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. G

    I Rewriting of equality in conditional probability distribution

    I don't get $$\frac{P[x<X<x+dx|N=n]}{dx}=f_{X|N}(x|n)$$ Can someone derive why? I would believe that $$f_{X|N}(x|n)=\frac{f(x,N)}{p_n(N)}$$ but I don't get how that would be the same. And I don't get that $$\frac{P[x<X<x+dx|N=n]}{dx}=\frac{P[N=n|x<X<x+dx]}{P[N=n]}\frac{P[x<X<x+dx]}{dx}$$ Can...
  2. J

    Probability that a bulb lasts for at least 7 months

    Homework Statement Homework Equations I don't even know how to start this. Should I use Poisson's distribution. Or assume f(x) = exp(-x) And then since mean is given we have integration from 0 to infinite xf(x) is mean. Not sure what this will give. The Attempt at a Solution Let f(x) be...
  3. J

    Probability that both people are from group X

    Homework Statement Homework Equations Probability = number of favourable events / all possible events The Attempt at a Solution Group X Y Total people Indians 10 8 18 (total 18 Indians in both group) Total People 25 20 45 (total 45 people in both...
  4. D

    Probability of spin-flip due to scattering

    Homework Statement A beam of spin-1/2 particles scatters off of a target consisting of spin-1/2 heavy nuclei. The interaction between the particle and nucleus is given by $$V(\vec{ r})= V_0~\delta (\vec{r})~ \vec{S}_1. \vec{S}_2$$ 1) Averaging over initial spin states, find the differential...
  5. jocarren

    I Probability & Time: Is There a Connection?

    Hi, I'm new in the forums, actually I registered to ask this question (I have these wild, often ridiculous ideas). Is there a fundamental relationship betwen the probability of ocurrence of an event and the flow of time around that event? I mean, imagine an isolated space where some type of...
  6. Clifford Engle Wirt

    B Conditional Probability, Independence, and Dependence

    (Mentor note: link removed as not essential to the question.) The problem is: what is relevance anyhow? My questions are these: did I get the math right in the following? Is there a better, more acceptable way to lay out the sample space Ω and the two events F and E? Apart from the math...
  7. Clifford Engle Wirt

    What are the connections between philosophy, art, and information theory?

    Hello Physics Forums Community, I'm Cliff, a DBA working at present for a financial company, with a background in philosophy and art. (So expect many of the questions I will be asking to be at the 'math for English majors level' :-) ) I have been recently been working with Fred I. Dretske's...
  8. Aslet

    I Probability of a Stochastic Markov process

    Hi everyone! I'm approaching the physics of stochastic processes. In particular I am studying from "Handbook of stochastic processes - Gardiner". This book defines a stationary process like: $$ p(x_1, t_1; x_2, t_2; ...; x_n, t_n) = p(x_1, t_1 + \epsilon; x_2, t_2 + \epsilon; ...; x_n, t_n +...
  9. W

    QM: Probability of measuring momentum

    Hi all, My question is in reference to the following paper: https://arxiv.org/pdf/1202.1783.pdf In equation 3.8, the author computes an order-of-magnitude approximation of probability of measuring negative momentum from the following wavefunction: $$ \Psi_k =\sum_{k=1,2}...
  10. 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...
  11. P

    What Are the Correct Values for Probability and Momentum in a Quantum Well?

    Homework Statement A particle in one dimension is in the ground state of the potential well given by V(x)= 0 for |x|<L/2 and infinite otherwise. Let P+ be the probability that the particle is found to move along the positive x direction and p be the magnitude of the momentum for that state of...
  12. 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...
  13. T

    Intro to probability density QM

    Homework Statement Q: A particle is in a linear superposition of two states with energies: ##E_0##& ##E_1## $$|\phi>=A|E_0>+\frac{A}{\sqrt{3-\epsilon}}|E_1>$$ (a) What is the value of A ? Express your answer as a function of ##\epsilon## (b) Use your expression to plot A vs ##\epsilon## (c)...
  14. tomwilliam2

