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.

View More On Wikipedia.org
Recent contents Top users
  1. O

    Probability and Stats - Show the Sample Space

    Homework Statement An assembly line is observed until items of both types—good (G) items and items not meeting specification (N)—are observed. Show the sample space. Homework Equations Let G be Good Let N be Not Good The Attempt at a Solution S = {GN, GGN, GG...N, GG..., NG, NNG, NN...G...
  2. fisher garry

    A Calculate the moment of inertia of H with the probability from wavefunction

    the spin, S, for an electron is $$\frac{\hbar}{2}=5.27 \cdot 10^{-35} $$ $$\frac{2MR^2 \omega}{5}=\frac{2MRv}{5}$$ It is said that the speed of an electron is 2200 km per second and can be calculated in classical manners from electrostatic and accelerating forces on the electron from (1.11)...
  3. FallenApple

    A Testing if probability is the same for two groups

    Ok, so in a logistic regression context, I need to test if the probability of ##Y_{i}=1 ## is the same for two different groups at different ages where age is a continuous variable. This is actually complicated because of nonlinearity. Can I default to testing if the odds of ##Y_{i}=1 ## is...
  4. thebosonbreaker

    B Sweets in a bag probability problem

    Andrei has a bag of x sweets. He removes two sweets from the bag simultaneously (without replacement). He now removes a third sweet. The probability that the third sweet is red is (x/2) - 1. How many red sweets were in Andrei's bag to begin with? Could somebody please tell me if (and how) it is...
  5. H

    I Why probability current = 0 at infinity? Why must wavefunction be continuous?

    Q1. Why is the probability current ##j(x,t)=0## at ##x=\pm\infty##? (See first line of last paragraph below.) My attempt at explaining is as follows: For square-integrable functions, at ##x=\pm\infty##, ##\psi=0## and hence ##\psi^*=0##, while ##\frac{\partial\psi}{\partial x}## and hence...
  6. UsableThought

    Has anyone read "Willful Ignorance" by Herbert Weisberg?

    (To any passing moderator: Feel free to move this to "statistics" forum if you feel that would be more appropriate.) Although my "to read" list is already too long, I have lately been getting increasingly interested in learning the basics of conditional probability, including Bayesian analysis...
  7. F

    Transforming Correlated Standard Normals with Cholesky Decomposition

    Homework Statement Given correlation matrix $$M = \begin{bmatrix} 1 & .3 & .5 \\ .3 & 1 & .2 \\ .5 & .2 & 1 \\ \end{bmatrix}$$ And 3 independent standard normals $$N_1, N_2, N_3$$ using cholesky decomposition A) get the correlated standard normals B) and if...
  8. Mr Real

    B Understanding Probability: Why Do We Divide by 2 in Certain Situations?

    I recently read a question. It was: If 100 birds are sitting in a circle and all of them peck a bird either on their left or on their right randomly, what's the expected number of birds that will be unpecked? The answer to this is 25 birds as probability of not being pecked is 25%. The logic...
  9. M

    MHB Probability: Replacement & Without Replacement

    Can someone please explain the probability of replacement and without replacement with one simple example for each?
  10. M

    MHB Probability Definition

    Probability of an event happening = (Number of ways the event can happen)/(Total number of outcomes) Please, explain the above definition How is the above definition applied to the following question. A coin is tossed 100 times. How many heads will pop up? Solution: Let P = probability...
  11. M

    MHB Probability of AND vs. OR: Understanding the difference and formulas

    Can someone explain in simple terms the difference between the probability of AND and the probability of OR. Can you provide an example for each? Can you please explain the AND/OR formulas for each probability found in most textbooks?
  12. M

    MHB What is the Probability of the Rapture Happening in Our Lifetime?

    In the New Testament, the Apostle Paul introduced an event that has come to be known as the rapture of the church. The rapture is an imminent event, which means it could happen at any moment in time. What is the probability that the rapture of the church will take place in our lifetime? Set up...
  13. M

    MHB Probability of Turning into a Bird

    Substitute teacher, Baraba Rose, convinced a class of kindergarden kids that she would turn into a bird before the school day ends. What is the probability that this event will take place? Solution: Let P = probability P(turning into bird) = 0 The answer is 0 because human beings cannot turn...
  14. M

    MHB What is the Probability of No Rain Tomorrow?

