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. malawi_glenn

    I Probability distribution of random events

    Hi Imagine we have a lottery, with chance of winning 1 in 1000 (1/1000). I have made computer simulations in order to find confidence levels for winning. At 1000 bought lottery tickets, the confidence of winning is 64.1% and 2000 bought lottery tickets the confidence of winning is 87.1% By...
  2. C

    MHB Find Probability of two polygons

    There are five hexagons. The edges of each hexagon have been colored with one of three colors randomly. If you pick two hexagons randomly without replacement, what is the probability that they are the same? (Rotation is okay). The total space or denominator is 3^(2×6), therefore we have...
  3. cliffhanley203

    B How might determinism affect probability?

    The probability, I was taught, for red to appear at roulette (on a European table, with a single green zero) is 18/37; and the probability for non-red to appear (black or green zero) is 19/37. If we live in a deterministic universe (I appreciate that that’s a big if for some, not so much for...
  4. B

    I Lowest probability for physical possibility

    My question is whether there is a lowest possible probability for something to possibly (physically) occur on a cosmic basis? That is, is there a threshold 'lowest' probability below which something cannot occur? I'm not referring to 'zero' as the lowest probability. That's obvious. Rather, a...
  5. A

    I Liouville equation with Dirac delta as probability density

    I would like to know the solution to Liouville equation ∂ρ/∂t=-{ρ,H} given the initial condition ρ(t=0)=δ(q,p) where δ(q,p) is a dirac delta centered in some point (q,p) in phase space. I have the feeling, but I'm not sure, that the solution is of the form ρ(t)=δ(q(t),p(t)) where q(t) and...
  6. F

    Probability of winning 10 times in 12 matches

    <Moderator's note: Moved from a technical forum and thus no template.> Hi all, I'm looking at an exercise in probability and I have a little doubt. So the exercise goes like this: "Two players are playing against each other. They have the same probability of winning a single game. In order to...
  7. J

    I What is the Difference between the lottery and QM?

    It seems to me that so far in quantum mechanics, because we have yet to establish probability patterns for, say, what the spin of a photon is, we have made it some mystical thing that we can only know by measuring it. I don't really buy that; is the result of a lottery impossible to know until...
  8. L

    I Estimating a mean from games of ruin

    Imagine a gambler playing a casino game with fixed bet, fixed odds, no skill, and a starting bankroll, ##M_0##. She plays until she can no longer afford to bet and records only how many bets she was able to make, ##N_0##, until she could not afford to bet. Each day she goes back to the casino...
  9. L

    A Convergence of a subsequence of a sum of iid r.v.s

    ##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...
  10. A

    MHB Show that the probability that he will never stops gambling is zero

    Can u help me with this question pls Assume that a gambler plays a fair game where he can win or lose 1 dollar in each round . His initial stock is 200 dollar. He decides a priory to stop gambling at the moment when he either has 500 dollars or 0 dollars in his stock. Time is counted by the...
  11. B

    I Probability of a random walk reaching the point X; maximal c

    https://ibb.co/guBuPd As the title indicates, I want to calculate the Probability of a stock price reaching a determined point, by considering the system as a random walk model, and after that, to compute the so called "maximal curves". I found the whole explanation in this article...
  12. L

    I Shock crossing probability for isotropic particle flux

    Hi there, I am currently trying to understand the theoretical frame work of diffusive shock acceleration. I am having trouble understanding a step in the derivation given by drury 1983 (http://www.oa.uj.edu.pl/user/mio/Ast-Wys-En/Literatura/drury.pdf). In the derivation of eq. 2.47 it is stated...
  13. entropy1

    B Random variable reflecting its probability

    If we have a series of, say, twenty coin tosses, then each discernable specific series of outcomes has equal probability to occur. However, there is only one discernable specific series consisting of twenty 1's, while there are many more discernable series consisting of ten 1's and ten 0's. So...
  14. CDL

    I Quantum Scattering Differential Probability

    I am reading Griffiths' Introduction to Quantum Mechanics, specifically the chapter on scattering. He is discussing the scenario where an incoming beam of particles scatter off an azimuthally symmetric target. At large separation ##r## from the scattering centre, the wavefunction for incoming...
  15. T

    MHB Probability of winning dice game

    Hey, so I've got this problem that I'm trying to figure out. I've worked out something that I think is probably right through simulation, but I'm not really sure how to tackle it from a purely mathematical probability perspective. So, would anyone know how I should approach this? I've tried a...
  16. renec112

    Probability distribution momentum for particle

    Homework Statement A particle with mass m is moving on the x-axis and is described by ## \psi_b = \sqrt{b} \cdot e^{-b |x|}## Find the probability distribution for the particles momentum Homework Equations ## \Phi (p)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \Psi(x,0) \cdot e^{-ipx} dx##...
  17. Aleoa