    I Probability: 4 arbitrary points on a sphere

    This is a well-known problem I think: take four arbitrary points on a sphere as the vertices of a tetrahedron. What is the probability that the centre of the sphere will be located within the tetrahedron? A friend gave me this puzzle to solve, and I came up with the answer P = 1/8. This is the...
  15. Gonv

    Probability of finding a particle in a region

    Homework Statement A particle is restrained to move in 1D between two rigid walls localized in ##x=0## and ##x=a##. For ##t=0##, it’s described by: $$\psi(x,0) = \left[\cos^{2}\left(\frac{\pi}{a}x\right)-\cos\left(\frac{\pi}{a}x\right)\right]\sin\left(\frac{\pi}{a}x\right)+B $$ , determine...
  16. S

    B Loophole on theorem related to Conditional Probability

    The theorem says The probability that an event B occur after A has already occurred is given by P(B/A) =P(A intersection B) /P(A) But applying thus to a problem like the probability of occurrence of all 3 tails on 3 coins when tossed if 1 tail has already occurred is P(B/A) =(1/8)/(7/8)=1/7...
  17. N

    MHB Conditional Probability - Faulty Plumbing

    This question has been driving me crazy. A large industrial firm uses three local motels to provide overnight accommodations for its clients. From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton and 30% at Lakeview. What is the...
  18. tomwilliam2

    I Shopping List Game: Probability Question

    I have a probability question which has cropped up while playing a game called "Shopping List" with my sons. The game is played like this: you pick a fixed shopping list of 10 items, and three other players do the same. You have all of the items on cards, face down on the floor in-between the...
  19. Kara386

    Roughly estimate the probability that 2 photons are from Higgs

    Homework Statement Make a very rough estimate of the probability that two high energy photons with an invariant mass of 126GeV are decay products of the Higgs. Use information found elsewhere (so I need to find this info preferably on the internet). Homework EquationsThe Attempt at a Solution...
  20. mfb

    Insights Intransitive Dice with a Twist - Comments

    mfb submitted a new PF Insights post Intransitive Dice with a Twist Continue reading the Original PF Insights Post.
  21. S

    MHB Understanding Sample Proportions and the Binomial Distribution

    Hi, I am doing a past paper but I am kinda stuck on one of the questions. These are the answers I have: 2a. 225/260 = 0.8654 2b. 32/260 * 4/32 = 0.01407 2c. 32/260 * 28/32 + 228/260 * 221/228 = 0.9577 Then for 2d, I have no idea what to do. Am i suppose to draw one of those probability...
  22. E

    Probability of penetrating a potential barrier

    Homework Statement The probability for a particle of energy E<<V0 to penetrate a potential barrier of height V0 and width d is approximately \frac{16E}{V_0}exp\left[\frac{-2d\sqrt{2m(V_0-E)}}{\hbar}\right]. An electron moves between two potential barriers of height V0 and 2v0 that are of widths...
  23. I

    I Meaning of Dirac Delta function in Quantum Mechanics

    If I have a general (not a plain wave) state $$|\psi\rangle$$, then in position space : $$\langle \psi|\psi\rangle = \int^{\infty}_{-\infty}\psi^*(x)\psi(x)dx$$ is the total probability (total absolute, assuming the wave function is normalized) So if the above is correct, does that mean...
  24. P

    MHB Probability Homework Question 2

    Suppose that X, Y are uncorrelated random variables which are each measurements of some unknown quantity $\mu$. Both random variables have $\mu_{X} = \mu_{Y} = \mu$, but $\sigma^2_{X} > \sigma^2_{Y}$. Determine the value of $\alpha$ in [0, 1] which will minimize the variance of the random...
  25. P

    MHB Exam P Question for my Probability Homework

    A machine consists of two components whose lifetimes have a joint density function: $f(x,y) = \left\{ \begin{array}{ll} \frac{1}{50} & \quad x \geq 0, y \geq 0, x+y \leq 10 \\ 0 & \quad Otherwise \end{array} \right.$ The machine operates until both...
  26. Pushoam

    Probability of getting an ace in case of a loaded die

    Homework Statement Homework EquationsThe Attempt at a Solution 12)a) a) Since both events are independent : P( 33) = 9 / 441 b) Let’s have 4: throwing 4 Not 4 : not throwing 4 P ( 4 and not throwing 4 ) = P( 4) P ( not throwing 4) = (17*4)/441c) Let’s consider the sample...
  27. T

