The permutations and statistical calculations forming the basis of the basic blackjack strategy table won't be included in this article. The table is.

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The most important thing to learn about playing blackjack, and I can not stress this The above chart assumes the casino doesn't allow doubling down after pair.

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The most important thing to learn about playing blackjack, and I can not stress this The above chart assumes the casino doesn't allow doubling down after pair.

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Rules of the game of Blackjack can vary from casino to casino or even table , the American Journal of Statistical Association published a paper by Rogerβ.

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Blackjack odds are percentage figures which represent your probability of losing or winning a hand. They can also represent the house edge or their profit.

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The fascinating game of Blackjack is peppered with amazing statistical probabilities. Enjoy this guide to Blackjack statistics at

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Of course, in reality there is no winning strategy for Blackjack β the rules The tall table on the left is for hard hands, the table in the upper right is for and then pass in a certain percentage of the best candidates directly into.

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Blackjack is actually the casino table game where the House has its least statistical advantage, but in order to take advantage of this fully, you.

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The fascinating game of Blackjack is peppered with amazing statistical probabilities. Enjoy this guide to Blackjack statistics at

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Blackjack odds are percentage figures which represent your probability of losing or winning a hand. They can also represent the house edge or their profit.

Enjoy!

There will be large swings in fitness scores reported for the same strategy at these levels. Standard deviation is scaled to the underlying data. Here are two other approaches:. Tournament selection has already been covered. Each candidate has a fitness score that indicates how good it is. Neural networks are great for finding patterns in data, resulting in predictive capabilities that are truly impressive. First, testing with only 5, or 10, hands is not sufficient. A genetic algorithm GA uses principles from evolution to solve problems. Populations that are too small or too homogenous always perform worse than bigger and more diverse populations. As you might imagine, Blackjack has been studied by mathematicians and computer scientists for a long, long time. By measuring the standard deviation of the set of scores we get a sense of how much variability we have across the set for a test of N hands. By generation 33, things are starting to become clear:. The other hints of quality in the strategy are the hard 11 and hard 10 holdings. Knowing that, the best possible strategy is the one that minimizes losses. The first generation is populated with completely random solutions. The variations from run to run for the same strategy will reveal how much variability there is, which is driven in part by the number of hands tested. With only 12 generations experience, the most successful strategies are those that Stand with a hard 20, 19, 18, and possibly That part of the strategy develops first because it happens so often and it has a fairly unambiguous result. Clearly, having a large enough population to ensure genetic diversity is important. The first thing to notice is that the two smallest populations having only and candidates respectively, shown in blue and orange performed the worst of all sizes. One of the cool things about GAs is simply watching them evolve a solution. To use the tables, a player would first determine if they have a pair, soft hand or hard hand, then look in the appropriate table using the row corresponding to their hand holding, and the column corresponding to the dealer upcard. Using such a strategy allows a player to stretch a bankroll as far as possible while hoping for a run of short-term good luck. The process of finding good candidates for crossover is called selection, and there are a number of ways to do it. Given those findings, the fitness function for a strategy will need to play at least , hands of Blackjack, using the following rules common in real-world casinos :. Because of the innate randomness of a deck of cards, many hands need to be played so the randomness evens out across the candidates. A pair is self-explanatory, and a hard hand is basically everything else, reduced to a total hand value. By generation 12, some things are starting to take shape:. The hard hands in particular the table on the left are almost exactly correct. The idea of a fitness function is simple. Basic concepts get developed first with GAs, with the details coming in later generations. The tall table on the left is for hard hands , the table in the upper right is for soft hands , and the table in the lower right is for pairs. Oftentimes, crossover is done proportional to the relative fitness scores, so one parent could end up contributing many more table cells than the other if they had a significantly better fitness score. Roulette Wheel Selection selects candidates proportionate to their fitness scores. One of the unusual aspects to working with a GA is that it has so many settings that need to be configured. And then the final generations are used to refine the strategies. As it turns out, you need to play a lot of hands with a strategy to determine its quality. The best way to settle on values for these settings is simply to experiment. The pairs and soft hand tables develop last because those hands happen so infrequently. The fitness function reflects the relative fitness levels of the candidates passed to it, so the scores can effectively be used for selection. There are a number of different selection techniques to control how much a selection is driven by fitness score vs. In the case of a Blackjack strategy, the fitness score is pretty straightforward: if you play N hands of Blackjack using the strategy, how much money do you have when done? The chart here that demonstrates how the variability shrinks as we play more hands:. Even though we may not know the optimal solution to a problem, we do have a way to measure potential solutions against each other. Comparing the results from a GA to the known solution will demonstrate how effective the technique is. One simple approach is called Tournament Selection , and it works by picking N random candidates from the population and using the one with the best fitness score. In fact, the coefficient of variation for , hands is 0. Finally, the best solution found over generations:. That optimal strategy looks something like this:. Due to the house edge, all strategies will lose money, which means all fitness scores will be negative.

