hand at a time.

If you make the same bet on each of two or more hands simultaneously, your risk will be higher yet.

free blackjack bodog easy way to put it all together is to relate all the numbers to your big bet.

Your hourly win rate will be 10% to 50% of your big bet.

The standard deviation applicable to an hour of play will be four to eight times your big bet.

More on risk later in this chapter.

Expected Value Expected value means the average of all possible outcomes.

To find the expected value, find all of the possible outcomes positive and negative and the probability of each outcome's occurrence.

Multiply each possible outcome by its probability and sum up the products.

Your sum is the expected value.

For example, suppose you have a natural against an ace and you have seen no other cards in a one-deck game.

What is the expected value of insurance in this situation?

If the dealer has a lOin the hole, your insurance bet wins double.

If the dealer has any other card in the hole, you lose your insurance bet.

If you insure all of your naturals, you lose eight cents on every dollar of insurance.

The above calculation is for single deck, but multiple-deck calculations yield the same loss of eight cents per dollar of insurance.

The gambler insures learn more here natural to lock in a certain winner; you should be looking at the -8% expected value.

This is a simple example of expected value.

For most blackjack decisions, calculating the probabilities is more complicated.

The definition of expected value is the same - the sum of: possible outcomes times the probabilities of those outcomes.

Thinking in Terms of Expected Win Learn to associate expected value with your bet.

Thinking in this manner will eliminate any temptation to bet large when the dealer has the advantage.

Whether you won or lost the previous hand does not alter the value of this hand.

If the expected value says that you are giving the dealer 50 cents on this hand, then you are giving away 50 cents even if you have just gotten a natural or have won three hands in a row.

The same logic applies to other casino games.

The dealer's advantage on the pass line at craps is 1.

Thinking in this manner will take all the fun out of gambling and help turn you into an investor.

If you play two hands, in total they are worth a dollar and a half.

You must fight one hand at a time, but the object is to win the war; the war is decided by expected values.

After you have mastered the strategy, a satisfying way to defme the battle is by use of a target number of hours played rather than a target number of dollars won.

To play the target number of hours is to win the battle.

Risk You can win at this game, but you will not win every session.

Results between those two numbers are what you can expect to see about two-thirds of the time.

It will not be rare to see a swing of two standard deviations, e.

Suppose you play blackjack with the benchmark betting scheme for 40 hours.

What is the standard deviation that describes your ups and downs?

The answer is easy to find: Simply multiply the one-hour standard deviation by the square root of the number of 190 PROFESSIONAL BLACKJACK hours you play.

For the expected win, multiply the number of hours times the expected win per hour.

Finding Standard Deviation Statisticians use standard deviation and variance to describe the dispersion of possible outcomes around the expected value.

One way to find the https://partysutra.com/blackjack/double-down-in-blackjack-crossword.html is to subtract the expected value from each possible outcome, square these differences, multiply each square, in tum, by its probability of occurring, and sum.

The sum is called the variance, and the square root of the variance is called the standard deviation.

Recall the insurance example, in which the insurance bet is lost if the dealer's hole card is other than 10, and double money is won if the hole card is 10.

The expected value was found to be -8%.

The standard deviation for this example is calculated as follows.

From each possible outcome, the expected value is subtracted.

Then they are multiplied by their probabilities.

Then those products are summed.

The variance is found to be 1.

The standard deviation is the square root of the variance, or 1.

Over a large number of decisions of this type, say, more than 192 PROFESSIONAL BLACKJACK a couple of hundred, the distribution of possible outcomes approaches the normal.

Normal Curve The normal distribution is useful.

It is symmetric and has a standard shape.

You can tell from the normal distribution as shown in table 83 how likely a given outcome is if you have an expected value and a standard deviation.

The expected value plus or minus one standard deviation contains the actual outcome about 68% of the time.

