An Elementary Introduction to the Theory of Probability

This book explains concepts simply and clearly. It is the exact opposite of a typical American math book which is filled with useless formalism and endlessly repeated examples. This book conveys the core concepts of probability quickly and seemingly effortlessly.

This brief text, which was written for high school students in the Soviet Union following World War II, is an illuminating introduction to probability theory that does not require a foundation in calculus. The authors develop the theory by generalizing from examples, most of which are taken from military or industrial applications. This gives the reader insight into how mathematicians develop theorems by abstracting from problems arising in the real world. The theorems are proved rigorously except in the final chapter on normal distributions. Formal proofs about normal distributions require advanced mathematics not familiar to the intended audience. Probability theory is developed in the first section of the text. The authors define probability. They explain the addition rule and how it simplifies when events are mutually exclusive. Likewise, after they obtain the multiplication rule in terms of conditional probabilities, they explain how it simplifies when events are mutually independent. The authors discuss Bayes' formula for the probability of a hypothesis given that a given event has been observed using several examples. They then prove Bernoulli's formula for the most probable number of occurrences of an event when there are a large number of trials. The second section of the text is on random variables. The authors discuss laws of distribution, mean values, variance and standard of deviation, and how these quantities are used to measure the dispersion of a random variable. Their development culminates in Chebyshev's law of large numbers. In the final chapter on normal distributions, the authors informally discuss their properties and show how they can be used to solve problems. In a brief conclusion, the authors discuss other developments in probability theory that are beyond the scope of this text. This text is an excellent introduction to probability theory. I recommend it highly for the insights it offers. However, it does not contain exercises. To learn mathematics, one must solve problems. Therefore, I suggest that you read this text in conjunction with a problem book on probability or a text on probability that does contain exercises such as Samuel Goldberg's Probability: An Introduction.