    There is a 30 percent chance of rain tomorrow. What is the probability of no rain tomorrow. Solution: Let P = probability P(no rain tomorrow) = 1 - P(rain tomorrow) P(no rain tomorrow) = 1 - 0.30 P(no rain tomorrow) = 0.70 or 70 percent Right?
  15. M

    MHB What is probability and how does it relate to precalculus?

    Most students struggle with probability because questions usually involve FUZZY word problems. I am reviewing precalculus right now. In the later chapters of precalculus, a touch of probability is introduced. Can someone explain the basics of probability? I know, for example, that all...
  16. grquanti

    I Jump probability of a random walker

    Hello everybody. I have a Markowian homogeneous random walk. Given the transition matrix of the chain, I know that ##P[ X(t) = i | X(t-1) = j ] ≡ P_{j→i}=T_{ij}## where ##T## is the transition matrix and ##X(t)## is the position of the walker...
  17. binbagsss

    Probability , expectation, variance, cross-term vani

    Homework Statement I have a variable ##s_i## with probability distribution ##w(s_i)## ##(\Delta(s_i))^2## denotes the variance ##=<(s-<s>)^2>=<s^2>-<s>^2## I want to show ## \sum\limits_{i\neq j} <\Delta s_i> < \Delta s_j> =0 ## where ## < > ## denote expectation My book has: ## <\Delta...
  18. N

    MHB Prove Probability: Step-by-Step Guide

    Prove the following I literally have no idea where to start or what to do.
  19. redtree

    A Deriving Probability Amplitude from Markov Density Function

    1. Given a Markov state density function: ## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ## ##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...
  20. T

    I Is nuclear decay probability always constant?

    I have heard that the probability of an unstable nucleus decaying is always constant. Is there any way to change this probability?
  21. Daniel Petka

    B Probability Amplitude: Understanding Photons

    https://www.physicsforums.com/attachments/197770The path of a photon is a perfect straight line not a sine wave, right? (if the probability amplitude is zero)https://www.physicsforums.com/attachments/197771
  22. J

    MHB Probability Distribution Problem

    Suppose that cars pass a certain intersection at a rate of 30 miles per hour. What is the probability that during a three-minute interval, no cars will pass the intersection? I am really just wondering which distribution to use. I thought is should be Poisson because it is asking for events...
  23. A

    Convergence in distribution example

    Homework Statement Homework Equations [/B] Definition: A sequence X_1,X_2,\dots of real-valued random variables is said to converge in distribution to a random variable X if \lim_{n\rightarrow \infty}F_{n}(x)=F(x) for all x\in\mathbb{R} at which F is continuous. Here F_n, F are the...
  24. F

    I Maximum likelihood w/ histogram, zero probability samples

    I'm trying to replicate a machine learning experiment in a paper.The experiment used several signal generators and "trains" a system to recognize the output from each one. They way this is do is by sampling the output of each generator, and then building histograms from each trial. Later you...
  25. G

    Odd Well / Quantum Tunneling

    Homework Statement An electron with a total energy of Eo = 4.4 eV is in the potential well shown above. 1) Find the ratio of the wavelength in Region III to the wavelength in Region I. λ III / λI = 1.772) Given that the wave function of the electron vanishes at the left boundary of Region...
  26. I

    Atomic Physics - Orbital Angular Momentum Probability

    Homework Statement Consider an electron in a state described by angular wavefunction $$\psi(\theta,\phi)=\sqrt{\frac{3}{4 \pi}}\sin \theta \cos \phi$$ Here θ and φ are the polar and azimuthal angles, respectively, in the spherical coordinate system. i. Calculate the probability that a...
  27. Saracen Rue

