Card Counting Chart

Blackjack Card Counting Studies

  1. The basis of any card counting system is playing perfect basic strategy, since making simple mistakes will undermine whatever advantage you hope to gain through counting. Even if you have to use a basic strategy chart to help you at the table to help you, always remember that you need to master the fundamentals before moving on to more advanced.
  2. Card Counting for the Dragon 7. The Dragon 7 is a side bet that pays 40:1 if the banker hand wins with three cards and a total score of seven. Typically, it has an average house advantage of about 7.6 percent, so we’re not always going to take the bet.

Card counting is a casino card game strategy used primarily in the blackjack family of casino games to determine whether the next hand is likely to give a probable advantage to the player or to the dealer. Card counters are a class of advantage players, who attempt to reverse the inherent casino house edge by keeping a running tally of all high and low valued cards seen by the player.

This page contains links to and explanations of severalcharts which have been created with CVData and provide illustrationsof many Blackjack principles. Note: Some of these studiesare quite advanced. You do not need to understand these chartsto count cards. Below is a quick index of sections describingthe charts. Each section has one or more links to the chartimages.

Ease of Use vs.Efficiencies of Various Strategies

In an attempt to visually illustrate the differences inease of use and efficiencies between strategies, I’ve createda 3D Scatter Chart. The chart consistsof 14 balloons suspended above an x-z grid. The x-axis isBetting Correlation. The z-axis is playing efficiency. Thestring from each balloon intersects the grid at the BC andPE for that strategy. The height of the balloon (y-axis) isthe ease of use of the strategy. Thereby, each balloon indicatesall three variables. The ideal system (impossible to obtain)would be at the top, right, back. Note, the two strategiesin the center (Omega II and Uston APC), are very high PE,Ace-Neutral strategies. If Ace side counts are kept for thesestrategies, they would move substantially to the right placingthem closer to the ideal combination of efficiencies. However,they drop in height as they become more difficult to use.Be careful of the parallax problem. Balloons closer to thefront appear not to be as high as they are.

Advantage andUnits Won/Lost vs. True Count

The count is better for you at extremely high counts withtwo esoteric exceptions (described at the end.) I’ve attacheda combination chart which shows Advantageand Units Won/Lost vs. True Count. The green area shows thelosses at negative values and gains at positive values. Ofcourse, the big gains and losses are at relatively low plusor minus counts, because this is where the majority of handsexist. The red line shows advantage. It is very smooth forthe majority of counts, but goes wild at the very high andlow counts. This is despite the fact that this is a simulationof one billion hands and the data has been smoothed (witha quadratic B-spline algorithm.) Problem is, there just aren’tthat many hands at the extreme counts and the variance isobscene. Of course, if you play long enough, you will experiencea few wild counts. Your results at those counts are essentiallyrandom. Unfortunately, the human mind is more likely to remembersuch events, even though they have no meaning. This is whypeople watch X-files and other silly TV shows.

Esoterica

  • 1. If you are playing single deck, and you and allother players play without any variation whatever, thencertain wild TC’s will only occur with certain dealt cardsequences. This will result in automatic wins or lossesat specific extremely high or low counts. The odds of runninginto this situation are approximately zero.
  • 2. If you are side-counting Aces and there are noneleft, you’ve got a problem with a high count.

Advantage by Typeof Hand

Counting

I've been experimenting with topology maps in an attemptto better show statistics by type of hand. The attached AdvantageSurface Chart shows advantage for the various first twocards. X-axis is type of hand (all hard hands, soft handsand pairs). Z-axis is dealer up card. Y-axis is eventual advantagegiven six deck, Hi-Lo, 1-8 spread.

Time Spent inAdvantage Situations Balanced vs. Unb.

Comparing the percentage of time that two systems indicatespecified advantages is problematic because the counts arenot continuous. Different systems result in different levelsof advantage percentage. However, I took a shot at it:

The first chartdisplays the time spent at certain advantages for K-O andHi-Lo.

Chart

The second chartshows the advantage at each count (Running Count forK-O and True Count for Hi-Lo.)

