What is Probability theory: Definition and 104 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. 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...
  2. 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...
  3. 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...
  4. V

    Probability theory and quantum mechanics

    In probability theory a sample space is a set containing all possible outcomes of an experiment and an event is a subset of the sample space (an element of its power set). I think it would be natural to think of the basis of the vector space representing a quantum system as a sample space and...
  5. M

    Probability Theory: Poisson Distribution

    Homework Statement A random variable has a Poisson distribution with parameter λ = 2. Compute the following probabilities, giving an exact answer and a decimal approximation. P(X ≥ 4) Homework Equations P(X = k) = λke-λ/k! The Attempt at a Solution P(X ≥ 4) = Ʃk = 4∞...
  6. T

    Problem with probability theory and random variables

    Hello. I have a problem with probability theory task. The task is: X and Y is independent random variables with same density function fx=fy=f. What will be probability of P(X>Y). This P(X>Y) reminds me a cdf: P(X>Y)=1-P(X<Y)=1-cdf of X. Cdf of x is equal to integral ∫f dx from -inf to...
  7. T

    Problem with probability theory task

    Hello. I have problem with probability theory task. Sorry for my english but i'll try to define the task. There are four classmates. Ana, Beta, Ceta and Deta. During the break all of them tiffed (probability is p) or became best friends (probability is 1-p). And all with each other tiffed or...
  8. micromass

    Prob/Stats An Introduction to Probability Theory and Its Applications by Feller

    Author: William Feller Title: An Introduction to Probability Theory and Its Applications Amazon Link: https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20 https://www.amazon.com/dp/0471257095/?tag=pfamazon01-20 Prerequisities: Table of Contents for Volume I: Introduction: The Nature of...
  9. T

    Construction of coupling and maximal coupling (probability theory)

    Homework Statement Let U, V be random variables on [0,+\infty) with probability density functions f_U(x)=2e^{-2x} and f_V(x)=e^{-x}. 1. Give a coupling of U and V under which \{U\geq V\} with probability 1. 2. Give a maximal coupling of U and V. Homework Equations Cumulative distribution...
  10. A

    Probability theory. quick question regarding conditionalizing the binomial dist

    Hello, so suppose we have B(n,p), where n is discretely uniformly distributed on the integers of the interval (1,5) Is the expected value 3p, and is the variance 3p-p^2 ? I arrived at those answers by treating n as another variable, so np/5 summed over all n is 3p, and similar logic for...
  11. N

    [probability theory] prove E(X|X) = X

    Homework Statement Well, not really, but in essence that's the part I'm having trouble with. The actual question is The equality seems obvious enough, but I'm unsure how to actually prove it... Homework Equations N/A The Attempt at a Solution So it seems I have to prove that P( \{ E(Xh(Y)|Y)...
  12. B

    Please check my work: Probability Theory

    Let K be a standard normal random variable. Find the densities of each of the following random variables: X= |K| Y = K2 I get: fX(x) = √(2/π) e-x2/2 and fY(y) = 1/√(2*π) 1/√y e-y2/2
  13. D

    Probability Theory- Standard Normal Distributions

    Homework Statement Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall...
  14. BWV

    Does QM ever violate classical probability theory?

    reading this http://uk.arxiv.org/abs/1004.2529 about supposed parallels between the mathematical structure of probability in QM and some problems in economics question is that are there really any violations of classical probability theory, such as Pr(A) > Pr(A \cup B) in QM? The supposed...
  15. Q

    Start Learning Probability Theory Now

    Probability theory... It may make on sense but I want to understand probability intuitively...!...can anyone explain...?how should i start...?
  16. R

    Why are sigma fields significant in probability theory?

    As the title. Why are sigma fields important in probability? The only one reason I can think of is that sigma fields are used as domain, e.g. borel fields uses sigma fields instead of power set. However, are there any other significances of sigma fields in probability theory? Thanks for your...
  17. K

    Probability theory question (mini max functions)

    Homework Statement EX=EY=5, VarX=1, VarY=sigma^2 >1 Z=aX+(1-a)Y, 0<=a<=1 find a that minimizes VarZ, and another a that maximize VarZ Homework Equations The Attempt at a Solution Not even sure where to begin *EX=5, VarX=1 thus EX^2 = 26 marginal px(x) =26/x^2 = 5/x but...
  18. A

    How do you think about probability theory?

    I find the probability theory I'm doing in college very difficult until I start wording it all out in my head. If I word it out then there's no confusion about what P(A) represents and what P(B|A) represents etc. but if I don't word it out then I have trouble thinking about it. I think its the...
  19. D

    Probability theory - Poisson and Geometric Random Variable questions

    Homework Statement [/b] There are two problems I need help with, which are posted below. Any help is appreciated. 1)Let X have a Poisson distribution with parameter λ. If we know that P(X = 1|X ≤ 1) = 0.8, then what is the expectation and variance of X? 2)A random variable X is a sum of...
  20. M

    What should I study to best learn probability theory?

