What is Probability theory: Definition and 102 Discussions

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.
Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

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  1. cianfa72

    A Karhunen–Loève theorem expansion random variables

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  2. Demystifier

    I A variation of the sleeping beauty problem

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  3. gentzen

    A Concrete examples of randomness in math vs. probability theory

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  4. A

    I The third central moment of a sum of two independent random variables

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  5. P

    I Basic Probability Theory Question about Lebesgue measure

    Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?
  6. C

    I Help with probability problem: Probability that one random Gaussian event will happen before another one

    For concretness I'll use atoms and photons but this problem is actually just about probabilities. There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom...
  7. J

    B How helpful is probability theory?

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  8. V

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  9. B

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  10. Jatex

    I Radon-Nikodym Derivative and Bayes' Theorem

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  11. Killtech

    I Quantum physics vs Probability theory

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  12. Demystifier

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  13. W

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  14. W

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  15. W

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  16. W

    Probability Theory: Need help understanding a step

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  17. W

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  18. W

    Probability Theory: Multinomial coefficients

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  19. F

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  20. S

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  21. Sarina3003

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  22. Pouyan

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  23. Pouyan

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  24. S

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  25. Aslet

    I How Does the Inclusion-Exclusion Principle Relate to Probability Theory?

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  26. D

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  28. A

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  29. Hughng

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  30. Demystifier

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  31. mr.tea

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  32. B

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  33. J

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  34. J

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  35. J

    Inequality involving probability of stationary zero-mean Gaussian

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  36. J

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  37. BobblyHats97

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  38. Matejxx1

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  39. R

    Prove/Disprove: p(a∩b) ≤ q^2 with a,b Independant

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  40. P

    Calculating permutations for a normally distributed variable

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  41. M

    Upper level probability theory over summer

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  42. K

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  43. M

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  44. M

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  45. J

    Limit involving extinction probability of branching process

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  46. A

    Probability theory question

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  47. B

    Partner study for grad students on winter break: Success stories?

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  48. micromass

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    As I realized recently, the probability theory as used in quantum mechanics does not follow Kolmogorov's axioms. I am interested in a book that treats probability theory as it is done in quantum mechanics. Is this treated in books on quantum logic? Any other good book on the mathematical...
  49. R

    MHB Probability Theory: Q1, Q2, and Q3

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  50. V

    Confused about formal definitions of probability theory

    I think the first thing that is confusing me is the terminology. There are too many similar terms (e.g. probability measure, probability distribution, probability density function, probability mass function) What are the general concepts and what are the instances of those concepts? Like, are...
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