What is Probability: Definition and 1000 Discussions

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

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  1. Juanda

    I Adding random numbers: Tolerance analysis

    I am not a fan of random and statistics. I know it is extremely useful and probably the mathematical branch more applicable to real life to understand the world around us but I am a Calculus and Vectors boy. This problem though I find interesting. I would like to find a generalized solution for...
  2. T

    A combinatorial probability question

    I've a small molecule ABCD made of building blocks A,B,C,D. The molecule can get cleaved at any 'bond' between the building block one bond at a time. Researchers have compiled probabilities from dissociation studies of large number of molecules( made up of many other building blocks) and the...
  3. W

    B Are both of these the same thing? (logical NOT and the complement rule)

    Hi everyone, This is an example of binary variable called as logical NOT https://www.fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/chap2s1.html?scroll=sseclognot ...and this is the complement rule of probability...
  4. D

    B Roulette wheel physics and probability

    Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is...
  5. Hennessy

    Help Calculating probability between 2 limits for the ground state

    TL;DR Summary: Looking for help on a Intro to QM Problem Hi All, THIS IS A GRADED PIECE OF WORK AT MY UNIVERSITY PLEASE DO NOT JUST GIVE ME THE ANSWER , I have made this post to see if what i've calculated seems reasonable, it sounds unlikely as 0.4 - 0.5L is in the middle of the well. The...
  6. W

    B Why did it suddenly become subtractive? (Example of Bayes’ Theorem)

    From https://corporatefinanceinstitute.com/resources/data-science/bayes-theorem/#:~:text=Formula for Bayes' Theorem&text=P(A|B) –,given event A has occurred Example of Bayes’ Theorem Imagine you are a financial analyst at an investment bank. According to your research of publicly-traded...
  7. T

    I A non intuitive probability?

    In Aubrey Clayton's book" Bernoulli's Fallacy" which documents the conflict between frequentists and Bayesian interpretations of probability, he describes a problem that was proposed in the 19th century that gives a counterintuitive result. The Problem: "Infer the state of a bag of 3 balls...
  8. K

    I Are different interpretations of probability equivalent?

    There appear to be at least three concepts of probability. In my words frequencies in a history (theretical and measured) reasonable expectation propensity to outcomes There may be more. I am wondering whether these are actually different meanings, which could affect how probability is used in...
  9. K

    I Why is there no consensus about the meaning of probability in MWI?

    As there appears to be no consensus about the meaning of probability in a deterministic model, I am asking what the sticking point is? That's all really.
  10. MAXIM LI

    Limit of probabilities of a large sample

    My first thought as well but I think the problem is deeper than that. I think that as the n tends towards infinity the probability of the the sample mean converging to the population mean is 1. Looking at proving this. By the Central Limit Theorem the sample mean distribution can be approximated...
  11. J

    I Probability, observers and the multiverse

    Not sure if I'm putting this in the right place! I have a question about probability and conscious observers. Aside from other arguments for and against a multiverse, does the idea that a multiverse could contain a vast number of consicous observes make it more likely, given that we find...
  12. WMDhamnekar

    I Computing the expectation of the minimum difference between the 0th i.i.d.r.v. and ith i.i.d.r.v.s where 1 ≤ i ≤ n

    Problem :Let ##X_0,X_1,\dots,X_n## be independent random variables, each distributed uniformly on [0,1].Find ## E\left[ \min_{1\leq i\leq n}\vert X_0 -X_i\vert \right] ##. Would any member of Physics Forum take efforts to explain with all details the following author's solution to this...
  13. chwala

    Solve the given problem involving conditional probability

    Phew! took time to figure this out...i guess there may be a way to use combinations or markov process i do not know... anyway, it was pretty straightforward, we have the ##P_r(w) = \dfrac{n-3}{n}## from box ##X## and this will result in ##P_r(w) = \dfrac{4}{n+1}## in box ##Y##. Together i...
  14. MichPod

