What is Probability theory: Definition and 101 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. Demystifier

    I A variation of the sleeping beauty problem

    Some time ago we had a discussion of the sleeping beauty problem https://www.physicsforums.com/threads/the-sleeping-beauty-problem-any-halfers-here.916459/ which is a well known problem in probability theory. In that thread, there was no consensus whether the probability of heads is 1/2 or 1/3...
  2. gentzen

    A Concrete examples of randomness in math vs. probability theory

    A recent question about interpretations of probability nicely clarified the role of the Kolmogorov axioms: [... some excursions into QM, negative probabilities, and quasiprobability distributions ...] Conclusion: the Kolmogorov axioms formalize the concept of probability. They achieve this by...
  3. A

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

    Is it true that when X and Y are independent, E ({X+Y}3) = E (X3)+E(Y3)?
  4. 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?
  5. 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...
  6. J

    B How helpful is probability theory?

    I am interested in stomach acid and heat expansion, for instance the stomach will become heated due to an athelete competing. The heat causes atheletes to live shorter than people who don't have their body heated so often. I do a lot of differential equations and number theory, but I was...
  7. V

    Expected Value of Election Results

    I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong? Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house. There are 4 cases: winning the senate and house (##S \cap H##), winning just...
  8. B

    Mixed random variables problem

    I got (a) and (b) but I'm still working on (c). The solutions can be found here for your reference: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/lecture-9/MIT6_041SCF13_assn05_sol.pdf. But...
  9. Jatex

    I Radon-Nikodym Derivative and Bayes' Theorem

    I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt: \begin{equation} \label{eq1} \begin{split} \frac{\mathrm d\mu_{\Theta\mid X}}{\mathrm d\mu_\Theta}(\theta \mid x) &= f_{\Theta\mid X}(\theta\mid x) \mathrm \space...
  10. Killtech

    I Quantum physics vs Probability theory

    Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems...
  11. Demystifier

    Insights The Sum of Geometric Series from Probability Theory - Comments

    Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.
  12. W

    Probability Theory: Order statistics and triple integrals

    Homework Statement Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##. a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##. b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...
  13. W

    Probability theory: Understanding some steps

    Homework Statement Hi all, I have some difficulty understanding the following problem, help is greatly appreciated! Let ##U_1, U_2, U_3## be independent random variables uniform on ##[0,1]##. Find the probability that the roots of the quadratic ##U_1 x^2 + U_2 x + U3## are real. Homework...
  14. W

    Probability Theory: Simultaneous picks

    Homework Statement [/B] Hi all, I have an issue understanding the concepts pertaining to the following problem, assistance is greatly appreciated. I understand the "flow" of the problem; first find the probability of obtaining balls of the same colour, then use the geometric distribution...
  15. W

    Probability Theory: Need help understanding a step

    Homework Statement Discrete random variables ##X,Y,Z## are mutually independent if for all ##x_i, y_j, z_k##, $$P(X=x_i \wedge Y=y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_k )$$ I am trying to show (or trying to understand how someone has shown) that ##X,Y## are also independent as a result...
  16. W

    Probability Theory, work check

    Homework Statement Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance. Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?Homework Equations The Attempt...
  17. W

    Probability Theory: Multinomial coefficients

    <Moderator's note: Moved from homework.> Hi all, I have an issue understanding a statement I read in my text. It first states the following Proposition (Let's call it Proposition A): The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n...
  18. F

    Is there an optimal distance between measurements for regression

    Suppose I am trying to approximate a function which I do not know, but I can measure. Each measurement takes a lot of effort. Say the function I am approximating is ##y=f(x)## and ##x \in [0,100]## Supose I know the expectation and variance of ##f(x)##. Is there a way to compute the confidence...
  19. S

    How were you exposed to probability theory in physics?

    Hi everyone. As a graduate student in statistics, I had taken a graduate course in measure-theoretic probability theory. In a conversation with the professor, he had remarked that if I wanted to pursue further research on some of the topics covered, it may be wise to do background reading or...
  20. Sarina3003

    Probability theory, probability space, statistics

    Homework Statement Homework Equations All needed are in the picture above (i hope so) The Attempt at a Solution to me it is extremely difficult because it is so complicated with many notations. Also, I actually don't know how to read the question properly to answer it Is E(beta) is the...
  21. Pouyan