The actual outcome will be worse than one standard deviation below the expected value 16% of the time, and the actual outcome will be better than one standard deviation above the expected value 16% of the time.

Here is a sample use of the normal curve.

Table 83 says that the area in the tail beyond 1.

Living With Risk When deciding how much to bet, pay special attention to the standard deviation because it quantifies the FINER POINTS 193 risk you will be taking.

You must be able to sleep with the risk.

You will have streaks where you lose 20, 30, or 40 big bets over the course of several days of playing blackjack.

If you do not ride easy with such unwholesome and sticky burrs, you will have some grisly dreams.

You must build confidence in the system and in your ability.

Getting a click here dollars ahead will help.

You must get accustomed to thinking of money as a means of keeping score rather than for what it will buy.

Think about buying later.

Compartmentalize such thinking away from professional matters.

If your only impression upon reading these dollar figures was that I was two automobiles ahead after three days and on the final day dropped one automobile to finish one automobile ahead, you are not yet ready to bet big yourself.

If you thought of the nwnber of bets instead of the purchasing power and you have the required capital, then go ahead and bet big.

Bankroll Advice First, decide how much money you are willing to risk at blackjack.

By that I mean the amount that if you lost it all you would give up the game.

That is your total stake for the purposes of this discussion.

The bet size that is mathematically optimal is discussed later in this chapter.

Personally, I prefer to bet 194 PROFESSIONAL BLACKJACK more conservatively than can be justified by mathematics.

This paragraph presents the approach to bet sizing I use myself.

I prefer to relate the biggest bet I am willing to make to my total stake.

For a short trip to casino country, a day or a weekend, a sufficient bankroll is 50 times whatever you select as your big bet.

For an extended blackjack trip, I recommend taking at least 100 times whatever you select as your big bet.

Doubling your bet size doubles your expected win, but also doubles risk and doubles capital requirements.

You can only increase your expected win without increasing capital requirements by playing more hours.

Playing two weekends a month will yield twice the expected win and twice the expenses on the same trip cash and the same total stake.

If you want to make a lot of money playing blackjack, you are going to have to play a lot and bet a lot.

You must be able to lose a lot of money.

You do not want to lose, but you will lose frequently.

You will win frequently too, and over the long haul, you are certain to win more than you lose, but the long haul may be a long time in coming.

You simply must be secure enough financially and emotionally to handle losses that will occasionally amount to the FINER POINTS 195 trip-cash numbers and may, in a very bad streak, add up to your total stake.

The smaller your big bet in relation to your bankroll, the lower the probability that you will be wiped out.

Your blackjack bankroll behaves much like a portfolio of stocks.

A mathematician would describe it as following a random walk with an upward drift.

Your possible outcomes from investing in blackjack are normally distributed.

Investors in the stock market do not have any assurance that their outcomes are normally distributed; their possible outcomes appear to come from a distribution that has a greater chance of extremely good or extremely poor outcomes.

This is a rewrite of an article I wrote for the March 1990 issue of Blackjack Forum.

The rules used are benchmark plus double after splits and no resplits.

Other sets of rules would yield similar results.

There was one player at the table, who used the high-low with strategy numbers from -1 to +6.

Whenever the count thanks. blackjack billy blood sweat and beer topic negative, the player switched tables.

Different bet schemes would yield different results.

The study was done for both six and eight decks.

With six decks, the shuflle came with a deck and a half left.

With eight decks, the shuffie came with two decks left.

Then the bankroll was brought back to its initial position, another 1000 hands were played, and another lowest point was registered.

This procedure was repeated until 20,000 lowest points had been tallied.

This was done for both six and eight decks.

Then the whole procedure was repeated, but with 4000 hands played before the lowest point is recorded and the bankroll was reset.

Thus a total of 200 million hands of blackjack were played for this study.

The results are shown in table 84.

Each row shows the probability of being down by a given amount or more at the lowest point of the session.

For example, after playing 1000 hands at eight decks, 19.

