This friendly guide is the companion you need to convert pure mathematics into understanding and facility with a host of probabilistic tools. The book provides a high-level view of probability and its most powerful applications. It begins with the basic rules of probability and quickly progresses to some of the most sophisticated modern techniques in use, including Kalman filters, Monte Carlo techniques, machine learning methods, Bayesian inference and stochastic processes. It draws on thirty years of experience in applying probabilistic methods to problems in computational science and engineering, and numerous practical examples illustrate where these techniques are used in the real world. Topics of discussion range from carbon dating to Wasserstein GANs, one of the most recent developments in Deep Learning. The underlying mathematics is presented in full, but clarity takes priority over complete rigour, making this text a starting reference source for researchers and a readable overview for students.;Cover -- Half-title -- Title page -- Copyright information -- Contents -- Preface -- Nomenclature -- 1 Introduction -- 1.1 Why Probabilities? -- 1.2 Events and Probabilities -- 1.2.1 Events -- 1.2.2 Assigning Probabilities -- 1.2.3 Joint and Conditional Probabilities -- 1.2.4 Independence -- 1.3 Random Variables -- 1.4 Probability Densities -- 1.5 Expectations -- 1.5.1 Indicator Functions -- 1.5.2 Statistics -- 1.6 Probabilistic Inference -- 1.A Measure for Measure -- 2 Survey of Distributions -- 2.1 Discrete Distributions -- 2.1.1 Binomial Distribution -- 2.1.2 Hypergeometric Distribution