Effect of Removing Cards
From the Deck
Before Beat the Dealer was published in 1962, professional players wereworking out counting systems. They knew they had an advantagewhen the remaining deck was rich in ten cards and Aces. They just didn’t know what the advantage was. Dr. Thorp was, through the useof high speed computers, able to quantify the effects of removal of anycard from a deck of cards. For example, when all of the fives wereremoved from a deck, the player had a 3.58% advantage. He alsofound the player had an advantage of 1.89% when four tens wereadded to a deck. He did calculations for all of the cards. This becamethe basis for the first two counting systems he developed - the FiveCount and the Ten Count.
Later, the effects of removing individual cards was studied, and thefollowing table was developed:
Card 2 3 4 5 6 7 8 9 10 A
Effect .38 .44 .55 .69 .46 .28 0.0 -.18 -.51 -.61
The numbers are the percent advantage to the player when one of eachdenomination is removed from a deck. If a five card is removed fromthe deck, the effect is +0.69% player advantage. In other words, theremoval is positive for the player. Conversely, remove a ten value cardand the effect is -0.51%. The removal has a negative effect on theplayer. If you remove an eight, there is neither a positive or negativeeffect. The eight is the only card that is completely neutral. It willalmost always be assigned a value of 0 in the count.
Next, turn your attention to the nine. The value of the nine is -0.18%. This is less than one half the value of the ten value card, and less thanone third of the value of the Ace. This means the nine is almostneutral, so if it were assigned a value of 0, there would be only a smallloss in accuracy.
Of the smaller cards, they all have a significant value, even though thereare differences in the effects of removal. The one with the least effect isthe seven card which has an effect less than half that of the five. If wewere creating a counting system and assigned the five a value of plusone, then it would be mathematically correct to assign the seven a valueof +0.4. Since this would be unmanageable in a counting system, weshould probably round down and assign it a value of 0. In doing so,the +0.28 value of the seven is partially offset by the -0.18 value of thenine.
In our developing card counting system there are 16 ten value cardsand four Aces with values of -0.51 and -0.61, respectively. Since theyare close in value, I am assigning them a value of minus one for thedeveloping system. Since a counting system needs a plus one value forevery minus one value to be in balance, we need 20 plus one valuecards. I’ll get them by assigning the rest of the cards plus one. So thevalue of the 2s, 3s, 4s, 5s, and 6s will be plus one. Now let’s constructa table so we can see what this evolving system looks like.
Card 2 3 4 5 6 7 8 9 10 A
Effect .38 .44 .55 .69 .46 .28 0.0 -.18 -.51 -.61
Value +1 +1 +1 +1 +1 0 0 0 -1 -1
What has evolved is the plus/minus system, also known as HI-LO, orPoint Count. You should note that not all cards are valued equally. This is the price we pay to keep the count simple. This is a Level One(some say one level) count. It is easier to add and subtract ones thanto add and subtract ones and twos (Level Two), or halves, or ones,twos and threes (Level Three System). With increased variation in theassigned values, there can be a more accurate reflection of the trueeffect of removal. Unfortunately, with this increased accuracy andcomplexity, errors in using them may offset any gains. This may leadus to an easy to use, simple Level One system. There is eventually aline we cross where the gains are minuscule for the extra effort in use. This is a major consideration you will have when you later choose asystem from the ones presented.
Earlier I mentioned the term balanced. A balanced system is one whereall of the card values add up to 0. In other words, there are the samenumber of plus value cards as minus value cards. A count would beginwith 0 and when the deck(s) is/are counted, the final count is 0. Since0 = 0, the system is balanced. An unbalanced system is one which hasmore of one value than the other. If there are four extra +1 cardscounted, say the 7s, and you start counting at 0, your final countwould be +4. This accounts for the extra four +1 cards in the countingsystem. Likewise, if you begin the count at -4, and count one deck,your final count would end up at 0. Since -4 does not equal 0, this isan unbalanced count. Later you will be presented with severalunbalanced systems. They will be discussed in detail then.
Lastly, there are side counts for some systems. The Aces are oftencounted alongside the main count. The purpose is to be able to adjustthe main count up or down to compensate for an excess or deficit ofAces. Aces dilute the tens in a plus/minus count. When a decisionregarding whether to insure or not needs to be made, ten cards are theonly ones that matter, so an adjustment may be made for the Aces. Ifthe tens are counted with a side count of Aces, one uses the side countto adjust up or down the count for betting decisions. This addsanother count to make when you already have a lot of things to do. More gifted individuals might try doing a side count to improve theperformance of a system. One can get both a high level betting systemand improve the insurance/playing performance of a system by adding aside count of Aces.