    Chess board probability

    Homework Statement 2 squares are chosen at random from a chess board.What is the chance that these 2 squares will share exactly 1 corner? Homework Equations P=favourable possibilities/Total possibilities The Attempt at a Solution So,the total no of possibilities should be 64C2. Now,for...
  28. Pushoam

    Probability of a student failing a prelim but passing the course

    Homework Statement Homework EquationsThe Attempt at a Solution [/B] The % of course passing students who fail the prelim test is 38%. So, the required probability is 38 %. Isn't the probability of a student failing the prelim when he his passing the course is known already = probability...
  29. M

    MHB Conditional probability prove or disprove

    Hey! :o Let $P$ be a probability measure on a $\sigma$-Algebra $\mathcal{A}$. I want to prove or disprove the following statements: $P(A\mid B)=1-P(\overline{A}\mid B)$, for $A, B\in \mathcal{A}$ $P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$ I have done the following...
  30. Muthumanimaran

    Finding the probability of 1s electron within a cubical volume

    Homework Statement How to calculate the probability of finding an 1s electron within 1 picometer cubic region located 50pm from the nucleus. Homework Equations The probability of an 1s electron within a spherical volume of radius 'a' from nucleus can be find using the expression...
  31. J

    A Help with this problem of stationary distributions

    I need help with this Consider an irreducible Markov chain with $\left|S\right|<\infty $ and transition function $p$. Suppose that $p\left(x,x\right)=0,\ x\in S$ and that the chain has a stationary distribution $\pi .$ Let $p_x,x\ \in S,$ such that $0<\ p_x<1$ and $Q\left(x,y\right),\ x\in...
  32. M

    MHB Probability measure: prove or disprove

    Hey! :o I want to prove or disprove that for a probability measure $Q$ over a $\sigma$-algebra $\mathcal{B}$ with $A,B\in \mathcal{B}$ the following hold: $Q(A\cup B)=1-Q(\overline{A}\cup \overline{B})$ $1-Q(\overline{A}\cap \overline{B})=Q(\overline{A})+Q(\overline{B})+Q(\overline{A}\cup...
  33. M

    MHB Calculate the probability and the number of ways

    Hey! :o I am looking the following exercise: At a tango course $12$ married couples participate. Find an appropriate measurable space and calculate the probability that exactly nine couples dance together. $5$ women sit at a round table. $3$ men come later. With how many ways can the men...
  34. W

    I Probability of Staying in x<0 for Superposition of 2 Gaussians

    Homework Statement I am supposed to find probability of staying in x < 0 for a superposition of two Gaussians. The wavefunction is something along the lines of: Homework EquationsThe Attempt at a Solution Usually, the step involved in finding probabilities for 1 particle is just to perform...
  35. mastrofoffi

    Prob/Stats Finding a Probability Book for Physics Students for a Complex Course

    Hello, I'm taking a course in Probability for which our only resource is our professor's notes but I find them somewhat hard to follow. He told us he wrote these notes because he could not find a book which contained all the topics we'll be doing, but I still hope someone here can help me find...
  36. P

    Poisson Random Variable probability problem

    Homework Statement X is a Poisson Random Variable with rate of 1 per hour, following the Poisson arrival process a. Find the probability of no arrivals during a 10 hour interval b. Find the probability of X > 10 arrivals in 2 hours c. Find the average interarrival time. d. For an interval of 2...
  37. B

    Where will the pendulum land -- Probability Question

    Homework Statement Imagine a number line -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 A pendulum begins at -1 and then swings towards the LEFT or RIGHT. There is no knowing how fast the pendulum is traveling or how much energy it has but will only swing LEFT or RIGHT. (I) Need to work out the...
  38. P

    Geometric Random Variable probability problem

    Homework Statement X is a geometric random variable with p = 0.1. Find: ##a. F_X(5)## ##b. Pr(5 < X \leq 11)## ##c. Pr(X=7|5<X\leq11)## ##d. E(X|3<X\leq11)## ##e. E(X^2|3<X\leq11)## ##f. Var(X|3<X\leq11)##Homework EquationsThe Attempt at a Solution Can someone check my work and help me? a...
  39. M