The third chartshows the frequency of hands at each count. Here, the redand green are charted using a logarithmic scale. The blueribbon in the back is the same data as the green ribbon plottedwith a standard scale. It is there to show why I had to usea logarithmic scale and to show the huge number of hands ata TC of zero in a system that truncates instead of rounds.

Cost of Errors

The Cost of Errors Chart is anattempt to show the cost of various types of errors. Fivecolumns are provided using data from five multi-deck, multi-playersims. The height of the columns represents advantage.

  • Column 1 displays the effect of playing errors. Thetotal height of the column represents the advantage withno errors. The green segment indicates the penalty resultingfrom one error per hundred hands. The blue section is theadditional penalty of another error per hundred. And onthrough five errors. The red pedestal is the advantage witha 5% error rate. The errors are serious; but not idiotic.Insure is reversed, surrender is reversed, split or doubleis changed to hit, hit and stand are reversed. But, a hard18 up is never hit, an eleven down is never stood, and adouble or split is never taken when it shouldn't.
  • Column 2 displays the effect of betting errors fora non-cover bettor. Again, succeeding circular slices ofthe bar show the effect of errors. The player is spreading1 to 8. Errors are: if should be a one bet, bet two; ifshould be a two, bet three; if should be a three througheight, but two.
  • Column 3 displays the effect of miss-estimating theremaining cards for TC calculation. The green shows theeffect of a 10% error and the blue shows the additionaleffect of an additional 10% error.
  • Column 4 shows the effect of using no indexes at all.The green section is the penalty when using perfect BS vs.-10 to +10 indexes. The count is still used for betting purposes.
  • Column 5 is the effect of errors for a cover-bettor.The betting is very conservative not allowing large increasesor decreases, no increases after losses, no decreases afterwins and no change after pushes. The error scenario is toocomplex to explain here. (OK, I’m too lazy.)

I did not include a column for flat betting, because therewouldn’t be one. You’d lose all of your advantage.

Double Diamond Blackjack

A question was raised as to the advantage of a new gamecalled Double Diamond Blackjack. This game pays extra on aDiamond BJ, but less on other BJ's. Also, several other fancyrules are added. The problem with such a game is the hugepenalty of reducing the BJ payoff. I've created a SurfaceArea Chart which displays the difference in winnings betweena normal single deck game and a game like Double Diamond (DD).The DD game I used was 6 card charlie, 5 card 2:1, DiamondBJ pays 2:1, Normal BJ pays even, but is automatic win, Doubleon any number of cards even after splits. The chart has alltwo card hand types on the x-axis, dealer upcard on the y-axisand difference in winnings on the z-axis. The z-axis is winningson the Diamond game minus winnings on a normal game.

Looking at the chart, the games are equal where the blueand green meet. Green is a slight advantage for DD, red isa serious disadvantage for DD. The green/blue splotchinessis due to the small number of hands run (160,000,000). Itindicates that the variance at that number of hands is actuallygreater than the difference in results between the two games.The solid green with upcard combinations totalling 5, 6, 7and 8 indicates a very slight gain in using the DD rules.This is due to the gain from double on any number of cards,6 card charlie, and 5 card 21. The huge slice through thestack shows the loss due to most BJ's paying even money. Thisis somewhat less at BJ vs. Ace because of the BJ automaticwin rule.

The point of the chart is to show the enormous penaltyof the BJ rule change versus the very slight gains by theoddball rules.

Components of Advantage

This chart provides two rows of StackedBars. There are 14 pairs of bars representing the advantagesthat can be gained using various strategies according to Griffincalculation techniques. The y-axis (advantage) is not quantifiedas it is relative. The rectangular columns in the back rowindicate the relative gains when playing multi-deck. The darkgreen signifies gain from betting and the red indicates gainfrom using indexes. A 1-8 spread is assumed. The circularcolumns in the front row indicate the relative gains whenplaying single deck. A 1-2 spread is assumed. Here, threecomponents are displayed. Again, betting and playing gainare shown. The, additional, blue segment indicates the gainfrom playing SD vs. MD.