    Maxwell aphorism about Probability

    In the first Volume of his lectures (cap. 6 first Paragraph), Feynman cites Maxwell : "The true logic of this world is in the calculus of probabilities". Considering the formal and rigorous definition of probability, very often misunderstood by not-scientists, what do you think is the deep...
  18. H

    Estimating the Probability of Nuclear Power Plant Damage from Rare Tsunamis

    Homework Statement The frequency of a tsunami large enough to threaten the safety of a nuclear power plant has been estimated to be 1 in 200 years. If the plant’s life is 65 years what is the probability it will be damaged by such a tsunami? What is this probability more commonly termed? [2...
  19. P

    Finding Probability Density Functions for Independent Random Variables

    Homework Statement Hello! I'm trying to understand how to solve the following type of problems. 1) Random variables x and y are independent and uniformly distributed on the interval [0; a]. Find probability density function of a random variable z=x-y. 2) Exponentially distributed (p=exp(-x)...
  20. Emir Shark

    Programs Returning to Physics: Study plan?

    Hi all, I am completely new to this forum. So allow me to introduce myself. I am currently paving my career as a mathematician, particularly in the field of probability theory and financial mathematics. I am currently pursuing a PhD in this subject and could not help but notice how closely...
  21. Simonel

    I Non-countable uniform spaces probability

    A point is chosen at random inside a circle.Find the probability 'p' that the point chosen is closer to the center of the circle than to its radius. This comes from the noncountable uniform spaces sections.
  22. evinda

    MHB Calculate Probability: Diff $Y-X \leq 1$

    Hello! (Wave) Suppose that $X$ has the uniform distribution on the interval $[0,2]$ and $Y$ has the uniform distribution on the interval $[2,4]$. If $X,Y$ are independent, I want to find the probability that the difference $Y-X$ is $\leq 1$. I have thought the following.The density function of...
  23. Mehmood_Yasir

    What Is the Expected Wait Time for the Last Pedestrian at a Traffic Light?

    Homework Statement Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A...
  24. M

    Probability of getting a Four-of-a-kind?

    Homework Statement Why is my method not getting the correct answer? What am I missing? Homework EquationsThe Attempt at a Solution I am trying to find the probabilities of different hands in a 5-card poker (Texas hold' em). While the document below shows the answer, I like to use a method...
  25. Aleoa

    I Probability of equally likely events

    "Why 2 equally-likely events has each a probability of 0.5 ?" If i explain this saying that is due to the frequences as N goes to infinity, I'm saying a tautology cause it's implied in the definition of probability. So, going to a deeper level, why the probabilities of each of 2...
  26. Mehmood_Yasir

    How Does the PDF of Wait Time Vary for Pedestrians at a Traffic Signal?

    Homework Statement Pedestrian are arriving to a signal for crossing road with an arrival rate of ##\lambda## arrivals per minute. Whenever the first Pedestrian arrives at signal, he exactly waits for time ##T##, thus we say the first Pedestrian arrives at time ##0##. When time reaches ##T##...
  27. Mehmood_Yasir

    Probability and pedestrian wait time density function

    Homework Statement Pedestrians approach to a signal at the crossing in a Poisson manner with arrival rate ##\lambda## arrivals per minute. The first pedestrian arriving the signal starts a timer ##T## then waits for time ##T##. A light is flashed after time T, and all waiting pedestrians who...
  28. Tajeshwar

    What Is the Probability of Drawing a Specific Marble Color Sequence?

    Homework Statement An Urn contains two white marbles and one black marble. A marble is drawn from the Urn without replacement and put aside without my seeing it. Then a second marble is drawn, and it is white. What is the probability that the unknown removed marble is white, and what is the...
  29. 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...
  30. Tajeshwar

    Probability of draw with replacement

    Homework Statement An urn contains 50 marbles – 40 blue and 10 white. After 50 draws, exactly 40 blue and 10 white are observed. You are not told whether the draw was done “with replacement” or “without replacement.” What is the probability that the draw was done with replacement? Homework...
  31. T

    Find the probability of energy value after a given measurment

    Homework Statement I am having a issue conceptualising the problem as I believe the answer is 0. Part c) Homework EquationsThe Attempt at a Solution My answer is 0 and it for the following reason. A the being say time t=0 the system is in some arbitary state, then when I got to measure the...
  32. N

    A How to calculate the probability of error in an AWGN channel?

    Hello, I found a paper on the calculation of probabilities of error. However, I didn't know how to plot the graph using equation 3 and 4. These are the list of equations: And this is the graph: I hope that anyone in this forum may guide me to plot this graph. Thank you.
  33. Tajeshwar

    How Do You Calculate the Probability of Wanting to Watch a Football Game?