    Probability Density Function problem

    Homework Statement Presume the relation ##\frac{x}{x+y^2}-y=x## is defined over the domain ##[0,1]##. (a) Rearrange this relation for ##y## and define it as a function, ##f(x)##. (b) Function ##f(x)## is dilated by a factor of ##a## from the y-axis, transforming it into a probability density...
  28. A

    I Question: Proposed Solution to Two Envelope Paradox

    Su, Francis, et. al. have a short description of the paradox here: https://www.math.hmc.edu/funfacts/ffiles/20001.6-8.shtmlI used that link because it concisely sets forth the paradox both in the basic setting but also given the version where the two envelopes contain ( \,\$2^k, \$2^{k+1}) \...
  29. M

    Probability observed value not in range for prediction

    Homework Statement Hello all, I created a predictive model from a data set of observed values and am looking for probabilities for accuracy. Data set A (observed) and data set B (predictive model) have a correlation of 84 % using linear regression. Data set A and B are both normally...
  30. K

    I Finding the average time with given probability

    Hi all, I am thinking a problem of drawing a ball in a sealed box. Assuming there is a box, contains plenty red and white balls, the number of red and white balls are unknown but let's assume there will be ##p## chance to draw a red ball and ##q=1-p## chance to get a white one. Those...
  31. S

    Difference between probability waves & electromagnetic waves?

    What I know: A ripple/wave in a field gives rise to a particle. For example, a ripple in electric field creates a photon. Question: Is this the same principle as probability wave which when observed reveals a particle?
  32. G

    Probability Density in an infinite 1D square well

    Homework Statement The wave function of a particle of mass m confined in an infinite one-dimensional square well of width L = 0.23 nm, is: ψ(x) = (2/L)1/2 sin(3πx/L) for 0 < x < L ψ(x) = 0 everywhere else. The energy of the particle in this state is E = 63.974 eV. 1) What is the rest energy...
  33. math4everyone

    Question about Probability of operating time of transistors (MTBF)

    Homework Statement Homework Equations $$f_X(x)=\lambda e^{-\lambda x}$$ $$F_X(x) = 1-e^{-\lambda x}$$ $$\mu = \frac{1}{\lambda}$$ The Attempt at a Solution a) $$f_{X,Y}(x,y) = f_X(x)f_Y(y) = \frac{1}{800} e^{-\frac{1}{800}x} \frac{1}{1000}e^{-\frac{1}{1000}y}$$...
  34. N

    A Comparing Kullback-Leibler divergence values

    I’m currently evaluating the "realism" of two survival models in R by comparing the respective Kullback-Leibler divergence between their simulated survival time dataset (`dat.s1` and `dat.s2`) and a “true”, observed survival time dataset (`dat.obs`). Initially, directed KLD functions show that...
  35. durant35

    I Many worlds and high-amplitude anomaly branches

    A question came up to my mind while thinking about probabilities and Born rule in the context of the Everettian approach. It is often said that anomalies/maverick branches where the experiments go horribly wrong and crazy stuff happens have a negligible amplitude/measure so they really don't...
  36. M

    Probability virus question at different infection rates

    Homework Statement From various studies, it is known that once an individual is infected with a virus, they become infectious at rate λ. The individual will recover at rate λ, independent of the time it took for them to become infectious. Let X be the total amount of time an individual has this...
  37. Euler2718

    I Hypergeometric Distribution Calculation in Libreoffice

    Given this libreoffice command: HYPGEOM.DIST(X; NSample; Successes; NPopulation; Cumulative) >X is the number of results achieved in the random sample. >NSample is the size of the random sample. >Successes is the number of possible results in the total population. >NPopulation is the size...
  38. P

    B Not sure what this problem would be called....

