What is Probabilities: Definition and 396 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. B

    Why didn't radioactive decay probabilities cause the same uproar as QM

    It is equally puzzling why we are confined to probability amplitudes for RD as in QM measurements. Newtonian determinism is undermined in both, so why were there still Newtonian determinists around when QM hit the scene? We still have deterministic equations for both ofc but they are limited to...
  2. G

    Prove expression for N using inclusion-exclusion principle

    Well ##\cup_{i} E_i## is just the event that at least one color is not used, so its probability is given by ##1- (1/N)^N##. Now if I is a subset of {1,...,N} where ##\left | I \right | = l## then ##Pr(\cap_{i\in I} E_i) = (1-l/N)^N## (I'm guessing this is where I'm making a mistake?). So then we...
  3. W

    I Conditional probabilities of conditioned probabilities

    I know that ##P(A,B) = P(A|B) \ P(B)##. But If i should like to define conditional probabilities for already-conditioned probabilities ie. $$P(A,B|C)$$ how should I do it? Writing something like ##P(A,B|C) = P(A|B|C) \ P(B|C)## seems nonsensical, and I've seen stuff that suggests ##P(A,B|C) =...
  4. entropy1

    I Getting the probabilities right

    If we have a jar with 3 blue balls and 7 white balls, we say that the probability of blindly getting a blue ball out of that jar is 30%. If we have a jar with 2 blue balls and 8 white balls, we say that the probability of blindly getting a blue ball out of it is 20%. Now if we carry out 10...
  5. D

    On being alive now and Bayesian probabilities....

    Hi everyone! I am new here, I have been reading you for many years and I just registered hoping that you can help me out with these disturbing thoughts, as they are taking away too much of my time :( Let's say my lifespan is around 80 years. The lifespan of the universe is of the order of...
  6. R

    Probability question: picking colored marbles out of a bag

    there are 5 marbles in a bag: 4 are blue, and 1 is red. Solution there are 5 in total 4 are Blue so 4/5 or80% Is it right? thanks
  7. user366312

    Finding conditional and joint probabilities from a table of data

    Let, alpha <- c(1, 1) / 2 mat <- matrix(c(1 / 2, 0, 1 / 2, 1), nrow = 2, ncol = 2) chainSim <- function(alpha, mat, n) { out <- numeric(n) out[1] <- sample(1:2, 1, prob = alpha) for(i in 2:n) out[i] <- sample(1:2, 1, prob = mat[out[i - 1], ])...
  8. user366312

    I How can I compute Markov transition probabilities from given data?

    Suppose, I have a Markov Chain as follows: ##S=\{1, 2\}## ## \alpha = \begin{bmatrix} 0.5&0.5\end{bmatrix} ## ## P = \begin{bmatrix} 0.5&0.5\\ 0&1 \end{bmatrix} ## And, the following data regarding 5 steps of Markov Chain taken 12 times: Steps 1 2 3 4 5...
  9. I

    I Boltzmann Distribution and microstate probabilities

    For a canonical ensemble the probability of occupying a certain microstate varies depending on the energy, however I thought that every microstate has an equal chance of being occupied. So what part of the canonical ensemble have I misunderstood?
  10. J

    A Is "Equal a priori Probabilities" the correct scientific approach?

    Hello, My understanding of the scientific approach when faced with an unknown sequence: Hypothesis: There is an order/signal/bias/code = Goal to achieve Null hypothesis: Absence of bias/order/code = Equal Probability = Target to destroy. Science tries to break the code. In order to prove the...
  11. jk22

    B Are there unverifiable assertions about probabilities

    Suppose we obtain a probability of ##1/\sqrt{2}## from QM for example. This will be never verifiable since experiments can only give rational numbers even more : finite digits. Does this mean that such a theory cannot be real in some sense since it would need an everlasting expetiment ? Are...
  12. K

    I Isobar states decay probabilities

    Hello! Given an excited state of a nucleon, such as the ##\Delta## baryon (and here I mean all its 4 version ##\Delta^{++}##, ##\Delta^{+}##, ##\Delta^{0}##, ##\Delta^{-}##), the decay channel is (in this case) to a pion and a nucleon. I was wondering, is the decay probability the same for all 4...
  13. J

    I David Deutsch (1985) attempt to solve the incoherence problem

    Can anyone elaborate on Deutsch's attempt to solve the incoherence problem? He postulates a continuously infinite set of universes, together with a preferred measure on that set. And so when a measurement occurs, the proportion of universes in the original branch that end up on a given branch...
  14. K

    What are the probabilities involved with package shipping insurance?