    I have the choice of taking, next spring, a course on linear algebra (theoretical, covers linear operator theory) or a course on applied math (covers asymptotic analysis and pdes). I cannot take both because i would be taking too many classes at once, and the classes i am already planning on...
  21. R

    Probability Theory - Expectation Problem

    Homework Statement Discrete random variables X and Y , whose values are positive integers, have the joint probability mass function , (, ) = 2−−. Determine the marginal probability mass functions () and (). Are X and Y independent? Determine [], [ ], and [ ]. The Attempt at a Solution...
  22. M

    What is the Correct Probability in Unorthodox Weather Prediction Methods?

    Here is a simple problem but with a lot of hidden difficulty: We have a weather station somewhere in the country, with the aid of satelitte information and it gives us the probability of rain every day. Suppose it is an average value p, for the season. At some other place there is an...
  23. S

    Conditional Expectation Question (Probability Theory)

    Homework Statement (Question is #6 on p.171 in An Introduction to Probability and Statistics by Ruhatgi & Saleh) Let X have PMF Pλ{X=x} = λxe-λ/x!, x=0,1,2... and suppose that λ is a realization of a RV Λ with PDF f(λ)=e-λ, λ>0. Find E(e-Λ|X=1) The Attempt at a Solution The...
  24. F

    Associative Law for Multiplication - Probability Theory and Boolean Algebra

    Hi, I'm having some trouble solving one of the problems from my homework assignment. Homework Statement Prove: P((AB)C) = P(A(BC)) Where A,B,C are either true or false. Homework Equations We can't do this by using a truth table, we can use the following equations: P(A + B) = P(A) + P(B) -...
  25. S

    Probability Theory proof help?

    Homework Statement Let X be a geometric(p) random variable for some p ∈ (0, 1). For n, k ∈ N, show that P(X = n + k |X > n) = P(X = k). Now, explain this property using the interpretation of X as the first successful trial among independent trials each of probability p. Homework...
  26. F

    A good book for probability theory? Any recommendations?

    Hi all. I am searching for a good textbook on probability theory. I finished my B.S. in mathematics last fall and have covered real analysis and abstract algebra so I want a book that goes into actual mathematical theory and not just a basic book about probability like you would learn in...
  27. D

    Probability Theory: Is A Certainty with C & T?

    Homework Statement A - something that is a known possibility C - correct conditions conducive to produce A T - many trillion years in which C continues Homework Equations Is it true that with the above factors the probability of "A" is "1", or very close to 1? The Attempt at a...
  28. F

    Minimized Risk (Probability Theory)

    Homework Statement The rv X has pdf f(x) = cx, for 1 < x < 2, with f(x) = 0 otherwise. Compute the 3 M’s, and also compute the minimized risks in each case. Mode, median, mean. Homework Equations The Attempt at a Solution I think I have computed the 3 M's mode of x = 2 median...
  29. R

    [probability theory] simple question about conditional probability

    Hi all, I've got this very simple problem: I know it is an elementary problem, but I never really got into that bayes' theorem, which I need to use here, right? I would be grateful for simple and plain explanation. thanks for your time, rahl.
  30. A

    Probability Theory ; Binomial Distribution?

    Homework Statement Now you and your fiend play a different game. You flip your coin until it comes up heads the first time. Let X denote the number of flips needed. Your friend rolls its die until it comes up "3" or "5". The first try let Y denote the number of rolls needed. Assume X and Y are...
  31. B

    Advanced probability theory books?

    I'm interested in learning the calculus of general random variables, i.e. those that do not necessarily have a density or mass function - such as mixtures of continuous / discrete / Cantor-type variables. There seem to be several different approaches: 1. Via densities, using delta...
  32. V

    |R| ≠ |R²| in probability theory

    According to Cantor, http://books.google.com/books?id=Er1r0n7VoSEC&pg=PA98&dq=plane+line+cardinality+%22set+theory%22&as_brr=0" example). Do you see the contradiction |R| = |R2| vs. |R|/|R2| = 1/∞? Am I missing something?
  33. W

    Could any one suggest one book on probability theory to me?

    i am working on theoretical physics i hope i can get a book on probability theory i hope this book is a bit advanced to have detailed discussion of characteristic functions and their applications. i do not like the statistics applications. I hope that part is as small as possible...
  34. V

    What is the best book for learning probability theory as a beginner?

    i have to study following topics in probability theorem and iam totally a beginner so please suggest me a book Probability, conditional probability, random variables, Expected Value, Specific discrete and continuous distributions, e.g. binomial, Poisson, geometric, Pascal, hypergeometric...
  35. M