    Homework Statement I am given the following 3 joint probabilities: p(I am leaving work early, there is a football game that I want to watch this afternoon) = .1 p(I am leaving work early, there is not a football game that I want to watch this afternoon) = .05 p(I am not leaving work early...
  34. CDL

    Composite Spin 1/2 System Probability Question

    Homework Statement Two spin ##\frac 12## particles form a composite system. Spin A is in the eigenstate ##s_z = + \frac 12## and Spin B is in the eigenstate ##s_x = + \frac 12## What is the probability that a measurement of the total spin will give the value ##0##? Homework Equations I know...
  35. J

    A Sum of independent random variables and Normalization

    Hi, Lets say I have N independent, not necessarily identical, random variable. I define a new random variable as $$Y=Σ^{N}_{i=0} X_{i}$$ does Y follow a normalized probability distribution?
  36. Mathman2013

    Elementary probability question: dice roll

    Homework Statement Lets say you role a single honest dice 12 times. Where succes is either if the die shows 4 og 1 eye. My question is what is the probability of sucess or failure? Homework EquationsThe Attempt at a Solution Is it binomial or conditional probability?
  37. L

    MHB A question about the law of total probability

    Dan put n different kinds of cheese in a row randomly, so that between two kinds of cheese there is a space. Then he puts a pickle on one of the n-1 spaces between the cheeses randomly. What is the probability of goat cheese and parmesan cheese (2 from n kinds of cheese) to be in the different...
  38. J

    A Sample Test | Component Lifetime

    Hi, I'm currently working on a project for which I have to determine the Life-Time of a certain mechanical component within a certain confidence interval. By sampling a small number (let's say n = 10) of these components and measuring the number of hours until failure, I want to determine this...
  39. L

    MHB Question -Sample space in probability

    I would like to know how to solve the following question: Throw a cube until you get the number 6, then stop throwing. a) What is the sample space of the experiment? b) Let's call the event to throw the cube n times En. How much elements from the sample space are within En? **The cube is a...
  40. microsansfil

    I How the probability amplitude is estimated in practice

    Hi, I would like to know how the amplitude of probability is estimated/determinated in practice, for a given experiment. In this example 1.3.2 Analysis of Experiment 2 it is assumed that the probability for each of the two possible states are equiprobable. Than from the experimental results...
  41. E

    A Simulation from a process given by "complicated" SDE

    Actually this is more of a simulation question but since PF doesn't have Simulation category I ask here. I need to simulate a path from a proces given by this Stochastic DE: $$ dX_t = -a(X_t-1)dt+b\sqrt{X_t}dB_t $$ where ##B_t## is wiener process/brownian motion and a and b are just some...
  42. 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...
  43. zonde

    I Do these Bell inequalities rely on probability concept?

    This question came up in another thread. I will post again the link to Nick Herbert's proof here: https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138 I don't see where the probability shows up in Nick Herbert's proof. For discussion of Eberhard's proof I...
  44. Saracen Rue

    Determine what value of a allows for largest probability

    Homework Statement There are two possible solutions for 'a' for which ##f(x)=|x^x-x^a|## is a probability density function. Determine the value for a which produces a PDF with the largest probability of random variable 'x' falling within two standard deviations either side of the mean.Homework...
  45. M

    I Probability function for discrete functions

    My textbook says that if ##X: \Omega \to \mathbb{R}## is discrete stochast (I.e., there are only countably many values that get reached), then it suffices to know the probability function ##p(x) = \mathbb{P}\{X =x\}## in order to know the distribution function ##\mathbb{P}_X: \mathcal{R} \to...
  46. M

    MHB Find the probability that each group has an equal amount of odd and even numbers

    A set of numbers 1,2,...,4N gets randomly divided into two groups with equal amount of numbers. Calculate the probability:7 a) Each group has an equal amount of odd and even numbers, b) All numbers that are divisible by N, to fall in only one of the groups, c) All numbers that are divisible by...
  47. nothing909

    Probability Question -- Choosing red and white coins from two tables

    Homework Statement table 1 has 3 red and 5 white coins. table 2 has 4 red and 2 white coins. a coin is chosen at random without knowing its color from table 1 and placed onto table 2. then as coin is taken from the second table. what is the probability that the coin chosen from the second table...
  48. M

    MHB Probability that we get totally at most 30 times "head"

    Hey! :o A fair coin is tossed $ n $ times; $ X_i = 1 $ denotes the event that "head" appears in the $ i $-th toss. a) How are the single toss $X_i$, $=1, \ldots , n$, distributed? b) How many toss are needed so that the proportion of "head" $\overline{X}_n$ is in the interval $0, 45 <...
  49. K

    B Probability of a photon passing through a filter

    In quantum physics they say that the probability P for a photon to pass through the filter depends on the angle Φ between the photon and the filter polarization axis: P = cos^2(Φ) And if I'm not wrong, when the photon passes through that filter (like illustrated in the image above) the...
  50. Amlan mihir

    What is the probability? Is it independent?

    Homework Statement The Attempt at a Solution ; [/B] I think it is independent event so , the probability would be 0.5 . correct if me i am wrong please.
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