Blackjack in general - This is a general article on the game of blackjack. It includes a demonstration showing that blackjack probabilities change depending upon composition of undealt cards, among other factors.
Basic Strategy - This article explains what basic strategy is, the difference between composition dependent basic strategy and total dependent basic strategy and why total dependent strategy is generally used. It presents a simplified total dependent strategy called Generic Basic that is simpler to learn because it applies to a broader range of conditions while being very close in expected value to traditional total dependent strategy. The few minor exceptions to Generic Basic can be ignored at a small loss in EV or later added at player's convenience.
c++ code to output all possible unbusted player hands - Uses nested for loops, arrays, and integer division to identify and output all unbusted player hands.
c++ code to compute 1, 2, or 3 splits - This is my original algorithm for computing splits. It is limited to 1, 2, or 3 allowed splits.
c++ code to approximate computation of any number of splits in blackjack - Computing pair splits in blackjack is not an easy problem. This is a relatively simple recursive algorithm which could theoretically be used to compute any number of allowed splits. In general the number of shoe states required equals (allowed splits.) For example, 3 allowed splits requires just 3 shoe states. Once the expected value(s) of drawing to a single pair card have been computed for these 3 states, split EVs can be easily computed following the algorithm's execution. Results are very close to the true values for full shoes. Variation increases as number of undealt non-pair cards lowers. Variation only occurs in resplits. There is no variation at all for 1 allowed single split.
Computing counting system probabilities - Shows example of computing HiLo Count probabilities at mid-shoe for a single deck for a running count of 0 (RC = 0). Methodology could be extrapolated to any counting system.
Composition Dependent Basic Strategy Versus an Up Card of 10 - Shows how to play composition dependent basic strategy versus a dealer up card of 10 for 8 or fewer decks.
Betting - This is an introduction into determining how much to bet when a positive expected value opportunity exists. It includes a link to a very informative article on the consequences of various betting strategies.
Card Counting Theory - This article shows some of the things considered in developing a card counting system for those that may be interested.
Counting System Implementation - Explanation of why tags are used to simplify card counting and how to implement a tag based counting system. Explains the difference between balanced and unbalanced systems, how to true count a balanced system, and why unbalanced systems are sometimes used.
Alternative to true count - Generally decisions using a balanced count such as HiLo are based on true count (= 52*(current running count)/(current number of cards remaining)) while those using an unbalanced count are based on running count only. This shows when to buy generic insurance using HiLo for 1 to 8 decks simply using running count.
Contrast in approaches to counting system variation: running count versus true count
Generally balanced count variation is measured by true count while the measure for an unbalanced count is running count. However, either approach is workable for a count absent a user defined initial running count offset.
Example data: overall EV using basic strategy
6 decks, dealer stands on soft 17, NDAS, 1 allowed split all ranks, 1 card to split aces
(Same data is used to plot) 1) true count vs. pen and 2) running count vs. pen
for overall expected value (using basic strategy) range of -4% to +4%
HiLo Count
KO Count
Insurance in Blackjack - Determining when and when not to take insurance in blackjack. Links to computed data for HiLo and KO counting systems for 1, 2, 4, 6, and 8 decks.
Insurance Count - Shows a simple counting system that is dependent upon the density of tens remaining to be dealt. The expected value of an insurance bet is solely dependent upon the probability of drawing a ten value card (10, Jack, Queen, King.) Therefore, this count can be used to determine when it is positive EV to insure with 100% accuracy. Also, this count can be used as the basis for developing more advanced counting systems.
Method to Derive Perfect Insurance (for any counting system) -
Possible Number of Shuffles in Blackjack - How many ways can a deck (shoe) be shuffled relative to blackjack?
A few words about variance..... - Basic variance
Computing Variance of an Individual Player Hand - How variance of a player hand that does not involve pair splitting could be computed either excluding or including insurance bet. Pair splits are more complicated.
Simulation / Combinatorial Analysis
Card Counting Dynamics - The probability of drawing any rank from a freshly shuffled shoe is exactly the same value whether a player is side counting every rank and always knows what the exact shoe composition is or whether a player is using a counting system and only knows the running count of his counting system and a(n) (approximate) number of cards remaining to be dealt. As cards are randomly dealt and exposed the player that is keeping track of the exact shoe composition can always compute the exact probability of drawing any rank. The player that is using a counting system can also compute a probability of drawing any rank, but this calculation is based upon less than perfect information. This first shows how all random shoe compositions could be enumerated and used to compute values relative to any counting system. In order to do this a very large number of subset compositions would need to be processed. Second it shows that rank probabilities based only on a counting system can be computed and possibly be used to create a combinatorial analyzer that is based upon only a given counting system.
Effect of Number of Cards Remaining in Blackjack and Floating Advantage - This page looks at the effect of simply reducing the number of cards remaining prior to each round. It also looks at effect on expected value for less than a full shoe for AND a shoe composition that is similar to a full shoe composition in that the probabilities for drawing each rank are equal or nearly equal to that of a full shoe.
Example of varying number of neutral cards in HiLo Count - HiLo is the most used blackjack card counting system. Basic use is to vary bet according to HiLo count and play cards according to basic strategy. More advanced is to also vary card play strategy depending on HiLo count by using indexes. HiLo's neutral cards are {7,8,9} and have no effect on its count. This is an example of how a counter's playing strategy indexes vary for a player hand of a pair of eights based on presence of less or more {7,8,9} when half of the cards of a 6 deck shoe remain. It is a use of my count enumeration program.
Refining the Tens (Insurance) Count - Shows how the Tens (Insurance) Count can be used as a basis to devise counts that can better estimate overall expected value.