Effect of Removing Cards
From the Deck
Before Beat the Dealer was published in 1962, professional players were
working out counting systems. They knew they had an advantage
when the remaining deck was rich in ten cards and Aces. They just
didn’t know what the advantage was. Dr. Thorp was, through the use
of high speed computers, able to quantify the effects of removal of any
card from a deck of cards. For example, when all of the fives were
removed from a deck, the player had a 3.58% advantage. He also
found the player had an advantage of 1.89% when four tens were
added to a deck. He did calculations for all of the cards. This became
the basis for the first two counting systems he developed - the Five
Count and the Ten Count.
Later, the effects of removing individual cards was studied, and the
following 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 each
denomination is removed from a deck. If a five card is removed from
the deck, the effect is +0.69% player advantage. In other words, the
removal is positive for the player. Conversely, remove a ten value card
and the effect is -0.51%. The removal has a negative effect on the
player. If you remove an eight, there is neither a positive or negative
effect. The eight is the only card that is completely neutral. It will
almost 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 than
one third of the value of the Ace. This means the nine is almost
neutral, so if it were assigned a value of 0, there would be only a small
loss in accuracy.
Of the smaller cards, they all have a significant value, even though there
are differences in the effects of removal. The one with the least effect is
the seven card which has an effect less than half that of the five. If we
were creating a counting system and assigned the five a value of plus
one, then it would be mathematically correct to assign the seven a value
of +0.4. Since this would be unmanageable in a counting system, we
should 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 the
nine.
In our developing card counting system there are 16 ten value cards
and four Aces with values of -0.51 and -0.61, respectively. Since they
are close in value, I am assigning them a value of minus one for the
developing system. Since a counting system needs a plus one value for
every minus one value to be in balance, we need 20 plus one value
cards. I’ll get them by assigning the rest of the cards plus one. So the
value of the 2s, 3s, 4s, 5s, and 6s will be plus one. Now let’s construct
a 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, or
Point 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 than
to add and subtract ones and twos (Level Two), or halves, or ones,
twos and threes (Level Three System). With increased variation in the
assigned values, there can be a more accurate reflection of the true
effect of removal. Unfortunately, with this increased accuracy and
complexity, errors in using them may offset any gains. This may lead
us to an easy to use, simple Level One system. There is eventually a
line 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 a
system from the ones presented.
Earlier I mentioned the term balanced. A balanced system is one where
all of the card values add up to 0. In other words, there are the same
number of plus value cards as minus value cards. A count would begin
with 0 and when the deck(s) is/are counted, the final count is 0. Since
0 = 0, the system is balanced. An unbalanced system is one which has
more of one value than the other. If there are four extra +1 cards
counted, say the 7s, and you start counting at 0, your final count
would be +4. This accounts for the extra four +1 cards in the counting
system. 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 is
an unbalanced count. Later you will be presented with several
unbalanced systems. They will be discussed in detail then.
Lastly, there are side counts for some systems. The Aces are often
counted alongside the main count. The purpose is to be able to adjust
the main count up or down to compensate for an excess or deficit of
Aces. Aces dilute the tens in a plus/minus count. When a decision
regarding whether to insure or not needs to be made, ten cards are the
only ones that matter, so an adjustment may be made for the Aces. If
the tens are counted with a side count of Aces, one uses the side count
to adjust up or down the count for betting decisions. This adds
another count to make when you already have a lot of things to do.
More gifted individuals might try doing a side count to improve the
performance of a system. One can get both a high level betting system
and improve the insurance/playing performance of a system by adding a
side count of Aces.
From the Deck
Before Beat the Dealer was published in 1962, professional players were
working out counting systems. They knew they had an advantage
when the remaining deck was rich in ten cards and Aces. They just
didn’t know what the advantage was. Dr. Thorp was, through the use
of high speed computers, able to quantify the effects of removal of any
card from a deck of cards. For example, when all of the fives were
removed from a deck, the player had a 3.58% advantage. He also
found the player had an advantage of 1.89% when four tens were
added to a deck. He did calculations for all of the cards. This became
the basis for the first two counting systems he developed - the Five
Count and the Ten Count.
Later, the effects of removing individual cards was studied, and the
following 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 each
denomination is removed from a deck. If a five card is removed from
the deck, the effect is +0.69% player advantage. In other words, the
removal is positive for the player. Conversely, remove a ten value card
and the effect is -0.51%. The removal has a negative effect on the
player. If you remove an eight, there is neither a positive or negative
effect. The eight is the only card that is completely neutral. It will
almost 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 than
one third of the value of the Ace. This means the nine is almost
neutral, so if it were assigned a value of 0, there would be only a small
loss in accuracy.
Of the smaller cards, they all have a significant value, even though there
are differences in the effects of removal. The one with the least effect is
the seven card which has an effect less than half that of the five. If we
were creating a counting system and assigned the five a value of plus
one, then it would be mathematically correct to assign the seven a value
of +0.4. Since this would be unmanageable in a counting system, we
should 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 the
nine.
In our developing card counting system there are 16 ten value cards
and four Aces with values of -0.51 and -0.61, respectively. Since they
are close in value, I am assigning them a value of minus one for the
developing system. Since a counting system needs a plus one value for
every minus one value to be in balance, we need 20 plus one value
cards. I’ll get them by assigning the rest of the cards plus one. So the
value of the 2s, 3s, 4s, 5s, and 6s will be plus one. Now let’s construct
a 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, or
Point 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 than
to add and subtract ones and twos (Level Two), or halves, or ones,
twos and threes (Level Three System). With increased variation in the
assigned values, there can be a more accurate reflection of the true
effect of removal. Unfortunately, with this increased accuracy and
complexity, errors in using them may offset any gains. This may lead
us to an easy to use, simple Level One system. There is eventually a
line 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 a
system from the ones presented.
Earlier I mentioned the term balanced. A balanced system is one where
all of the card values add up to 0. In other words, there are the same
number of plus value cards as minus value cards. A count would begin
with 0 and when the deck(s) is/are counted, the final count is 0. Since
0 = 0, the system is balanced. An unbalanced system is one which has
more of one value than the other. If there are four extra +1 cards
counted, say the 7s, and you start counting at 0, your final count
would be +4. This accounts for the extra four +1 cards in the counting
system. 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 is
an unbalanced count. Later you will be presented with several
unbalanced systems. They will be discussed in detail then.
Lastly, there are side counts for some systems. The Aces are often
counted alongside the main count. The purpose is to be able to adjust
the main count up or down to compensate for an excess or deficit of
Aces. Aces dilute the tens in a plus/minus count. When a decision
regarding whether to insure or not needs to be made, ten cards are the
only ones that matter, so an adjustment may be made for the Aces. If
the tens are counted with a side count of Aces, one uses the side count
to adjust up or down the count for betting decisions. This adds
another count to make when you already have a lot of things to do.
More gifted individuals might try doing a side count to improve the
performance of a system. One can get both a high level betting system
and improve the insurance/playing performance of a system by adding a
side count of Aces.