So, what does this chart illustrate? Nothing new; buta few concepts that should be kept in mind:

  • Playing gain is equal to or more important than bettinggain in SD as opposed to MD where betting gain is substantiallymore important. However, both are important in MD.
  • Spread can make up for the loss in MD advantage, orfor the pessimist, spread is necessary to make up for theloss in MD advantage.
  • The differences between systems are dwarfed by thedifference in spread. That is, we spend altogether too muchtime thinking and debating about which system is best andnot enough time talking about how to maximize the spreadwithout getting tossed. This is the simple point of thechart.

Disclaimers: No simulations were run. Results are calculatedfrom Griffin formulae. Side counts, number of indexes andcover plays are ignored. PE calculation is questionable forunbalanced counts.

First Base Penalty

For some time, we have been aware that it is better tosit at third base in single deck, face down games. Commonsense tells us that we get to see more cards and can makebetter playing decisions. In an extreme case (seven players),the advantage difference between seats 6 and 7 is about 0.05%.You lose another .05% per seat as you move toward first base.However, the difference in advantages between first and secondseat is much worse. First seat can be as much as .16% worsethan second seat. As this is a severe penalty, I decided totake a look. First, I looked at the winnings by true count.I created a chart which shows the winnings for first seatand second seat by true count. [link]The chart shows that the winnings are identical for all countsbelow 4. But, at a TC of 4, second seat does better. At 5-8,better and better. After that,. It evens out. OK, we now knowthat there is something about these particular counts thatwe should examine. I then decided to look at hand types. Itook the winnings for the second seat and broke them up intoan array of all possible two card hands vs. dealer up cards.This is an array of 330 values. I also created the same arrayfor the second seat. I subtracted the second array from thefirst array and charted the remainders. The result is a combinationsurface area/contour chart that indicates the hands wherethe first seat has a problem. [link]Eureka! First seat has a serious problem with two tens againsta 3, 4, 5 and 6. Tens vs. 6 is particularly severe. All otherhand results are about the same. Common thinking would haveexpected many differences along the lines of the Illustrious18.

So, we have a problem with 10’s against 3-6 at TC’s of4-8. Guess what, the indexes for splitting tens at 3-6 are4-8 (Hi-Lo.) So, why is there a major problem with splittingtens in seat one? Well, if you think about it, there is aquirk in seat one. Remember, we are playing SD, face down,seven seats. That means, two rounds. Only round two is importantas that is where you are betting. To split tens, you musthave two tens and the dealer must have a low card. If youare sitting at seat one, the only cards that you can see afterthe start of the round are two high cards and one low card.This means that the playing count will now be the count atthe start of the round minus 1. If the round starts at a TCof +3, any seat has the possibility of splitting tens againsta 3. That is, any seat except for seat one. Seat one cannotbecause the count will always be one less than the count atthe start of the round or +2. 9% of the time, you will startround two at a true count of +3. 2.74% of the time, you willstart at a true count of +4. This means that 6.26% of thetime, every player has the possibility of splitting tens againsta 6 in the second round, except for the player in seat one.(26% of all gain is in round two at a starting TC of +3 inthis example.) The same holds for the other ten split opportunities,at reduced percentages. Therefore, seat one, and only seatone, has an automatic reduction in opportunity.

By the way, if you go through the same process betweenother seat pairs, you get the charts that you would expect.That is, the tens peak is muted and the other Illustrious18 decisions start to poke out from the plane.

I don’t consider this analysis complete and welcome comment.

Card Counting Chart

Exact vs. EstimatedTC Calculation

This section summarizes sims of nine billion hands withvarious methods of desk estimation. With the parameters thatI used, TC calculation using exact (to the card) deckdepth gave a .829% advantage and $17.29 win rate. When estimatingthe number of decks, generally, the worse the method of estimation,the lower your advantage, but the higher your win rate. Thisis due to overbetting. To show where this overbetting occurs,I chose a common method of deck estimation (287-312 cards=6decks, 235-286=5 decks, etc.) and compared it to exact depth.Advantage is .810% and win rate $17.32 (very slightly higherthan using exact remaining cards.) I created a chart showingthe average bet on the Y-axis and deck depth on the X-axis.In general, average bet increases as deck depth increasesbecause there are more high TC's. The average bet increasessmoothly when TC calculation is performed with exactremaining cards. However, the increase is lumpy when the remainingdecks are estimated. If you look at the chart (link is below)you will see how the sloppy estimate shows lumps of higherbetting. The lumps increase in volume as deck depth increasesbecause of the higher percentage of large TC's. These lumpsin the graph signify the areas of overbetting. The area ofthe largest lump is the area of highest risk.