    The situation is, I purchase an insurance for loss of a package during shipment, the insurance fee is 0.5% of the claimed actual value of the package, but this is optional, you don't need to purchase insurance but you will not get compensated if the package is lost.. Assume that the shipping...
  15. S

    A Calculating Bivariate Normal Probabilities

    Hello good people of PF, I came across this problem today. Problem Statement Given bivariate normal distribution ##X,Y \sim N(\mu_x=\mu_y=0, \sigma_x=\sigma_y=1, \rho=0.5)##, determine ##P(0 < X+Y < 6)##. My Approach I reason that $$ P(0 < X+Y < 6) = P(-X < Y < 6-X)$$ $$ =...
  16. E

    B Generating Stats and Probabilities

    Hey everyone, So lately, my friends and I have been playing a card game (magic, because we're nerds :P) and we've been trying to rank our decks (as well as our probabilities to win a game, a match, etc). I've begun working on a spreadsheet with some initial data points (about 30 in total). The...
  17. W

    I Probability that a Random String is a Word

    Hi, Say L is a human language (e.g. German, Chinese, etc.) and w is a string in L of length n>1. Is it known for different languages what the probability is that w is a word in L? And if S is an ordered set of strings, the probability that S is grammatically correct in L? I mean, I know or have...
  18. W

    Particle spin probabilities

    Homework Statement A beam of spin ##\frac{1}{2}## particles is prepared in the state: ##|\psi> = \frac{3}{\sqrt{34}}|+> + \frac{5i}{\sqrt{34}}|->## a) What are the possible results of a measurement of the spin component ##S_z##, and with what probabilities would they occur? b) Suppose that the...
  19. A

    Geometric Law of Probability with Dice

    Homework Statement We have a normal 6 sided dice marked from 1 to 6. There is an equal chance to get each number at every roll. Let's put 1&2 as A type, 3&4 as B type and 5&6 as C type. We roll the dice over and over until we get a number of every type. Let X be the number of rolls. We are...
  20. Vital

    I Conditional probability choosing from the objects

    Hello. I am reading an online stats book, and there is the following question, which I solved incorrectly, and I think I understand what is my mistake, but I will be grateful for your explanation, if I have incorrectly detected the logic behind my mistake. I am weak at math (trying to improve it...
  21. jk22

    B Probabilities with a Bell's state

    If we consider a singlet state : $$(|+-\rangle-|-+\rangle)/\sqrt{2}$$. And operators $$A=B=\textrm{diag}(1,-1)$$ I saw in a lecture that we can consider $$A\otimes B$$, it has multiple eigenvalues 1 and -1. It was then said : we can choose orthonormal basis vectors in each eigenspace. Hence...
  22. 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...
  23. Minish

    Finding the probabilities of macrostates for paramagnetic dipoles

    Homework Statement Hi! So I am given two different microstates of a system with 10 dipoles in a magnetic field B. I am asked to find the microstate that belongs to the macrostate with the highest probability, and to give that probability. I am also asked to find the same but with the highest...
  24. Mehmood_Yasir

    Pedestrian at a road crossing

    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...
  25. 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...
  26. C

    Expectation values and probabilities for spinors in a well

    Working on a homework at the moment involving spinors. The algebra isn't hard at all, I just want to make sure my understanding is right and I'm not doing this incorrectly. 1. Homework Statement An electron in a one-dimensional infinite well in the region 0≤x≤a is described by the spinor ψ(x)...
  27. CDL

    I Probabilities Associated with Sudden Changes in Potential

    Hi, I have a question about calculating probabilities in situations where a particle experiences a sudden change in potential, in the case where both potentials are time independent. For example, a tritium atom undergoing spontaneous beta decay, and turning into a Helium-3 ion. The orbital...
  28. M

    Probabilities and random variables

    Homework Statement In a given society, 15% of people have the sickness "Sa" , from them 20% have the sickness "Sb". And from those that don't have the sickness "Sa", 5% have the sickness "Sb" 1-We randomly choose a person. and we define: A:"the person having Sa" B:"the person having Sb"...
  29. M

    MHB Probabilities about length of songs

    Hey! :o We have a playlist with $2000$ songs. The length of the songs on the playlist are on average $3.5$ minutes (i.e. $3$ minutes and $30$ seconds) with a standard deviation of $1.7$ minutes. 1) Can we find the probability that a randomly chosen song is longer than $4.5$ minutes...
  30. S

    Expectation Value and Probabilities of Spin Operator Sy

    Homework Statement (a) If a particle is in the spin state ## χ = 1/5 \begin{pmatrix} i \\ 3 \\ \end{pmatrix} ## , calculate the expectation value <Sy>(b) If you measured the observable Sy on the particle in spin state given in (a), what values might you get and what is the probability of...
  31. L

    B Given a success rate of 1% and 100 tries....

