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#### Overview of this book

Are you looking to start developing artificial intelligence applications? Do you need a refresher on key mathematical concepts? Full of engaging practical exercises, The Statistics and Calculus with Python Workshop will show you how to apply your understanding of advanced mathematics in the context of Python. The book begins by giving you a high-level overview of the libraries you'll use while performing statistics with Python. As you progress, you'll perform various mathematical tasks using the Python programming language, such as solving algebraic functions with Python starting with basic functions, and then working through transformations and solving equations. Later chapters in the book will cover statistics and calculus concepts and how to use them to solve problems and gain useful insights. Finally, you'll study differential equations with an emphasis on numerical methods and learn about algorithms that directly calculate values of functions. By the end of this book, you’ll have learned how to apply essential statistics and calculus concepts to develop robust Python applications that solve business challenges.
Preface
1. Fundamentals of Python
Free Chapter
2. Python's Main Tools for Statistics
3. Python's Statistical Toolbox
4. Functions and Algebra with Python
5. More Mathematics with Python
6. Matrices and Markov Chains with Python
7. Doing Basic Statistics with Python
8. Foundational Probability Concepts and Their Applications
9. Intermediate Statistics with Python
10. Foundational Calculus with Python
11. More Calculus with Python
12. Intermediate Calculus with Python

# Law of Large Numbers

There are many schemes and systems that people claim can make you a big winner at the casino. But what these people fail to see is the reason why casinos are lucrative money-makers; the odds are always on the casino's side, ensuring that the casino will come out ahead and always win (in the long run). What the casinos have come to depend on is something called the law of large numbers.

Before we figure out how the casinos always make themselves winners in the long run, we need to define several terms. The first is sample average, or sample mean. The sample mean is what everybody thinks of when they think of the average. You calculate the sample mean by adding up the results and dividing by the number of results. Let's say we flip a coin 10 times and it comes up heads 7 times. We calculate the sample mean, or the average number of heads per flip, like so:

Figure 9.1: Formula for sample mean

The sample average is typically denoted...