    Probability of X<4 for X~Bi(5,0.2)

    For a random value X~Bi(5,0.2) evaluate the probability of X<4. Any ideas? What is Bi?
  36. R

    (True/False) Basic Probability Theory

    Dear all, I have a question. Suppose we have 3 events X,Y,Z defined as having 200 heads, 400 heads & 600 heads obtained in tossing a fair coin for 800 times. Then, P(Z)=P(X+Y)=P(600)=P(200+400)=P(X)+P(Y)=P(200)+P(400) The answer is false but I view it otherwise. My argument is based...
  37. D

    Probability theory: a regenerative process

    Homework Statement http://img291.imageshack.us/img291/4844/14820448wb9.png The Attempt at a Solution First of all I'm trying to find the expected time of a cycle. In a cycle two things can happen: 1) the car lives long enough to reach A with probability 1-F(A) 2) the car fails to...
  38. T

    How hard is a intro to probability theory

    My background is calculus through differential equations and linear algebra at a community college. How hard is an intro to probability theory at a school like UCLA?http://www.math.ucla.edu/ugrad/courses/math170ab/170Aoutline.shtml I'm thinking about taking it this fall.
  39. E

    Learn Probability Theory: Find a Book to Self-Study

    What book should I get to learn probability theory by self-study? I bought Shiryaev's Graduate Text in Mathematics and the problem is that it just develops so much theory but then provides few exercises, making it really hard to self-study. I probably should have expected this though. So, I am...
  40. P

    Entanglement Explained with Epistomological Probability Theory

    I'm interested in published papers, if any, addressing the issue of whether Bell type experiments can be explained simply with classical, epistomological probability theory. In particular, can the expected "quantum" result (i.e. the probability that photons A and B will both pass through their...
  41. M

    Probability Theory 2: Finding E[X_n], Var(X_n)

    Homework Statement An individual traveling on the real line is trying to reach the origin. However, the larger the desired step, the greater is the variance in the result of that step. Specifically, whenever the person is at location x, he next moves to a location having mean 0 and variance...
  42. M

    Probability Theory 2: Finding Mean and Variance of X_n on Real Line

    An individual traveling on the real line is trying to reach the origin. However, the larger the desired step, the greater is the variance in the result of that step. Specifically, whenever the person is at location x, he next moves to a location having mean 0 and variance \beta x^2. Let X_n...
  43. S

    How Do You Prove Basic Probability Theory?

    Hello There, this is my first post. I would like an information: Which is the basic way to prove probability theory?.. I mean prove the probability that a situation in influenzed from the situation before and/or calculate the probability that in n it will have a determinate situation...?? How...
  44. F

    Proving Independence: Probability Theory of A and B in a Simple Proof"

    If A and B are independent, prove that \bar A , \bar B are independent. Could someone help me start this. It's due tomorrow. I managed to prove that \bar A [/tex] is independent with B and that \bar B is independent with [itex] A , but I can't get the last one (the question I put...
  45. B

    Calculating Probability of Even Numbers with Loaded Die

    Homework Statement A die is loaded so that the numbers 2 through 6 are equally likely to appear but that 1 is 3 times as likely as any other number to appear. What is the probability of getting an even number?Homework Equations This is where I become lost - I'm not sure how to get the...
  46. F

    Probability Theory - conditional

    Question: Deer ticks can carry both Lyme disease and human granulocytic ehrilichiosis (HGE). IN a study of ticks in the Midwest, it was found that 16% carried Lyme disease, 10% had HGE, and that 10% of the ticks that had either Lyme disease or HGE carried both diseases. (a) What is the...
  47. F

    What Constitutes the Sample Space in a Birthday Probability Problem?

    Question: Find out the birthday (month and day but not year) of a randomly chosen person. What is the sample space of the experiment. How many outcomes are in the event that the person is born in July? Attempt: We first must define the sample space. This is how I did it, S = {JAN1...
  48. F

    Probability Theory: Shuffling a Deck of Cards

    EDIT: Please disregard, or delete. I got it. Stumped on this question: Shuffle a deck of cards and turn over the first card. What is the sample space of this experiment? How many outcomes are in the event that the first card is a heart? Attempt at a solution: D = \{ deck of 52...
  49. R

    Understanding Probability Theory: P(min(X, Y ) > x) Explained

    Probability theory... So we were given some practise problems for the exam... in it, we got a question that we have NEVER seen in class nor can I find it in the textbook. The question is: Find P(min(X, Y ) > x) and hence give the probability density function of U = min(X, Y ). Okay...
  50. P

    Calculating Probabilities for Independent Bernoulli Trials

    A Question Reads: "Suppose that the random Variable X is the number of failures before the first success in a series of independent Bernoulli trials with success probability p" a) derive the probability mass function of X b) what is the probability that X < x where x is a positive integer...
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