    Probability theory and statistics

    Homework Statement The time (minute) that it takes for a terrain runner to get around a runway is a random variable X with the tightness function fX = (125-x)/450 , 95≤x≤125 How big is the probability of eight different runners, whose times are independent after 100 minutes: a) Everyone has...
  22. Pouyan

    A paradox in probability theory and statistics

    Homework Statement In a vessel is a 5 cent coin and two 1-cent coins. If someone takes up two randomly chosen of these coins, and we let X be the total value of the coins taken, what is the probability function for X? Homework Equations I know that X has a value {2,6} The Attempt at a...
  23. S

    Proving the Continuity From Below Theorem

    Homework Statement Prove the continuity from below theorem. Homework EquationsThe Attempt at a Solution So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to...
  24. Aslet

    I A simple question about probability theory

    Hello everyone! I'm studying the physics of complex systems and I'm approaching probability theory. I understand that we need a ## \sigma-algebra ## and the Kolmogorov axioms in order to define a probability space but then I bumped into the following relation: $$ p(A_1 \cup A_2 ) = p( A_1 ) + p(...
  25. D

    Probability theory and statistics for Robotics and ME

    I study control theory and robotics. Recently I figured out that I have a much deeper understanding of probability and statistics compared to my colleagues. Is this 'talent' valuable in my field and if so, where? We used this theory to define white noise, but nothing more...as of now. Also I am...
  26. bhobba

    A State Space and Probability Theory

    Hi All This is in relation to the folllowing paper: https://arxiv.org/pdf/1402.6562.pdf See section 3 on examples where standard probability theory is discussed. Is it valid? To me its rather obvious but I had had a retired professor of probability say probability theory doesn't have a state...
  27. A

    I Question: Proposed Solution to Two Envelope Paradox

    Su, Francis, et. al. have a short description of the paradox here: https://www.math.hmc.edu/funfacts/ffiles/20001.6-8.shtmlI used that link because it concisely sets forth the paradox both in the basic setting but also given the version where the two envelopes contain ( \,\$2^k, \$2^{k+1}) \...
  28. Hughng

    Undergraduate Probability Textbook with Multiple Choice Practice Problems

    Would you please introduce me some textbooks that offer multiple choice or true-false questions at the end of each chapter so I can practice before my final exam? Basically, that class I am taking right now cover many central topics in Undergraduate Probability course such as: • Moments and...
  29. Demystifier

    I Is probability theory a branch of measure theory?

    In another math thread https://www.physicsforums.com/threads/categorizing-math.889809/ several people expressed their opinion that, while statistics is a branch of applied mathematics, the probability theory is pure mathematics and a branch of analysis, or more precisely, a branch of measure...
  30. mr.tea

    Prob/Stats Probability with measure theory

    Hi, I am looking for a book for studying probability theory using measure theory. This is the first course I am taking of probability. Notions and theorems from measure theory are part of this course. As it turns out, this is a catastrophic disaster, and the textbook for this course is also not...
  31. B

    Prob/Stats Seeking a good introductory book in probability theory?

    Dear Physics Forum personnel, I would like to seek your recommendation on a good, introductory textbook in the probability theory (non measure-theoretic treatment) that contains both the applied and theoretical treatment of the subject. My goal is to advance into the measure-theoretic...
  32. J

    Autocorrelation function of a Wiener process & Poisson process

    Homework Statement 3. The Attempt at a Solution [/B] ***************************************** Can anyone possibly explain step 3 and 4 in this solution?
  33. J

    Limit of a continuous time Markov chain

    Homework Statement Calculate the limit $$lim_{s,t→∞} R_X(s, s+t) = lim_{s,t→∞}E(X(s)X(s+t))$$ for a continuous time Markov chain $$(X(t) ; t ≥ 0)$$ with state space S and generator G given by $$S = (0, 1)$$ $$ G= \begin{pmatrix} -\alpha & \alpha \\ \beta & -\beta\...
  34. J

    Inequality involving probability of stationary zero-mean Gaussian

    Homework Statement Let $$(X(n), n ∈ [1, 2])$$ be a stationary zero-mean Gaussian process with autocorrelation function $$R_X(0) = 1; R_X(+-1) = \rho$$ for a constant ρ ∈ [−1, 1]. Show that for each x ∈ R it holds that $$max_{n∈[1,2]} P(X(n) > x) ≤ P (max_{n∈[1,2]} X(n) > x)$$ Are there any...
  35. J