    I A probability of field amplitude in QFT

    Per quantized scalar field (quantized Klein-Gordon equation), suppose we act on a vacuum state |0> with some set of creation operators to have some particles. How then can we calculate a probability density for the field to have a particular value ##\psi_0## (upon measurement) at a specific...
  15. B

    B Entanglement & Superposition Probabilities

    I cannot find a clear answer on the following beginner’s question on some QM fundamentals: Suppose we have two particles, A and B. Let’s say we generated these as (or otherwise entangled them as) an entangled pair with opposite/orthogonal states. Perhaps horizontally and vertically polarized...
  16. M

    Multi-event probability puzzle - is my answer correct?

    Here is my attempt. Beginning state: Bag B : B, B, O Bag A : R, R, G, V, Y Final state: Bag B: B, B, O, + G/V/Y Bag A: remaining balls First possible exchange that would have exactly 3 different colors in each bag is: Move 1: P ( Arjun moves either the green, violet, or yellow ball to...
  17. R

    I Variation of the Liar's Paradox

    A variation of the Liar's Paradox occurred to me: "Statistics are wrong 90% of the time". This statement seems to refute itself, but does so in a less straightforward way. I would appreciate any insights! And what about, "Statistics are wrong 50% of the time"? (Even odds.)
  18. E

    I TOUGH Textual Criticism math statistics question

    Okay let me rephrase this math question and frame it. It is math dealing with ancient Biblical texts and textual criticism. . Codex 01 (350AD) agrees with the MT (mjority text) about 87% of the time. Codex 03 (350AD) Agrees with MT about 87% of the time. 01 and 03 agree with each other...
  19. C

    A Are these results possibly the outcome of a probabilistic calculation?

    Hello, I am seeking the opinion of specialists in statistics and probabilities to evaluate the results of a forecast on a univariate time series derived from an experiment at hazcard.com. They are presented as obtained from a logic that includes no probability calculation, no learning, no use of...
  20. A

    B Definition of a random variable in quantum mechanics?

    In a line of reasoning that involves measurement outcomes in quantum mechanics, such as spins, photons hitting a detection screen (with discrete positions, like in a CCD), atomic decays (like in a Geiger detector counting at discrete time intervals, etc.), I would like to define rigorously the...
  21. S

    Prob/Stats Probability book for basic understanding

    I want to learn topics related to combinatorics, probability theory, discrete and continuous random variables, joint pdf and cdf, limit theorems and point estimation, confidence intervals and hypothesis testing. Any recommendations for books to learn those topics? High school level or...
  22. S

    Finding probability related to Poisson and Exponential Distribution

    My attempt: (i) ##\lambda =3## (ii) (a) ##P(N_{2} \geq 1=1-P(N_{2} =0)=1-e^{-6} \frac{(-6)^0}{0!}=0.997## (b) ##P(N_{4} \geq 3)=1-P(N_{4} \leq 2)=0.999## (c) ##P(N_{1} \geq 2) = 1-P(N_{4} \leq 1)=0.8## Do I even understand the question correctly for part (i) and (ii)?(iii) The expectation of...
  23. Memo

    Combination, partial permutation

    a) p=(4C1*6C2)/(10C3)=0.5 b) p=(4C1*6C2)/(10C3) + (4C2*6C1)/(10C3) + (4C3*6C0)/(10C3)=0.83 Please check if my answer is correct. Thank you very much.
  24. Hamiltonian

    Drawing marked balls from an urn

    part (a) was straightforward ##\mathcal{P} = \frac{20}{200} = 0.1##. Instead of directly trying to find the probability of the 20th drawn ball being marked I decided to start with finding the probability of the second ball drawn being marked and then after figuring that out moving to the cases...
  25. S

    Probability related to cumulative distribution function

    I have tried to answer all the questions but I am not that sure with my answer. That's the graph of ##F_X (x)## (I think) (i) P (X ≤ i) = ##\frac{i^2}{N^2}## and P(X < i) = 0 All of these are based on the graph (ii) P(X = i) = P(X ≤ i) - P(X < i) = ##\frac{i^2}{N^2}## Are my answers...
  26. tworitdash

    I Parameters of a distribution of a physical variable

    Pardon me if this is a very silly question. Although my research involves a lot of probability distributions, I consider myself a fledgling statistician. When people assign a probability distribution to a variable in a physical process, is it inherently assumed that the parameters of this...
  27. WMDhamnekar

    I How to obtain moment bound from the importance sampling identity?

