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. T

    I Understanding the probability density function

    Hi all This is not a homework question but something work related which I am having difficulty understanding which I was hoping someone from the community could help me with. I am trying to understand how to interpret & create the probability density function plot from a set of data. For...
  2. M

    Finding k in a probability density function

    Homework Statement Let X, Y, and Z have the joint probability density function f(x,y,z) = kxy2z for 0 < x, y < 1, and 0 < z < 2 (it is defined to be 0 elsewhere). Find k. Homework Equations Not sure how to type this in bbcode but: Integrate f(x,y,z) = kxy2z over the ranges of x (zero to...
  3. E

    Intro statistics question: probability of intersection

    Homework Statement If event A equals event B, then the probability of their intersection is 1. True or False? Apparently the correct answer is False. The Attempt at a Solution If A=B then they should overlap entirely and their intersection should be 1? The only way I see this working is if...
  4. Pushoam

    I Microstate, Macrostate, Probability

    If I consider a system of only two coins, then there are four microstates, each equally probable. But the probability that the system will have one head and one tail is the most. Describing the system by its all possible configurations is describing it in terms of its microstates. Here, the...
  5. S

    Probability (Permutation Combination)

    Homework Statement There are 22 students in a class. The professor will divide the class into 4 groups. Group 1 and 2 have 5 members each whilst Group 3 and 4 have 6. Given that the teacher forms the group at random, find the probabilities of : A = event where Paula, Trina, Gia all belong in...
  6. L

    How many points of zero probability in a finite well?

    Homework Statement An electron is trapped in a finite potential well that is deep enough to allow the electron to exist in a state with n=4. How many points of (a) zero probability and (b) maximum probability does its matter wave have within the well? Homework Equations For infinite potential...
  7. Delta2

    I Probability of an event in n tries

    IF the probability for an event to happen in one try is m/n , what is the probability for the event to happen at least one time in n successive tries. In each try the probability of event is m/n. I care mainly for the formula that I suppose it involves m and n but if you kind enough to provide...
  8. R

    Finding the probability of an electron in the forbidden region

    Homework Statement For the hydrogen atom in the first excited state, find the probability of finding the electron in a classically forbidden region. Homework Equations E1 = -Z2μe'4/2n2ħ2 ψ200 = ∫ψ(r)2dr from the limit to ∞ The Attempt at a Solution https://imgur.com/a/fWIMq I feel like this...
  9. R

    Probability of a measured energy for a hydrogen atom

    Homework Statement https://imgur.com/a/8deZc Homework Equations P(E) = ∫φ*(r)ψ(r)dr from -∞ to ∞ The Attempt at a Solution To find the probability, I know I have to use this equation: P(E) = ∫φ*(r)ψ(r)dr from -∞ to ∞ My question is, what is the energy eigenstate, φ*(r)? Is it the measured...
  10. T

    Probability of employees traveling to work

    Homework Statement According to a survey, of those employees living more than 2 miles from work , 90% travel to work by car . Of the remaining employees, only 50% travel to work by car . It's known that 75% of employees live more than 2 miles from work . Find the probability of that an...
  11. C

    B Particle pair creation probability

    I know that a high frequency light beam is more likely to generate a virtual electron-positron pair than a low frequency one. Can this probability depend on the reference frame? It seems there is a paradox. How do we explain it?
  12. M

    MHB Calculating Probability for Random Variables with Two Dice

    Hey! :o Two usual six-sided dice are thrown. We are given the probability space $\Omega:=\{(1,1),(1,2),..., (1,6), (2,1),..., (6,1)\}$ and the probability function $ P:Pot (\Omega)\rightarrow [0,1], \ P (\{(a,b)\})=\frac{1}{36}$. Calculate for the following random variable $X_i:\Omega->R$ the...
  13. B

    I Probability of creation of virtual particles

    Is it correct to assume that all known particles may be created as virtual particles in the vacuum? If so, is there a higher probability of a particular particle being produced than say some other particle type. For example, is an electron more likely to be created as a virtual particle than a...
  14. redtree

    B The expectation value of superimposed probability functions

    I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one). If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...
  15. T

    Probability of bulb malfunction

    Homework Statement consider 8 packs of bulb and let x be the number of bulbs in a pack that 'fail' the first time they are used . If 0.02 of all bulbs of this type fail on their first use and each 8-pack is consider random sample , what is the probability that anyone 8-pack has no bulb fail on...
  16. T

    Calculating Binomial Probability for Car Color Preferences

    Homework Statement Car colour preferences changes over year . In this year , suppose that 10% of the car are randomly selected , let the sample of cars are 20 . Find the probabilities between thre and five cars ( inclusive ) are black ... I am aksed to do this question using binomial ...
  17. P

    MHB James' question about a continuous probability distribution

    Since it's a PDF, that means the entire area under the curve must be 1, so $\displaystyle \begin{align*} \int_0^1{ a \left( x^2 + b \right) \,\mathrm{d}x } &= 1 \\ a \left[ \frac{x^3}{3} + b\,x \right] _0^1 &= 1 \\ a \left[ \left( \frac{1^3}{3} + b\cdot 1 \right) - \left( \frac{0^3}{3} + b...
  18. R

    Are Coin Toss Events A, B, and C Independent?