    Birth and death process -- Total time spent in state i

    Homework Statement Let X(t) be a birth-death process with parameters $$\lambda_n = \lambda > 0 , \mu_n = \mu > 0,$$ where $$\lambda > \mu , X(0) = 0$$ Show that the total time T_i spent in state i is $$exp(\lambda−\mu)-distributed$$ 3. Solution I have a hard time understanding this...
  36. BobblyHats97

    If X ∼ Uniform(−1, 1) find the pdf of Y = |X|

    This question is killing me. I know the graph is non-monotonic so i have to split up finding F(Y) for -1<Y and Y<1 but then what do I do with the modulus? >.< Any help would be greatly appreciated! Thank you so much x
  37. Matejxx1

    Probability and Events (I don't quite understan the answer)

    Homework Statement An only child Urška puts 3 pieces of paper into a bag : on each piece of paper is written a name of one of her classmates :David,Niko,Dejan (those are the 3 names): She then randomly picks 2 pieces of paper from the bag and checks them Match the statements on the right with...
  38. R

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

    Prove or disprove the following statement: If p(a)=p(b)=q then p(a∩b)≤q2 We know nothing know about event a , b. The Attempt at a Solution I tried this but don't know correct or not Can some one help me let a, b are independent event 0<q<1 then p(a∩b) = p(a) p(b) = q*q = q^2 [/B]
  39. P

    Calculating permutations for a normally distributed variable

    For three dice, you can have 6 * 6 * 6 = 216 permutations (order matters). The dice has a uniform probability distribution of p(x) = 1/6. Easy. I'm trying to get an estimate of how many permutations you can have if a variable has a normal probability distribution. So for example, if a...
  40. M

    Upper level probability theory over summer

    I need another class for a 6 week summer semester and I'm curious if probability theory is generally a class you wouldn't want to cram in 6 weeks with another class? The only college level probability I've done was in a discrete math course but I'm fine with other areas since I also took...
  41. K

    Calculating log liklihood: Zero value of likelihood function

    Hello, I am analysing hydrology data and curve fitting to check the best probability distribution among 8 candidate distribution. (2 and 3 parameter distributions) The selection is based on the lowest AIC value. While doing my calculation in excel, how is it suggested to treat very low (approx...
  42. M

    Structure of generated sigma algbra

    I am think what is the structure of generated ##\sigma##-algebra. Let me make it specific. How to represent ##\sigma(\mathscr{A})##, where ##\mathscr{A}## is an algebra. Can I use the elements of ##\mathscr{A}## to represent the element in ##\sigma(\mathscr{A})##?
  43. M

    Extension of measure on sigma-algebra

    Suppose ##\mu:\mathcal{F}\rightarrow[0,\infty)## be a countable additive measure on a ##\sigma##-algebra ##\mathcal{F}## over a set ##\Omega##. Take any ##E\subseteq \Omega##. Let ##\mathcal{F}_{E}:=\sigma(\mathcal{F}\cup\{E\})##. Then, PROVE there is a countable additive measure...
  44. J

    Limit involving extinction probability of branching process

    Let x(a) be the extinction probability of a branching process whose offspring is Poisson distributed with parameter a. I need to find the limit as a approaches infinity x(a)e^a. I tried computing x(a) directly using generating functions, and I found that it's the solution to e^(a(s-1))=s, but...
  45. A

    Probability theory question

    Homework Statement Alice attends a small college in which each class meets only once a week. She is deciding between 30 non-overlapping classes. There are 6 classes to choose from for each day of the week, Monday through Friday. Trusting in the benevolence of randomness, Alice decides to...
  46. B

    Winter break math study

    I'm a grad student studying electrical/computer engineering. Since I have a month of winter break coming up soon, I want to use it to study some more about probability theory and stochastic processes. Has anyone previously done a self study or partner study over a break like this? If so, how did...
  47. micromass

    Is *-Algebra the Key to Understanding Quantum Probability Theory?

    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...
  48. R

    MHB Probability Theory: Q1, Q2, and Q3

    Q1. There are n cells and each cell contains k balls. One ball is taken from each of the cells. Find the probability that the second lowest label from the balls drawn is m. Q2. Game played by two friends: each player picks two balls. The person who gets the first white ball in the second draw...
  49. 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...
  50. F

    Basic Probability Theory (Equaly Likely Principle)

    Homework Statement Five cards numbered 1 to 5 are shuffled and placed face down on a table. Two of the cards are picked at random. [Hint: find all of the possible outcomes of this experiment which form the sample space S and use the Equally Likely Principle.] Find the probability of the...