    MHB Defining a Probability Distribution with Measure Spaces and Delta Functions

    Hey! :o Let $M$ be a measure space and $(a_i)_{i\in \mathbb{N}}\subset M$. I want to show that for positive $p_1, \ldots , p_n$ with $\displaystyle{\sum_{i=1}^np_i=1}$ by $\displaystyle{Q=\sum_{i=1}^np_i\delta_{a_i}}$ a probability distribution is defined. Do we have to show that...
  40. Frankenstein19

    Calculate the probability that the particle's x coordinate

    Homework Statement For a 1.0 × 10-26 g particle in a box whose ends are at x = 0 and x = 2.000 Å, calculate the probability that the particle's x coordinate is between 1.6000 and 1.6001 Å if n=1 Homework Equations The Attempt at a Solution I know that since the interval between 1.6000 and...
  41. C

    I Probability vs radial density-confusion

    Hi everyone; A very stupid confusion here. When we want to talk about the most probable radius to find the electron in $1s$ orbital, why do we talk about the radial density and not the probability itself? For instance, the probability of finding the the electron at a radial distance $r$...
  42. K

    B Probability of Mine in Block E at Crossroads AB & CD

    I thought of this problem: The roads AB and CD have block E in common. There are 6 blocks in road AB and 13 blocks in road CD. Someone has planted a mine in some of the blocks. He gives us this information: 1. There's only one mine in road AB in one of its blocks. 2. There's only one mine...
  43. G

    Upper bound for probability when Bayes risk is zero

    Homework Statement Bayes' risk is ##L^*=0## for a classification problem. ##g_n(x)## is a classification rule (plug-in) such that ##g_n=0## is ##\eta_n(x)\leq 1/2## and ##g_n=1##$ otherwise. The function ##\eta##is given by ##\eta(x)=\mathbb{E}(Y|X=x)##. Then ##\mathbb{P}(g_n(X)\neq Y)\leq...
  44. J

    Find the expectation from probability

    Homework Statement Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected...
  45. J

    B What Is the Probability of Scoring in the 88th Percentile for a Trait as a Male?

    I scored in the 88th percentile in a certain personality trait and am trying to figure out the probability of that given that I'm male. I'm trying the likelihood that I would land in the 88th percentile given that I'm male. Definitions: T = trait, M = males, F = female. Given: P(T|M) = 0.3...
  46. T

    I Probability of Surviving Pill Selection

    I'll start off by saying this is technically a coursework question, just not mine, nor of the present. I work in a tutoring center and it's been a while since I have taken stats and don't quite understand this person's professor's reasoning. The general idea of the question was that there was a...
  47. M

    Probability density function for a given problem

    https://www.quora.com/How-can-I-find-the-probability-density-function-of-a-continuous-random-variable-in-a-given-problem/answer/Maxime-Denis-2 How can I find the probability density function of a continuous random variable in a given problem? A mass m swings at the end of a rope (of length L)...
  48. LarryS

    I Classical vs quantum wave amplitudes?

    In classical mechanics and EM, the energy carried by a wave is the amplitude squared. In QM the (complex) amplitude squared of the position-space wave function is the position probability density. Do physicists regard this as anything more than just an interesting coincidence? Has anybody...
  49. D

    Probability that two measurements have the same true value

    Homework Statement I am in a lab course studying Brownian motion. I have gathered data for for movement in two dimensions. I currently have a fairly large data set for ∆x and ∆y and have taken the mean and standard deviation of each. The lab asks what is the probability that these two sets of...
  50. G

    Minimize the sum of Type I and Type II errors

    Homework Statement Given X_1,\dots,X_n a simple random sample with normal variables (\mu, \sigma^2). We assume \mu is known but \sigma^2 is unknown. The hypothesis is \begin{cases} H_0: & \mu=\mu_0 \\ H_1: & \mu=\mu_1 > \mu_0 \end{cases} Determine the rejection region R...
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