    Let ##X## be a non-negative random variable and let a > 0. We want to bound the probability ##P\{X \geq a\}## in terms of the moments of X. - Define a function ##h(x) = \mathbb{1}\{x \geq a\}##, where ##\mathbb{1}\{\cdot\}## is the indicator function that returns 1 if the argument is true and 0...
  28. S

    A Calculation of average damage using analytical methods

    Hello! A friend shared a problem he recently solved. It goes as follows: Given: Each dagger strike deals either normal damage = 20 or critical damage = 80. After each strike, the probability of dealing normal and critical damage changes (initial probabilities are 90% and 10% respectively). The...
  29. Kinker

    I Impossible probability between theoretical possibility and reality

    Boltzmann's brain, entropy reduction, Poincaré's recursion theorem, the probability of oxygen molecules in a room gathering in one place, the probability of quantum tunneling of macroscopic objects, etc. are theoretically possible. But the probability of these events is very low. Additionally...
  30. N

    Probability using combinations or alternatives

    I have attempted the solution above and I am fairly sure that it is correct. My question is the following: What am I calculating if I multiply 4/52 x 3/51 x 4/50 x 3/49 = 144/ 6,497,400 = 0.002%. I got the correct answer by following the principles of combination probabilities, but intuitively...
  31. S

    Number of ways to partition n persons and probability to form n groups

    1) At first my answer was ##n! \begin{pmatrix} n+r-1 \\ r - 1 \end{pmatrix} ## But I think that's not correct because let say first group consists of person A and B, by multiplying with n!, I also consider first group to be B and A which is just the same as A and B so there is double counting...
  32. MathMan2022

    Basic probability question (1 boy + 2 girls of a group of 7)

    I know this problem can be done as follows. P(1 boy and two girls) = (C(2,1)*C(5,2))/C(7,3) = 20/35 My question can this be written as probility fractions? Meaning Lets say if that (2/5*3/7+1/2*1/7)/(3/7) But that doesn't give same result? what am I doing wrong?
  33. H

    B An Observed Extreme Probability Event

    At the 2005 World Bridge Championships in Estoril a hand was dealt with QT8642 in two different suits and with the remaining card even. The odds of this happening by pure chance are one in 531 trillion. We can say that AQT864 and KJ9753 would form an equivalent class of "hands spaced by two"...
  34. ARoyC

    Questions from Quantum Measurements

    [Mentor Note: Two similar thread starts merged] The questions are from MIT OCW. First of all, I cannot understand what is the meaning of the measurement outcome being 0. How can an eigenvalue be 0? I tried doing the problems guessing that by 0 they mean the posterior state will be |0>. The only...
  35. J

    B Multiply Probabilities vs. Sum of the Squares

    Hi! I'm getting confused by these two things. If I have two uncorrelated probabilistic events, and I want to know the probability of seeing them both land beyond 3.3 sigma (for example), do I multiply the probabilities .001*.001 or do I do sum of the squares sqrt(.001^2 + .001^2). I assume it is...
  36. shivajikobardan

    Comp Sci Monty Hall Game Show Problem in Javascript

    The following problem is sometimes called “The Monty Hall Game Show Problem.” You are a contestant on a game show and have won a shot at the grand prize. Before you are three closed doors. Behind one door is a brand new car. Behind the other two doors are consolation prizes. The location of the...
  37. M

    Hello friends!