    What is the probability of no success? 1 success? 2 successes? No success: 0.99^100 = 36.6% chance? 1 success: (0.99^99) + 0.01 = 37.97% chance? or (0.99^99) * (0.01)^1 = 0.369% chance ? You would think it would be fairly likely 2 successes: ? -Depends on formula for 1 success Can...
  32. T

    Electron spin probabilities

    Homework Statement After beta- decay electron and antineutrino comes out, electron is moving along z axis and it is moving with velocity v. It's spinor is ## \mid\chi\rangle=A\left(\frac{\sqrt{1+\frac{v}{c}}\sin\frac{\theta}{2}}{\sqrt{1-\frac{v}{c}}\cos\frac{\theta}{2}}\right) ## where A is...
  33. Observeraren

    B What is the sum of multiple probabilities

    If I have an asset that has a 10% chance to fail and I have ten of these assets in a basket, then what is the chance that one will fail in this basket? 10%?:partytime: What is the chance of 10 failing? 0,01%? Please also explain in some laymans terms. I am a total noob when it comes to...
  34. O

    MHB Probabilities involving dynamic sample space

    Sorry for the possible double post. I really need help with this...anyway let's assume we have chances of winning "something" anything...can be the lotto or whatever. We have A, B, C, D, E and each have a different chance of winning. We will also give them each a value, and the chance of winning...
  35. Danny Boy

    A Probability of obtaining general quantum measurement outcome

    The Fundamental Theorem of Quantum Measurement is stated as follows: Every set of operators ##\{ A_n \}## ##n =1,...,N## that satisfies ##\sum_n A_n^{\dagger}A_n = I## describes a possible measurement on a quantum system, where the measurement has ##n## possible outcomes labeled by ##n##. If...
  36. LarryS

    I Probabilities for degenerate eigenvalues?

    In non-relativistic QM, given a wave function that has a degenerate eigenvalue for some observable, say energy. There is a whole subspace of eigenvectors associated with that single degenerate eigenvalue. How is the measurement probability for that degenerate eigenvalue computed from the...
  37. aquaelmo

    Find the cdf given a pdf with absolute value

    Homework Statement Consider a continuous random variable X with the probability density function fX(x) = |x|/5 , – 1 ≤ x ≤ 3, zero elsewhere. I need to find the cumulative distribution function of X, FX (x). 2. Homework Equations The equation to find the cdf. The Attempt at a Solution FX(x)...
  38. G

    Logarithm and statistical mechanics

    Hello, I'll try to get right to the point. Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.
  39. BiGyElLoWhAt

    I Average profits and losses on a roulette table

    Recently, youtube suggested a video to me on roulette betting theory. The essence of the idea: Start with a small bet. If you win, bet the same, if you lose, double up. If you lose a second time, double again (or bet the sum of your previous bets) until you win. Start over at the small bet. The...
  40. Mehmood_Yasir

    I Conditional Expectation Value of Poisson Arrival in Fixed T

    Assume a Poisson process with rate ##\lambda##. Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...
  41. PhotonSSBM

    Probabilities for two spin systems interacting in isolation

    Homework Statement Consider a system A consisting of a spin 1/2 having magnetic moment ##\mu_0##, and another system A' consisting of 3 spins 1/2 each having magnetic moment ##\mu_0##. Both systems are located in the same magnetic field B . The systems are placed in contact with each other so...
  42. A

    Probabilities of the gene appearing that made humans smart?

    What exactly was the probability of the genes manifesting that give rise to human intelligence? The event must be incredibly rare seeing how there were several species of animals yet humans are the only one who can do complex math and language.
  43. H

    I Why aren't those two probabilities equal (exponential dist)

    I don't know how to type latex in the forums so I took a picture and uploaded it
  44. WeiShan Ng

    Average magnetic moment of the system

    I was reading the statistical physics textbook and was really confused with the notation: I don't understand the last part of the section. Why is that \sum_{\sigma = \pm1} \sigma P(\sigma) equals to \left< \sigma \right>? And what does \left< \sigma \right> actually mean? Is it the average...
  45. R

    Confusion between z and t values

    Homework Statement Hi, Alright so I have some confusion on when to use specific tests and the z vs t test. Given this example (not my homework) could someone please clarify. Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation...
  46. 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...
  47. M

    I Bootstrap percolation and percolation

    I read about bootstrap percolation and I would like to find links and similarities between bootstrap percolation and percolation (the initial model). I wonder if there is any result in percolation that is still valid in the bootstrap model.
  48. M

    MHB What is the Probability of Selecting a White Ball from Two Boxes?

    Hey! :o I am looking at the following: We have two boxes. In the blue box there are one black and three white balls. In the red box there are two black and four white balls. We choose one of the boxes and then we pick a ball from that box. So we have that $\Omega=\{B,R\}\times \{b,w\}$...
  49. stevendaryl

    B Probabilities associated with temporal uncertainty

    This seems like a simple matter, but apparently it is controversial: Is it meaningful to talk about probabilities for temporal uncertainty? If I find myself in a room without a clock, I might wonder what time it is. I know that I entered the room at 9:00, so it has to be later than that. I know...
  50. B

    I Are these alleged equal probabilities really equal?

    Let me explain myself. In the roulette game, you can bet on dozens, that means, betting to 12 numbers which are placed separately along the wheel. (12 out of 37 total numbers). But I had this curious doubt. if you bet on a sector of the roulette, composed by 12 numbers... the likelihood is...
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