Conclusions

The better your deck esitmation the smoother and moreaccurate your betting, improving exposure to risk but notincome.

Effect of a Back-Counteron your Play

Awhile back, I commented that I’d leave a table if I thoughtit was being stalked by a back-counter. Thought I’d sim theeffect. Ran two sims. First sim had three players. BS playersin seats one and two and a Hi-Lo player in seat three. Weare interested in seat three. Second sim was the same, buta fourth player Wonged in at a TC of +4 and left at the endof the shoe. Again, we are interested in seat three. Six decks,five deck penetration. Each player played 150 million handsexcept the back-counter who played 13 million. The attachedribbon chart (link below) graphs the winnings by TC for theHi-Lo player in each sim plus the back-counter. You will notethat the red ribbon (seat 3 in the second sim) and the greenribbon (seat 3 in the first sim) run evenly through the negativeTC’s. At about +3, the green player pulls ahead. That is,the Hi-Lo player at the table with the back-counter won lessmoney on positive counts. Overall, he lost about 0.15% advantage.

FIRST CHART- Winnings by TC.

OK, where is the lost advantage? The second chart hastwo series. The green series is the percentage of hands playedby seat three at the back-counter’s table of the hands playedby seat three at the back-counter-free table. The chart showsthat both seat three players played the same number of handsat negative TC’s, but at positive TC’s, the player disturbedby the back-counter played only 80% as many hands. This isdue to the back-counter eating cards in positive TC conditions.So far, no surprise. However, there is another effect. Thered series on this chart shows dollars bet instead of handsplayed. Again, the players at both tables bet the same perTC at negative TC’s. But, at positive TC’s the drop-off inunits bet is more severe than the drop off in hands played.Only 75% as many units are bet at high TC’s. That is, theaverage bet was lower at high TC’s. Why is this? Well, theHi-Lo player was using camouflage play. The spread was 1-8on both tables, but the player would never make large hand-to-handbet increases. Since the back-counter’s interference tendedto reduce the length of high TC consecutive hands, and reducedthe number of hands dealt per shoe in favorable situations,the Hi-Lo player had fewer opportunities to win enough handsin a row to pump his bet up to the optimum level.

CHART TWO- Hands played and Units bet by TC

This shows an important point about running a sim exactlyas you would play. It is not enough to show a simple 1-8 spreadsince realistic cover play may interact negatively with othercharacteristics of the sim.

Note: When just looking at the overall advantage, 150million hands is OK. But, when you break this down into smallergroups of hands (e.g. by TC), then you have fewer hands persituation and need more total hands to give good results.However, there is a short-cut that was used here. All lineswere smoothed with a 12 facet cubic B-spline formula. Thistakes information about neighboring data points (nearer pointscount more than farther points) and adjusts all points toproduce a smoother graph. This requires several hundred millioncalculations, but that’s only seconds on a Pentium. If youare looking for exact data, this is not valid. But, if youare looking at trends, it is quite accurate and fast. To performthis on a CVSIM chart, double-click on a series (e.g. groupof bars, a line, an area). The Format Series dialog box willappear. Click on the Options tab. Then, select a Smoothingformula at the bottom left. Click on Help to get informationon the options.

Advantage at VeryLow TC's

If you are using a huge number of indices, then your disadvantageat very low counts is slight. You have the ability to alteryour play which makes up for part of your disadvantage. However,these days, few people bother with the negative indices. Ifyou are using the Illustrious 18, then your advantage at verylow TC’s drops precipitously. The attachedchart shows advantage by TC for two players at the sametable. One uses the Ill. 18 and the other uses a full setof indices. Advantage at TC’s below -14 barely changes forthe full index player. Advantage for positive TC’s continuesto grow for both players. Does this mean that you should usea full set of indices? No, very little money is bet at thosevery low TC’s.

