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1: Probability Theory

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    15728

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    This chapter is devoted to a brief, and fairly low-level, introduction to a branch of mathematics known as probability theory.

    • 1.1: What is Probability?
      This page explains the scientific definition of probability, which quantifies the likelihood of specific outcomes from a general system by examining a large ensemble of similar systems. The probability \(P(X)\) of an outcome \(X\) is determined as the ratio of systems that display \(X\) to the total number of systems as it approaches infinity, with values ranging from 0 to 1 to indicate certainty or the absence of occurrence.
    • 1.2: Combining Probabilities
      This page explains probability calculations for distinct outcomes \(X\) and \(Y\) in a system \(S\). It covers mutually exclusive events, showing that their combined probability is the sum of individual probabilities \(P(X \mid Y) = P(X) + P(Y)\), and introduces the normalization condition that all possible outcomes must sum to one. Additionally, it details the computation of joint probabilities for independent events, highlighting that \(P(X \otimes Y) = P(X) \cdot P(Y)\).
    • 1.3: Mean, Variance, and Standard Deviation
      This page covers the concepts of mean and variance in statistics, using the example of student ages. It details the computation of the mean, defined for a variable \(u\), and discusses deviations from the mean. Variance is introduced as a measure of scatter around the mean, represented by \(\langle ({\mit\Delta} u)^2 \rangle\), while standard deviation, \(\sigma_u\), indicates the distribution's width around the mean.
    • 1.4: Continuous Probability Distributions
      This page explains probability density for continuous variables, highlighting that the probability of falling within a range is proportional to that range. It introduces the probability density function \(P(u)\), covering key concepts such as normalization, expected value \(\langle u \rangle\), and variance \(\langle({\mit\Delta} u)^2\rangle\), which are crucial for grasping the behavior of continuous random variables in probability theory.
    • 1.5: Exercises
      This page explores applications of probability in physics, covering diverse scenarios such as rolling dice, power dissipation in resistors, survival probabilities in Russian roulette, and probability density functions for vehicle speeds and radioactive decay. It illustrates critical statistical principles through practical examples and experiments.

    This page titled 1: Probability Theory is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.