    Hi, I am trying to write a function which would take two arguments: -> number of bits (which are a binary 0 or 1 value) N -> and acceptable number of mismatching bits M The function would statistically determine the probability of having M or less mismatching bits when randomly generating two N...
  39. T

    MHB Probability calculation with Bayesian Networks

    Given this base data (taken from Graphical Models )$P(C) = 0.5$ $P(\lnot C) = 0.5$ $P(R | C) = 0.8$ $P(R | \lnot C) = 0.2$ $P(\lnot R | C) = 0.2$ $P(\lnot R | \lnot C) = 0.8$ $P(S | C) = 0.1$ $P(S | \lnot C) = 0.5$ $P( \lnot S | \lnot C) = 0.5$ $P( \lnot S | C) = 0.9$ $P(W | \lnot S, \lnot...
  40. N

    B Why is state transition probability symmetric?

    Restricting to finite dimensional QP, suppose a system is in a state S1, an experiment is done, and state S2 is one of the eigenstates (assume all eigenvalues are distinct). The probability that the system transitions from S1 to S2 is p = Trace( S1*S2), using state operator notation. On the...
  41. R

    MHB Conditional Probability

    Dear All sorry for repeated post; There is a problem Problem: Three cards are drawn in succession from a deck without replacement. find the probability distribution for the number of spades. I have come with this solution. Let S1: appearance of spade on first draw S2: appearance of spade on 2nd...
  42. R

    MHB Calculating the Probability of Faulty Plumbing in Hotel Rooms: A Case Study

    Dear all Please help in solving the following problem. A large industrial firm uses 3 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 sheeraton and 30% at the Lakeview...
  43. A

    I Calculating the probability density of a superposition

    This ought to be some simple gap in my knowledge, but it bugs me nonetheless. Let me present the argument as I see it, I'm fairly certain that there is just some tiny part that I didn't learn correctly. Let us assume a wavefunction $$\Psi$$ is defined as a superposition of two wavefunctions...
  44. T

    MHB Probability Question: At Least One Customer Unable To Claim Car

    So, at a car rental company, 20% of car reservations are not claimed. There is a total of 22 cars and the manager takes 25 reservations a day. If all cars are claimed for a day, what is the probability that one or more customer who had reservations were unable to claim their car? I need to...
  45. Mr Davis 97

    Probability Integral Homework | Convergence Issue

    Homework Statement ##\displaystyle \int_{- \infty}^{\infty} \frac{1}{\sqrt{2 \pi}} x e^{- \frac{x^2}{2}} dx## Homework EquationsThe Attempt at a Solution So first off, obviously the answer is 0, because the integrand is odd and we have symmetrical limits of integration. However, when I make...
  46. durant35

    I Probability of not entering de Sitter

    As far as we know, the universe is undergoing accelerated expansion and heading towards empty de Sitter space. It is assumed that eventually the observable universe will be emptied out of matter and all radiation. Now if we take in account quantum mechanics, there's always non zero probability...
  47. M

    Sample space probability question

    Homework Statement [/B] Driving to work, a commuter passes through a sequence of three traffic lights. At each light he either stops, denoted by s, or continues, denoted by c. Assume that the outcome c or s for each traffic light is independent of the outcome of other traffic lights. (a)...
  48. D

    Finding general expression for probability current.

    The conservation of probability says: $$\partial_t J^{0} + \partial{i}J^{i} = 0$$ Use the Schrodinger equation to obtain$$ J^{i} (\vec r)$$. I have no idea where to start this kind of problem because the notation makes no sense to me. I would appreciate a hint or nudge in the correct direction.
  49. T

    MHB Probability of event given another event occurs twice in a row

    I have $P(B) = 0.4$ and $P(\lnot B) = 0.6$. $P(TS|B) = 0.7$ and $P(TS|\lnot B) = 0.25$ $P(B|TS) = 0.65116$ and $P(\lnot B|TS) = 0.34884$ (from bayes theorem). Now, if we get $B$ or $\lnot B$, and we get the same event twice in a row so we get $B$ then $B$ or $\lnot B$ then $\lnot B$, what...
  50. B

    Prob/Stats Is Probability by David Morin a good book?

    I had taken probability before but not in depth and I want to learn it again. Is David Morin's book good for a intermediate learner ? I don't want to waste time on easy stuff. The reviews are good but popular science books tend to omit proofs instead they just state the theorems and have easy...
Back
Top