    Post-grad, my background is in mathematical physics, probability/statistics, and information theory. I am here for discussion and collaboration on things I find interesting from time to time.
  38. Infrared

    Challenge Math Challenge - June 2023

    Welcome to the reinstatement of the monthly math challenge threads! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by...
  39. hjam24

    I Write probability in terms of shape parameters of beta distribution

    Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$ If we know that A wins, his score is specified by B's score; he...
  40. A

    I Probability to get a certain number or less by throwing two 6-faced die

    In a board game, I need to reach a certain place in a board divided into boxes, I move by throwing 2 six-faced dices. If my goal is 4 boxes away I need to obtain at least a combination of numbers that sums to 4, but any higher number is also a favorable outcome. I want to calculate the...
  41. T

    Chance of Selecting Twins from Group of 30 [Solved!]

    TL;DR Summary: Chance of picking 2 named people when randomly choosing 3 from a group of 30. For my daughter's homework question: There is a group of 12 girls and 18 boys. Two of them are twins (girl and boy). If I select three at random, what is the chance that the twins will be chosen? I...
  42. WMDhamnekar

    I How to determine if a set is a semiring or a ring?

    Let E be a finite nonempty set and let ## \Omega := E^{\mathbb{N}}##be the set of all E-valued sequences ##\omega = (\omega_n)_{n\in \mathbb{N}}F##or any ## \omega_1, \dots,\omega_n \in E ## Let ##[\omega_1, \dots,\omega_n]= \{\omega^, \in \Omega : \omega^,_i = \omega_i \forall i =1,\dots,n...
  43. M

    I Help Needed: Probability Problem in Risk Management Calculcations

    Hello all! I hope I have come to the right place, and I appreciate any help! My first disclaimer is that I am not a math professional of any sort, I am not bad at it but I just wanted to start with that so if I ask a stupid question, it's because I am ignorant mostly. What I work in is risk...
  44. C

    A Runners in a race, probability paradox

    There are a number n of runners in a race. We know their expected times from start to finish μ(i) and the corresponding standard deviations σ(i). The probability of runner 0 to finish first is given by this integral: It's from here: https://www.untruth.org/~josh/math/normal-min.pdf The 0 is...
  45. tworitdash

    I Making Sense of Notation Confusion in Statistical Digital Signal Processing

    I started my research in statistical digital signal processing two years ago, so I need to familiarize myself with all the notations people use in probability and statistics. I come from a deterministic science background. I name my variables based on what they mean. A velocity is a v , a...
  46. CoNiss

    A Calculating Probability of N Points on a Line Being Within Given Distance

    Probability of any random n points on a line being within a given distance Hi, I am a software engineer trying to solve the following problem analytically given a line segment in cm and n random points on it what is the probability that the distance between any 2 consecutive points on the...
  47. red65

    B Concerning the birthday problem in probability

    The problem is stated like this : There are k people in a room. Assume each person’s birthday is equally likely to be any of the 365 days of the year (we exclude February 29), and that people’s birthdays are independent (we assume there are no twins in the room). What is the probability that two...
  48. E

    Ensemble vs. time averages and Ashcroft and Mermin Problem 1.1

    The question is as seen below: My attempt (note that my questions are in bold below) is below. Please note that I am self-studying AM: (a) By the independence of any interval ##dt## of time and time symmetry, we expect these two answers are the same (Is there any way to make this rigorous?)...
  49. F

    I Sample space, outcome, event, random variable, probability...

    Hello, I am solid on the following concepts but less certain on the correct understanding of what a random variable is... Random Experiment: an experiment that has an uncertain outcome. Trials: how many times we sequentially repeat a random experiment. Sample space ##S##: the set of ALL...
  50. N

    B Head or Tails: The Question of Determinism and Probability

    I have a question that is bothering me. It is commonly accepted that when playing heads or tails with a fair coin and a large number of tosses are made, the probabilities of getting heads or tails are equal to 50% for each toss. However, the principle of determinism, which states that under the...
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