Hey guys, I got this stat project to do for my econometrics class and I want it to have something to do with blackjack, card counting, gambling etc.

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2. A Statistical Description. The decreased blackjack winnings for the two shifts occurred between. March and August of one year (the 'suspect' period); the forty.

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BlackJack has always been my favorite game because of a lot of misconceptions. Also, if you increase the number of games, the percentage times 6 Open Source Data Science Projects to Impress your Interviewer.

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Statistics was the thing that interested me,and here we hadaproblem that looked money from our project becauseourearly results werevery counterintuitive.”.

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Many statistical problems of practical interest are simply too complicated to explore analytically. In these cases, researchers often turn to simulation techniques in.

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2. A Statistical Description. The decreased blackjack winnings for the two shifts occurred between. March and August of one year (the 'suspect' period); the forty.

Enjoy!

Software - MORE

2. A Statistical Description. The decreased blackjack winnings for the two shifts occurred between. March and August of one year (the 'suspect' period); the forty.

Enjoy!

BlackJack has always been my favorite game because of a lot of misconceptions. Also, if you increase the number of games, the percentage times 6 Open Source Data Science Projects to Impress your Interviewer.

Enjoy!

Many statistical problems of practical interest are simply too complicated to explore analytically. In these cases, researchers often turn to simulation techniques in.

Enjoy!

Blackjack is one of the simplest games to play at a casino. Simulating Thousands of Blackjack Card Counting Strategies with Data Science Tools This also seems to provide a decent percentage of success as long as the strategy is followed with no deviations. State-of-the-art python project setup.

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Since the parents were selected with an eye to fitness, the goal is to pass on the successful elements from both parents. In fact, the coefficient of variation for , hands is 0. Back in the s, a mathematician named Edward O. The following items can be configured for a run:. 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. That score is calculated once per generation for all candidates, and can be used to compare them to each other. Imagine a pie chart with three wedges of size 1, 2, and 5. 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.

One of the great things about machine learning is that there are so many different approaches blackjack statistics project solving problems. Each candidate has a fitness score that indicates how good it blackjack statistics project.

The tall table on the left is for hard handsthe table blackjack statistics project the upper right is for soft handsand the table in the lower right is for pairs. The more hands played, the smaller the variations will be. 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 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. Using blackjack pairs and soft hand tables develop last because those hands happen so infrequently.

The best way to settle on values for these blackjack statistics project is simply to experiment. Tournament selection has already been covered. Once this fitness score adjustment is complete, Roulette Wheel selection is used.

The hard hands in particular the table on the left are almost exactly correct. Because of the innate randomness of a deck of cards, many hands need to be played so the randomness evens out across the candidates. Could we run with , or more hands per test? The idea of a fitness function is simple. 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 flat white line along the top of the chart is the fitness score for the known, optimal baseline strategy. Of course. 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. 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. Populations that are too small or too homogenous always perform worse than bigger and more diverse populations. By generation 33, things are starting to become clear:. Population Size. Neural networks are great for finding patterns in data, resulting in predictive capabilities that are truly impressive. 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. 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. There are a number of different selection techniques to control how much a selection is driven by fitness score vs. 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 solution is to use Ranked Selection , which 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. Varying each of these gives different results. It reduces variability and increases the accuracy of the fitness function. The fitness function reflects the relative fitness levels of the candidates passed to it, so the scores can effectively be used for selection. A cell in the child is populated by choosing the corresponding cell from one of the two parents. Due to the house edge, all strategies will lose money, which means all fitness scores will be negative. That evolutionary process is driven by comparing candidate solutions. The three tables represent a complete strategy for playing Blackjack. 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. A higher fitness score for a strategy merely means it lost less money than others might have. 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. As you might imagine, Blackjack has been studied by mathematicians and computer scientists for a long, long time. Knowing the optimal solution to a problem like this is actually very helpful. Once an effective fitness function is created, the next decision when using a GA is how to do selection. This is the very best solution based on fitness score from candidates in generation 0 the first, random generation :. 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. That gives us something called the coefficient of variation , which can be compared to other test values, regardless of the number of hands played. By generation 12, some things are starting to take shape:. Genetic algorithms are essentially driven by fitness functions. Here are two other approaches:. A genetic algorithm GA uses principles from evolution to solve problems. The other hints of quality in the strategy are the hard 11 and hard 10 holdings. Of course, in reality there is no winning strategy for Blackjack — the rules are set up so the house always has an edge. A pair is self-explanatory, and a hard hand is basically everything else, reduced to a total hand value. Using a single strategy, multiple tests are run, resulting in a set of fitness scores. Even though we may not know the optimal solution to a problem, we do have a way to measure potential solutions against each other. As it turns out, you need to play a lot of hands with a strategy to determine its quality. The chart here that demonstrates how the variability shrinks as we play more hands:. During that run, about , strategies were evaluated. Comparing the results from a GA to the known solution will demonstrate how effective the technique is. Roulette Wheel Selection selects candidates proportionate to their fitness scores. This works just like regular sexual reproduction — genetic material from both parents are combined. The goal is to find a strategy that is the very best possible, resulting in maximized winnings over time. 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. And then the final generations are used to refine the strategies. Running on a standard desktop computer, it took about 75 minutes. Basic concepts get developed first with GAs, with the details coming in later generations. 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. The process of finding good candidates for crossover is called selection, and there are a number of ways to do it. 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. 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. One of the cool things about GAs is simply watching them evolve a solution. The columns along the tops of the three tables are for the dealer upcard, which influences strategy. The soft hand and pairs tables are getting more refined:. There will be large swings in fitness scores reported for the same strategy at these levels. 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. The source code for the software that produced these images is open source. Knowing that, the best possible strategy is the one that minimizes losses. Once two parents are selected, they are crossed over to form a child. If you play long enough, you will lose money. One of the unusual aspects to working with a GA is that it has so many settings that need to be configured. Finally, the best solution found over generations:. The lack of genetic diversity in those small populations results in poor final fitness scores, along with a slower process of finding a solution. 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 :. But how many hands is enough? First, testing with only 5, or 10, hands is not sufficient. That optimal strategy looks something like this:. Standard deviation is scaled to the underlying data. Reinforcement learning uses rewards-based concepts, improving over time.