Sim particulars: Single deck, three players, 1.6 billionhands per player, four rounds per shuffle, SE at TC -30 was.14. AO II was used as it has an excellent set of SD indices.

Cut Card Effect

Thought I’d put together some charts to illustrate theCut-Card Effect. I created four charts from 2.6 billion single-deck,basic strategy hands. About half of the hands were fixed ateight rounds per deck and the other half dealt to a 75% penetration(6 to 9 rounds.) The first simplechart shows the advantage by hand depth. The red barsshow a even 0.2% advantage for the casino for all hand depthswhen dealing a fixed number of rounds. The green bars showthe enormous increase in the casino’s advantage in the laterounds when dealing with a cut card. The advantage is so great,that I had to use a logarithmic scale (0.2% to 14%). Fortunately,there are not many hands dealt at the 14% casino advantage.

Blackjack Card Counting Chart

The following three charts each show hand dealt quantities.Each chart has as it’s x-axis, all possible first two cardplayer combinations. The y-axis shows the dealer up-card.The z-axis shows the number of incidents of each of the firsttwo player cards vs. dealer up card..

Chart I: Thefirst chart shows the normal distribution of hand types. Thatis, the number of times that you will receive each of thepossible first two cards against each dealer up card.

Chart II:The second chart shows the distribution of hand types in thelast rounds when playing with a cut card. In this chart, thereexist more low cards since it is much more likely that youwill see additional rounds when large cards are dealt in theearlier rounds.

Chart III:This is essentially the difference between the two previouscharts. It shows the delta between the normal distribution of hands and the distribution of hands in the late roundswhen using a cut card. This is a surface area chart with aprojection of the colors to the base to more easily see theproblem areas. Red and orange areas show the types of handsmore likely to be seen in the late rounds. The chart showsa substantial increase in stiffs, particularly against dealerlow cards. Also, more low hands (5-12) against a dealer ten.There is a corresponding decrease in BJ’s, twenties, and 17-19hands against good dealer up cards.

I also have an oldchart which shows the advantage at each of the above handtypes. It can be seen that most of the hands where we haveseen increases due to the cut-card effect are poor advantagehands.

Of course, all that I’ve shown with all of the above iswhat was already known. The cut card adds hands when the deckis lean in tens. So, does this mean that you should avoidSD dealt to a fixed penetration. Yes, if you’re playing BS.But, if you’re counting, it’s not so clear. I’ve just startedworking on those charts, and it appears that counting overcomesthe effect even in the late rounds. At least at the depthsat which I’m currently testing.

The Effect of Numberof Players with Cover Betting

Normally, the number of players at a table has no effecton your advantage. However, when cover betting, this can change.I ran a total of five billion hands with cover betting asfollows:

  • No increase in bet after loss
  • No decrease after win
  • No bet change after push
  • Max increase or decrease two units
  • No cover plays
  • 1-8 Spread (1, 2, 4, 6, 8 at TC's of 1, 2, 3, 4, 5)
  • I allowed bet reset to one unit at shuffle as not resettingwould clearly hurt a full table player.
  • Five/six deck, strip rules

Advantages:

  • 1 player: 0.60%
  • 4 players: 0.45%
  • 7 players: 0.33%

I created a Bet Size by TC Chartfor the three players. X-axis is TC, y-axis is average amountbet (including double downs.) The red bars show the rapidincrease in average bet size for the head-on player. It nearlymatches the ideal. It drifts off very slightly at very highTC's because there are slightly fewer DD's at high TC's. Thegreen and blue bars show the players' at fuller tables muchslower and smoother increase in average bet size as they havemore difficulty raising there bets quickly as high TC's occurat lower hand depths. It also shows them overbetting at +1and +2 as they couldn't lower bets as quickly as desired.

I tried small sims with various number of players andno cover. There is no difference without cover. Also, theeffect of cover when playing head-on is negligible. I alsotried softening the cover by allowing a doubling or halvingof the bet and allowing bet increases after a lost split ordouble and bet decreases after a won split or double. Didn'tappear to change the results much, but I need to make moreruns in that area.