    Homework Statement Hi, I have this question that I've been pondering for a while, I keep flipflopping on what I think is right. I only need help on the last part on whether the events are independent or not, the rest of the text is backstory to the question. I know for events to be independent...
  19. K

    Is (-infinity, b) an event for any real number b?

    Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...
  20. SSGD

    I Positional Probability of Periodic Object Motion

    If you know the velocity of an object as a function of position Can you use a uniform distribution over one period and the object velocity to perform a change of variables for the positional probability. Example. X(t)=Asin(wt) V(t)=Awcos(wt) V(X)=+-Aw(1-(X/A)^2)^(1/2) P(t)=1/T T=Period Change...
  21. S

    MHB Probability of 3 people ending up on same team?

    There are 20 people and we are forming 2 teams of 10 people. 3 of the people (3/20) are women, 17 (17/20) are men. Names are drawn from random, in alternating order for teams. What is the probability that all 3 women end up on the same team? Initially I thought it was 25%. 100% probability for...
  22. Z

    Probability of Removing and not Removing

    Homework Statement A jar has 5 marbles, 1 of each of the colors red, white, blue, green and yellow. If 4 marbles are removed from the jar, what is the probability (i) that the yellow one was removed? (i) that the yellow one was not removed? Homework Equations probability = favorable...
  23. D

    B Understanding the probability cloud?

    I understand that the concept of a atom resembling a solar system is disregarded as being untrue because this is based on the Bohr model which doesn't represent how an electron would actually appear in its probability cloud. However, would it be possible that if there were an observer on the...
  24. M

    MHB What's the probability of selecting John and then James as disciples of Jesus?

    Jesus had 12 disciples. What's the probability of selecting John and then James? What is the set up?
  25. M

    MHB Probability of 2 Green Marbles

    Gilligan has 16 green marbles, 50 blue marbles and 60 red marbles. What is the probability of randomly selecting 2 green marbles? What is the set up?
  26. J

    A Probability density of dirac spinors

    The probability density of the dirac spinor is known to be ∑(Ψ)2 and I know how it is derived. However, I'm just wondering why it should be positive definite. Since the lower two components represent antiparticles, so shouldn't the probability density contribution of those two components be...
  27. G

    I Can you calculate probability with infinite sets?

    Suppose set A is defined as the even integers and set B is defined as for every even integer there are two odd integers, like so: {2,3,3,4,5,5,6,7,7 ... } Can you calculate that the probability of choosing an odd number is 66%?
  28. I

    Does the Radial Probability Function Indicate Angular Nodes?

    Homework Statement I have a quick question, Does radial probability function of finding electron (4πr2R2) show only radial nodes, or does it show angular nodes too. (I am like 90% sure that it shows radial nodes only) Homework Equations N/A The Attempt at a Solution N/A
  29. V

    I Probability of some sequence in a list of numbers

    Hello, would like to derive a length of list of random numbers in which I may find some special sequence of few numbers with some probability. For clearness I give an example: I have two generator of (pseudo) random numbers with same range of numbers, let's say (1-k). First generator give a...
  30. parshyaa

    I Can E1 and E2 be both independent and mutually exclusive events?

    Can you give a example to differentiate independent events from mutually exclusive events? Suppose there is a random experiment of rolling a die: E1 is a event of getting a multiple of 3 E1={3,6} E2 is a event of getting a multiple of 2 E2={2,4,6} Is E1&E2 are independent...
  31. Kelly Lin

    Probability of the polymer chain

    Homework Statement Homework Equations I want to check if I think it right! The Attempt at a Solution If N=1: ← or → (2 configurations/ each length is l) N=2: ← or → or ←← or ←→ or →← or →→ ------→----← (6 configurations/ folded polymer's length is l/2 andunfolded polymer's...
  32. Aslet

    I How Does the Inclusion-Exclusion Principle Relate to 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(...
  33. ElectricRay

