The Mathematics of Card Counting and Advantage Play in Blackjack
Mar 6, 2014
Card counting is a science and while a number of books and sites list the ways in which to beat the dealer at his game, what is the exact science behind smart strategies in blackjack and card counting?
To some blackjack is a game of chance, but to others it’s one of the few casino games where the outcome can be calculated using mathematical and observational skills.
In the 1980s and 90s, a team from MIT set out to beat the system using a tried and tested card counting system.
You might have heard of the Hi-Lo method as a strategy to win at blackjack, but what is the scientific basis behind card counting and why is it an effective way to beat the house?
The basics of card counting
Card counting is a blackjack strategy used by keen players. In principle it’s not too difficult to implement, but requires hours of training and practice.
There are different methods, with the Hi-Lo method being the most popular for its effectiveness and ease. Here, cards are assigned values based on where they’re located in the set.
Other card counting strategies:
• The KO method follows the same strategy as the Hi-Lo method, but the 7s are worth +1.
• Omega II assigns a worth of +2 to cards 4, 5 and 6, and the 10 and face cards a value of -2, where aces are worth 0.
• In Halves, the 2 and 7 cards are worth +5, the 5 is worth +1.5 and the 9 is worth -5.
Low cards ranging from 2-6 are assigned a value of +1, 10s, face cards and aces are -1, and all other cards are set to zero.
The card counter keeps a running tally of the cards that have been dealt, but the true count also depends on the decks used, where this is obtained from the run divided by the number of decks used.
The idea is that once the count goes above +2, then the game is in the player’s favor and bigger bets can be made.
Other methods include the High Opt 2 and Zen versions, which yield improved results but are harder and easier to miscount.
Also, the High Card Count is another possible strategy to use, since this is much easier where only the high cards are counted, but as a result, it’s still not as effective as the other techniques.
Playing for variance
In the gambling world, luck is a word that often gets tossed about, but the word “variance” can give us a mathematical difference of the expected advantage and the results produced.
Say you’re playing a game and applying your card counting technique of choice, and your hourly expected value is $25 per hour, then playing for 100 hours means your expectation is $2500.
However, in reality if you’re only $2000 ahead, you fall $500 below the expectation value, which is the variance of what you would expect to win, and can also work in reverse by having $3000 after 100 hours.
Variance does mean good news for players with effective strategies, since the longer you play, the closer you’ll get to balancing out the variance for winning.
Deviating the standard
Blackjack strategies might seem simple on the surface, but the game really does boil down to its mathematical elements.
It’s impossible to discuss statistics without referencing standard deviation. The standard deviation refers to how often an outcome will deviate from the mean or the average.
You’ll see this in many areas of statistical analysis, and it’s quite easy to determine. The standard deviation is essentially the square root of the variance.
How does the standard deviation affect your game? Well, there is no one answer, but if you can figure out the variance and the standard deviation, then you can assess to see if you’re winning a game or how much money you should play with.
The standard deviation can give you an idea of what to expect in terms of variance and help you predict your own game play.
Nothing but zero
The N-Zero is a good number to place your bets on, but is often forgotten by novice players.
This N0 gives you the theoretical number of hands that you should reach before hitting your statistical goal of being ahead by one order of standard deviation.
To explain this in dummies’ terms, 4xN0 gives the value needed to overcome the 2SD in your game play.
But the N0 is dependent on playing with the same set of rules and strategies. If this is met, then the $25 expected value per house gives N0 BY N0 = Variance/252.
To play to win in the long run, then play games and bet spreads at values lower than the N-Zero.
The risk adjusted return of Certainty Equivalence
When you take the product of taking the expected win rate and you adjust it to your level of risk in proportion to the inputted money and level of risk tolerance, you get the Certainty Equivalence.
This number will give you a mathematical litmus test on whether it is worth playing the game or not when compared to your bank size.
Even though you might assume a return expected value of $100 per hour, the CE might only show that the game is only worth $50 since your bankroll is low and you haven’t got the money to burn.
If the CE value goes negative, it means that you are over betting what you have in your account and it’s not worth the risk.
The number essentially gives you an idea on whether it’s worth taking a risk or not and can help you to make the smart call of when to play and when not to play.
The truth about blackjack is it’s a game of statistics and probability that can be dominated by mathematics. The question is, do you have what it takes to master the mathematical side of blackjack?