The Effect of Coveron Advantage by Penetration

Card

I put together an Effect of Coverchart to give some idea of the cost of various amounts ofcover betting. The results are from one half-billion roundsim. There were four players as follows:

Yellow: No cover Blue: No bet increases after a loss,no decreases after a win; but reset to one unit after a shuffleGreen: Same as above but also no bet change after a push andno jumping bets up or down by more than two units. Red: Sameas above but bet not reset to one after shuffle and InsuranceCover. (index of 4 for a BJ, 3 for a twenty and 2 for otherhands.)

All players had a spread of 1-8. A two unit bet was allowedat TC of +1 Which is earlier than in BJ Attack's sims as theheavy cover player probably wouldn't have a chance with slowerramping. The y-axis is advantage. X-axis is penetration from1% to 84%. Six decks, S17, DAS. TC accuracy was half-deck.All players played in all seats.

Note: The Red player had a disadvantage of .7% in thefirst hand. This is because he was not allowed to reset hisbet after a shuffle. The other players all had .38% disadvantageof the first hand. (Which was fortunate as that's what mycalculator says the BS advantage should be.)

I've also included a Percentage ChartThis chart shows what percentage of the total loss due tocover can be attributed to each type of cover, by penetrationlevel used by the Red (heavy cover) player. Red is the lossdue to Insurance cover and not resetting your bet after ashuffle. Green is the loss due to no jumping bets or changinga bet after a pass. Blue is the loss due to no increases aftera loss or decreases after a win. The Red area shows the largeeffect of not resetting the after shuffle bet for low penetrationgames. The Green area shows the effect of not being able tojump bets quickly at high penetration levels.

No surprises here. Cover is expensive.

Ameliorating thecost of cover

Given the high cost of cover play, I thought I'd lookat one way of softening the blow somewhat. I ran five billionhands with three types of players as follows:

  • Red Players: No cover at all.
  • Blue Players: Never more than double or halve bet.No change after push. Except reset bet to one unit aftershuffle.
  • Green Players: Same as above, but allow a bet increaseafter a Split or Double Down which lost or pushed.

The point of the sim is to see the gain from this onemodification to cover play. The logic behind the modificationis that after pushing a split or DD, you already have doublethe bet out. After losing double your money; it isn't unnaturalto bet the amount that you lost.

Results (Initial Bet Advantageand Win Rates):

  • No Cover - 0.937%, $8.70/hr
  • Full Cover - 0.555%, $4.15/hr
  • Mod Cover - 0.643%, $5.10/hr

The gain in advantage from the change was .09% or about23% of the cost of cover.

Chart - I've attached a WinRate by Hand Depth Chart. The x-axis is the Hand Depth.Y-axis is the cumulative Win Rate for hands up to the HandDepth and z-axis is the type of player.

Follow-up - These resultsbeg a question. Most players do not bother with soft doubleindexes as it has been shown that the gain in advantage isminor. However, soft doubles may be more useful with coverwhen using this modification. The point is to increase theexcuses to get more money on the table in positive situationswithout looking like you're jumping your bets. Of course youhave to decide whether making unusual soft doubles makes youlook more or less like a counter. I don't expect much gainhere.

Sim details - Six decks,five deck penetration, S17, DAS, six players, Hi-Lo, 1-8 spread,quick ramping (two units at +2). With slower ramping, theeffects would probably be greater than shown here.

Win Rate vs. Penetrationvs. Hands/Hour

The question was, if you a game has less penetration,but is faster, will I make as much money. The game was doubledeck, H17, DAS. For this I ran 26 sims (actually one CVCXsim) for penetrations from 50% to 75% in increments of onecard. The chart shows the win rate for each penetration. Fivepoints are displayed for 100, 125, 150, 175 and 200 handsper hour. This makes it easy to compare different penetrationsat different speeds. Note, the unit size and betting rampare different for each penetration as they are calculatedfor maximum bankroll growth. See the chart here: WinRate vs. Penetration vs. Hands/Hour

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