    Probability with two dice problem

    Homework Statement If we throw with two dice a and b, than we make two spaces call them space A and B. Space A = a+b and space B = a*b. Next the question there are a couple of questions: 1) What is the probability P(A > 7) 2) What is the probability P(B = odd) 3) What is the probability P( A n...
  34. Hawksteinman

    B Calculating Probability of Rain This Weekend

    I was reading a book about innumeracy and one of the chapters was on probability. This weather woman said 'there is a 50% chance of rain on Saturday, and a 50% chance of rain on Sunday, so the chance of rain this weekend is 100%' Obviously she was wrong, but it got me thinking how would one...
  35. S

    Hydrogen transition probability

    Hello! I have the following problem I'm trying to solve: Homework Statement An Hydrogen atom in the state |100> is found between the plates of a capacitor, where the electric field (weak and uniform) is: E(t) = \epsilon e^{-\alpha t / \tau}. Calculate the parameters of the potential...
  36. 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...
  37. T

    MHB How to derive $P(PH)$ without using a joint distribution table?

    Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?
  38. T

    MHB Is Marginalization Always Valid for Joint Probabilities?

    Given $$P(B \land C)$$ will it always be true that $$P(B \land C | A) P(A) + P(B \land C | \lnot A) P( \lnot A)$$ (regardless what $P(A)$ would be)? How can I prove this?
  39. B

    I Using physics to predict the outcome of a roulette wheel

    Hey guys! I have been working on this project during some time. First off, do you all think that this is possible? Of course, as we all may know, the roulette is a chaotic system, very sensitive to initial conditions, although deterministic. This means that small errors in the...
  40. T

    I What Does The Probability Density Function Tell You?

    Hello All I was wondering if someone could help explain what the probability density function tells you. I am trying to learn about surveying and the PDF keeps cropping up and I do not fully understand it. For example I have:- measured a single angle 15 times calculated my Standard Deviation...
  41. M

    MHB Is Overbooking Probability Calculated Correctly?

    Hey! :o I am looking at the following: An airline assumes that $5 \%$ of all passengers that have booked for a flight will not appear for departure. They therefore book a flight with $50$ seats by selling $52$ tickets. It is assumed that one passenger independently of the other cancels his...
  42. T

    MHB Probability Value Question (left-tailed test)

    I have a statistics problem that is probably not too difficult for someone who knows what they are doing, but I still need help with it. Here’s the scenario. There is a town with a highway built at the end of 2013. The mayor is concerned because of the number of traffic accidents on the...
  43. M

    I Linear regression and probability distribution

    I have some data that I want to do simple linear regression on. However I don't have a lot of datapoints, and would like to know the uncertainty in the parameters. I.e. the slope and the intercept of the linear regression model. I know it should be possible to get a prob. distribution of the...
  44. F

    Using a recursive algorithm to find the value of a game

    Homework Statement Imagine you are playing a game with me, of drawing balls from a box. There are two blue balls and two red balls. They are picked with equal probability, and are drawn without replacement. If you draw a blue ball, I give you $1. If you draw a red ball, you pay me $1.25. What...
  45. W

    Probability of a defective sample.

    Homework Statement In a manufacturing plant, a sample of a 100 items is taken from an assembly line. For each item in the sample, the probability of being defective is .06. What is the probability that there are 3 or more defective units in the sample? Homework Equations z = (x -...
  46. O

    I Confusion over using integration to find probability

    Hey everyone, first, let me say I understand the complement rule. Where I am confused is over the integration. My professor said that suppose you have a continuous cumulative distribution function F(x) = 1-e-x/10, if x > 0 (0, otherwise). And suppose you want to find P(X>12) you can use the...
  47. S

    I Definition of "equivalent" probability problems?

    Is there a precise definition for the statement that two differently worded probability problems are "equivalent"? One technique of (purportedly) solving a controversial probability problem is to propose an "equivalent" problem whose solution is not controversial. (e.g. The Sleeping Beauty...
  48. Jianphys17

    Other QM Probability: Is It Critical to Learn?

    Hello everyone ! I would like to know if prob. theory is critical to learning Qm !
  49. Seanskahn

    Probability of Random Numbers

    Homework Statement Let 0≤p≤1. Let there be k distinct numbers (they can be natural numbers) a1, a2, ... , ak, each repeating respectively b1, b2, ... , bk times. Let q < ∑r=1k br Determine the minimal values of b1 ... bk such that the probability of q numbers chosen out of ∑r=1k br numbers...
  50. S

    I Interpretation of probability density in QFT

    Hello! I am a bit confused about the interpretation of probability density in QFT. Let's say we have the Klein-Gordon equation. I understand that this is the field equation for a spin-0 charged particle. So if we find a solution ##\phi(x)## of the Klein-Gordon equation, as far as I understand...
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