Editing Card counting (section)

== Basics ==
Card counting is based on statistical evidence that high cards ([[ace]]s, 10s, and 9s) benefit the player, while low cards, (2s, 3s, 4s, 5s, 6s, and 7s) benefit the dealer. High cards benefit the player in the following ways:

# They increase the player's probability of hitting a Blackjack, which often pays out at 3 to 2 odds (although some casinos pay at 6 to 5).
# Doubling down increases expected value. The elevated ratio of tens and aces improves the probability that doubling down will succeed. The most common hand values that the player doubles down on are 11, 10, and 9; and drawing a high card to these will make a strong hand.<ref name="beat-19662">Thorp (1966), pp. 24–27, pp. 41–47, pp. 98–99, pp. 102–103, p. 110, p. 115.</ref>
# They provide additional splitting opportunities for the player.
# They can make the [[Blackjack#Insurance|insurance bet]] profitable by increasing the probability of dealer blackjack.
# They also increase the probability the dealer will bust, in the event that the dealer shows a low up-card (i.e. 2-6). This also increases the odds of the player busting, but the player can choose to stand on lower totals based on the count.

On the other hand, low cards benefit the dealer. The rules require the dealer to hit stiff hands (12–16 total), and low cards are less likely to bust these totals. A dealer holding a stiff hand will bust if the next card is a 10.<ref name="wiz-bjk2">{{cite web |author=Wizard of Odds |date=July 2011 |title=How to Play Blackjack |url=https://wizardofodds.com/games/blackjack/ |access-date=3 November 2011 |publisher=wizardofodds.com |archive-date=15 November 2011 |archive-url=https://web.archive.org/web/20111115231930/http://wizardofodds.com/games/blackjack/ |url-status=live }}</ref>

Card counters do not need unusual mental abilities; they do not track or memorize specific cards. Instead, card counters assign a point score to each card that estimates the value of that card. They track the sum of these values with a running count.<ref name="tob-19794">{{cite book |last1=Griffin |first1=Peter A. |title=The theory of blackjack: the compleat card counter's guide to the casino game of 21 |date=1999 |publisher=Huntington Press |isbn=978-0929712130 |edition=6th |location=Las Vegas, Nev.}}</ref> The myth that counters track every card was portrayed in the 1988 film ''[[Rain Man]]'', in which the [[savant syndrome|savant]] character Raymond Babbitt counts through six decks with ease, and a casino employee comments that it is impossible to do so.<ref>{{cite book |author1=Sklar, Jessica K. |url=https://books.google.com/books?id=Q-Q_2G8CiyYC&pg=PA151 |title=Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media |author2=Sklar, Elizabeth S. |date=29 February 2012 |publisher=[[McFarland & Company]] |isbn=978-0-7864-8994-7 |page=151 |access-date=26 August 2014 |archive-date=4 February 2021 |archive-url=https://web.archive.org/web/20210204001218/https://books.google.com/books?id=Q-Q_2G8CiyYC&pg=PA151 |url-status=live }}</ref><ref>{{cite web |title=Rain Man (1988) Plot Summary |website=[[IMDb]] |url=https://www.imdb.com/title/tt0095953/plotsummary |access-date=10 July 2014 |archive-date=17 July 2014 |archive-url=https://web.archive.org/web/20140717020855/http://www.imdb.com/title/tt0095953/plotsummary |url-status=live }}</ref>

=== Systems ===
Basic card counting systems assign a positive, negative, or zero value to each card. When a card is dealt, the count is adjusted by that card's counting value. Low cards increase the count; they increase the percentage of high cards in the deck. High cards decrease the count for the opposite reason. For example, the Hi-Lo system subtracts one for each 10, jack, queen, king, or ace and adds one for any card between 2 and 6. 7s, 8s, and 9s count as zero and do not affect the count.<ref>Axelrad (2010), p. 256.</ref>

A card counting system aims to assign point values roughly correlating to a card's ''effect of removal'' (EOR). The EOR is the estimated effect of removing a given card from play. Counters gauge the effect of removal for all cards dealt and how that affects the current house edge. Larger ratios between point values create better correlations to actual EOR, increasing the efficiency of a system. Such systems are classified as level 1, level 2, level 3, and so on. The level corresponds to the ratio between values.

The Hi-Lo system is a ''level-1'' count; the running count never increases or decreases by more than one. A ''multilevel'' count, such as Zen Count, Wong Halves, or Hi-Opt II, further distinguishes card values to increase accuracy. An advanced count includes values such as +2 and −2, or +0.5 and -0.5. Advanced players might also keep a ''side count'' of specific cards like aces. This is done where betting accuracy differs from playing accuracy.

Many side count techniques exist, including special-purpose counts used for games with nonstandard profitable-play options such as an over/under side bet.<ref>{{cite web |title=Blackjack Side Counts |url=https://www.qfit.com/blackjack-side-counts.htm |access-date=20 March 2009 |publisher=Qfit.com |archive-date=11 March 2009 |archive-url=https://web.archive.org/web/20090311045259/http://qfit.com/blackjack-side-counts.htm |url-status=live }}</ref>

Keeping track of more data with higher level counts can hurt speed and accuracy. Some counters earn more money playing a simple count quickly than by playing a complex count slowly.