The data in table 84 are simulation results; each true frequency could be slightly higher or lower.

The standard errors on the numbers in table 84 range from almost zero to 0.

For example, the standard error on the 19.

Having an edge means you will be ahead for certain if you play long enough, but it does not guarantee a win in the short run.

Comparison of the four columns of table 84 shows that there is more risk in playing 4000 hands than in playing 1000.

A surprising result is that risk seems to depend almost entirely on the number of hands played, and almost blackbelt in blackjack ebook at all on whether the play was against six or eight decks.

There was slightly more risk playing against six decks than playing against eight.

Big losses come from being unlucky on big bets.

The expected win is greater for six decks, one and a half cut off than for eight decks, two cut off; that greater profit opportunity greatly outweighs the increased risk.

Likewise, though there is more risk the longer you play, there is more expected win the more hands you play.

Higher bets are accompanied by more risk, but also of course by more expected win.

Table 84 is based on one person at the table, playing one hand at a time.

More people at the table probably would not change the risk much, but playing more than one hand at a time definitely would result in more risk.

Table 84 says that for 1000 hands, a bankroll of 50 big bets has a risk of ruin of about half a percent.

For a play of 4000 hands, a starting bankroll of 90 big bets is required to reduce the risk of ruin to less than half a percent.

Those numbers assume no cutting back in bet size in response to losses.

The odds are excellent that sometime in the near future your playing bankroll will be less than it is at this instant.

For example, table 84 shows that there is a 99% chance that at some point in the next 1000 hands your bankroll will be less than it is right now.

How far is your bankroll likely to fall if your trip is sort of average?

Remember that these are worst-point intermediate results, and not end-of-trip numbers.

If you want, you can compare the nadirs with the final distribution of results as described by the normal curve.

Comments On Risk Peter Giles says: The late Ken Uston was once quoted in the Review Journal as saying, "It's really tough to make a living at blackjack.

The fluctuations will really wipe out the average guy.

If I had to play by myself instead of on a teamI probably wouldn't be in it now.

I have chosen to not use an optimal betting system.

I would rather start with enough units to have a very low risk factor and keep the bets the same.

I am tired of having to make up losses by winning more bets than I lost.

I know the reverse is also true, but it still has a negative effect on players on a team.

The players would show up every day in a bad mood, and it may well have affected their games.

Cutting their bet sizes meant they had to win twice as many bets to get back to even in dollars.

I am still trying to determine how many units one is safe with.

What is generally recommended in books is, in my opinion, too risky.

This is one area in which it is hard to trust mathematics.

Donald Schlesinger says: I have the utmost respect for both Ken Uston and Peter Giles, but their pessimistic outlook on the individual card counter's plight is, in my opinion, too harsh.

Precise and careful money management protects the average guy from the wipeout that Kenny describes.

The game can be beaten and fluctuations can be overcome.

Your readers should take heart, keep the faith, and practice, practice, practice.

Marvin French says: I am back from Reno after a satisfying trip.

I enjoy the game much more when my bankroll goes steadily upward during the trip rather than ending high but with huge swings getting there.

Chance of Reaching All-Time High Unfortunately, most of the time you seem to be losing.

According to mathematician Peter Griffin, at most 1.

This can be tough on your nerves.

If you compare your present position with your all-time high, the feeling that you are losing will haunt you during 98.

One way to avoid getting depressed is to forget about your all-time high.

Chances of moving up are always greater than chances of moving down, but downward movements will continue to occur.

Your trend is still up.

You must hit a new high sometime if you keep playing well.

FINER POINTS 201 Avoid Going Broke: Use Proportional Betting By now you have probably won several hundred dollars betting small and are wondering whether you should increase your bet size.

Perhaps a proportional betting scheme has occurred to you; that is, you are considering betting a fixed proportion of your capital on each hand.

Gambling With an Edge guest Arnold Snyder #4

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