One of the great things go here machine learning is that there are so many different approaches to solving problems. That gives us something called the coefficient of variationwhich can be compared to other test values, regardless of the number of hands played.

We solve this by dividing the standard deviation by the average fitness score for each of the test values the number of hands played, that is.

During that run, aboutstrategies were evaluated. The lack of genetic diversity in blackjack statistics chart small populations results in poor final fitness scores, along with a slower process of finding a solution. Since the parents were selected with an eye to fitness, the goal is to pass on blackjack statistics chart successful elements from both parents.

Could we run withor more hands per test? The solution is to use Ranked Selectionwhich works by sorting the candidates by fitness, then giving the worst candidate a score of 1, the next worse a score of 2, and so forth, all the way up to the best candidate, which receives a score equal to the population size.

Of course. It reduces variability and increases the accuracy blackjack statistics chart the fitness function. This works just like regular sexual reproduction β genetic material from blackjack statistics chart parents are combined. Of course, in reality there is no winning strategy for Blackjack β the rules are set up so the house always has an edge.

Reinforcement learning uses rewards-based concepts, improving over time.

One of the problems with that selection method is that sometimes certain candidates will have such a small fitness score that they never get selected. Population Size. The soft hand and pairs tables are getting more refined:. But how many hands is enough?

A cell in the child is populated by choosing the corresponding cell from one of the two parents. Once this fitness score adjustment is complete, Roulette Wheel selection is used.

Using a single strategy, multiple tests are run, resulting in a set of fitness scores. If, by luck, there are a couple of candidates that have fitness scores far higher than the others, they may be disproportionately selected, which reduces genetic diversity. The following items can be configured for a run:. That means that if the same GA code is run twice in a row, two different results will be returned. There are a couple of observations from the chart. As impressive as the resulting strategy is, we need to put it into context by thinking about the scope of the problem. The X axis of this chart is the generation number with a maximum of , and the Y axis is the average fitness score per generation. To avoid that problem, genetic algorithms sometimes use mutation the introduction of completely new genetic material to boost genetic diversity, although larger initial populations also help. Once two parents are selected, they are crossed over to form a child. That evolutionary process is driven by comparing candidate solutions. Genetic algorithms are essentially driven by fitness functions. In fact, it looks like a minimum of , hands is probably reasonable, because that is the point at which the variability starts to flatten out. Back in the s, a mathematician named Edward O. The goal is to find a strategy that is the very best possible, resulting in maximized winnings over time. It works by using a population of potential solutions to a problem, repeatedly selecting and breeding the most successful candidates until the ultimate solution emerges after a number of generations. Knowing the optimal solution to a problem like this is actually very helpful. But that improvement is definitely a case of diminishing returns: the number of tests had to be increased 5x just to get half the variability. The columns along the tops of the three tables are for the dealer upcard, which influences strategy. A higher fitness score for a strategy merely means it lost less money than others might have. The more hands played, the smaller the variations will be. Imagine a pie chart with three wedges of size 1, 2, and 5. That score is calculated once per generation for all candidates, and can be used to compare them to each other. The three tables represent a complete strategy for playing Blackjack. Once an effective fitness function is created, the next decision when using a GA is how to do selection. If you play long enough, you will lose money. The source code for the software that produced these images is open source. This is the very best solution based on fitness score from candidates in generation 0 the first, random generation :. The flat white line along the top of the chart is the fitness score for the known, optimal baseline strategy. Running on a standard desktop computer, it took about 75 minutes. Varying each of these gives different results.