This table illustrates some example counting systems.<ref name="qfitcc2">{{cite web |title=Card Counting Strategies |url=https://www.qfit.com/card-counting.htm |access-date=10 March 2013 |publisher=Qfit.com |archive-date=10 March 2013 |archive-url=https://web.archive.org/web/20130310034023/http://qfit.com/card-counting.htm |url-status=live }}</ref><ref>{{cite book |last1=Archer|first1=John|title=The Archer Method of Winning at 21|date=1973|publisher= Henry Regnery Company |isbn=0-87980-328-2}}</ref>
{| class="wikitable"
|-
!Card Strategy
!2
!3
!4
!5
!6
!7
!8
!9
!10, J, Q, K
!A
!Level of count
|-
| Hi-Lo
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| 0
|align=center| 0
|align=center| 0
|align=center| −1
|align=center| −1
|align=center| 1
|-
| Hi-Opt I
|align=center| 0
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| 0
|align=center| 0
|align=center| 0
|align=center| −1
|align=center| 0
|align=center| 1
|-
| Hi-Opt II
|align=center| +1
|align=center| +1
|align=center| +2
|align=center| +2
|align=center| +1
|align=center| +1
|align=center| 0
|align=center| 0
|align=center| −2
|align=center| 0
|align=center| 2
|-
| KO
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| 0
|align=center| 0
|align=center| −1
|align=center| −1
|align=center| 1
|-
| Omega II
|align=center| +1
|align=center| +1
|align=center| +2
|align=center| +2
|align=center| +2
|align=center| +1
|align=center| 0
|align=center| −1
|align=center| −2
|align=center| 0
|align=center| 2
|-
| Red 7
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| 0 or +1
|align=center| 0
|align=center| 0
|align=center| −1
|align=center| −1
|align=center| 1
|-
| Halves
|align=center| +0.5
|align=center| +1
|align=center| +1
|align=center| +1.5
|align=center| +1
|align=center| +0.5
|align=center| 0
|align=center| -0.5
|align=center| −1
|align=center| −1
|align=center| 3
|-
| Zen Count
|align=center| +1
|align=center| +1
|align=center| +2
|align=center| +2
|align=center| +2
|align=center| +1
|align=center| 0
|align=center| 0
|align=center| −2
|align=center| −1
|align=center| 2
|-
| 10 Count
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| +1
|align=center| −2
|align=center| +1
|align=center| 2
|}

=== Design and selection of systems ===
The primary goal of a card counting system is to assign point values to each card that roughly correlate to the card's "effect of removal" or EOR (that is, the effect a single card has on the house advantage once removed from play), thus enabling the player to gauge the house advantage based on the composition of cards still to be dealt. Larger ratios between point values can better correlate to actual EOR, but add complexity to the system. Counting systems may be referred to as "level 1", "level 2", etc., corresponding to the number of different point values the system calls for.

The ideal system is a system that is usable by the player and offers the highest average dollar return per period of time when dealt at a fixed rate. With this in mind, systems aim to achieve a balance of efficiency in three categories:<ref name="tob-19794"/>

; Betting correlation (BC)
: When the sum of all the permutations of the undealt cards offers a positive expectation to a player using optimal playing strategy, there is a positive expectation to a player placing a bet. A system's BC gauges how effective a system is at informing the user of this situation.
; Playing efficiency (PE)
: A portion of the expected profit comes from modifying playing strategy based on the known altered composition of cards. For this reason, a system's PE gauges how effectively it informs the player to modify strategy according to the actual composition of undealt cards. A system's PE is important when the effect of PE has a large impact on the total gain, as in single- and double-deck games.
; Insurance correlation (IC)
: A portion of expected gain from counting cards comes from taking the insurance bet, which becomes profitable at high counts. An increase in IC will offer additional value to a card counting system.

Some strategies count the ace (ace-reckoned strategies) and some do not (ace-neutral strategies). Including aces in the count improves betting correlation since the ace is the most valuable card in the deck for betting purposes. However, since the ace can either be counted as one or eleven, including an ace in the count decreases the accuracy of playing efficiency. Since PE is more important in single- and double-deck games, and BC is more important in [[shoe (cards)|shoe]] games, counting the ace is more important in shoe games.

One way to deal with such tradeoffs is to ignore the ace to yield higher PE while keeping a side count which is used to detect an additional change in EV which the player will use to detect additional betting opportunities that ordinarily would not be indicated by the primary card counting system.

The most common side counted card is the ace since it is the most important card in terms of achieving a balance of BC and PE. In theory, a player could keep a side count of every card and achieve a near 100% PE, however, methods involving additional side counts for PE become more complex at an exponential rate as you add more side counts and the ability of the human mind is quickly overtasked and unable to make the necessary computations. Without any side counts, PE can approach 70%.<ref name="tob-19794" />

Since there is the potential to create an overtaxing demand on the human mind while using a card counting system another important design consideration is the ease of use. Higher-level systems and systems with side counts will obviously become more difficult and in an attempt to make them easier, unbalanced systems eliminate the need for a player to keep tabs on the number of cards/decks that have already entered play typically at the expense of lowering PE.<ref name="qfitcc2"/>