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. TheMathNoob

    Multiplication rule for conditional probabilities

    Homework Statement Selecting Two Balls. Suppose that two balls are to be selected at random, without replacement, from a box containing r red balls and b blue balls. We shall determine the probability p that the first ball will be red and the second ball will be blue I am confusing Pr(A|B) and...
  2. L

    Can you suggest a probability and statistics manual?

    Hello, I am a student in the Actuarial bachelor's degree. I've just finished my Stats and Probabilities class in which we completed the Introduction to Probability book by Ross. I really enjoyed this class and would like to continue studying this branch can you suggest a book that continues...
  3. A

    Two state system probabilities

    I started to study statistical physics from a book, and it starts with basics about statistics and probabilities (which are things mostly new for me). In the book there is the following statement: "The simplest non-trivial system which we can investigate using probability theory is one for...
  4. P

    MHB What are the odds of rolling 4, 5, 6, and 7 of a kind with 7 dice?

    Could somebody please tell me, in layman's terms, the answer to the following: When 7 standard six sided dice are thrown what are the odds of 4 showing the same number, 5 showing the same number, 6 and 7. Also the odds of 7 sixes being thrown. This is all to do with a game used to raise money...
  5. tomdodd4598

    Probabilities of the States of a Spin 1 Particle

    I have been following a series of Leonard Susskind's lectures called 'Quantum Entanglements' (Part 1). In general, he explains how to find the probabilities of measurements of spin ½ particles' states, both single particles and pairs of them. I have learned the following: how to use the 2x2 spin...
  6. ChrisVer

    Neutral Meson Oscillations (probabilities)

    I am looking into the probability for : \mathcal{P}(B^0 \rightarrow B^0). I said that if I start from a state |B^0> = \frac{1}{\sqrt{2}} (|B_L> +|B_H>) with L(ight)/H(heavy) are the mass eigenstates, then after some time t the state will evolve: |B^0(t) > = e^{-iHt} |B^0>= \frac{1}{\sqrt{2}} (...
  7. ChrisVer

    Toss coin probabilities (Bino vs Gauss approx)

    Hi, I calculated the probability to this scenario: getting between 3 and 6 Heads after tossing a coin for n=10 trials... The binomial probability for this is: P(3 \le k \le 6) = \sum_{k=3}^6 Bi(k;p,n)= \sum_{k=3}^6 \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} =\sum_{k=3}^6 \frac{10!}{k!(10-k)!}...
  8. gfd43tg

    Angular momentum values and probabilities

    Homework Statement Homework Equations The Attempt at a Solution For a angular momentum ##J##, there exists the eigenvalue problems $$J^{2} \mid j \hspace{0.02 in} m \rangle = j(j +1)\hbar^{2} \mid j \hspace{0.02 in} m \rangle $$ $$ J_{z} \mid j \hspace{0.02 in} m \rangle = m \hbar \mid j...
  9. A

    Explaining Probabilities in a deterministic world

    Let's assume we're living in a mechanical deterministic world. Now do you agree that any uncertainty is a result of lack of knowledge? We flip a symmetric coin. The equations of movement are deterministic, but the outcome is uncertain, with probability 50% tail or head. Thus, it's the initial...
  10. W

    Calculating measurement probabilities for the CHSH game

    Hi All, First, the context for this question can be found in Wilde's Quantum Shannon Theory text on the arXiv, specifically the section starting at the bottom of page 98, entitled "Entanglement in the CHSH Game". My particular question relates to Exercise 3.5.11 on page 100 of the same...
  11. D

    Uniform distribution- probabilities

    Hello, I am stuck at this exercise: 1. Homework Statement X ~ U(0, a), a > 0 and Y = min(X; a=2). - Find the cumulative distribution function of Y -Is the variable Y continuous ? Homework Equations 3. The Attempt at a Solution [/B] The density function for X is f(t)= 1/a if 0≤t≤a , 0...
  12. E

    MHB Interpreting Ratio of Two Probs: Can Higher Prop Survive?

    If two 5-year survival probabilities are p1=.55 and p2=.41 the ratio is .55/.41 = 1.34 but since probabilities are in [0, 1] should I take the log first? Which is the more appropriate way to interpret the ratio? the ratio of logs is Log(.55)/log(.41) = .671 Which is less than one although...
  13. D

    MHB Mutually Exclusive Probabilities: A vs. B

    Hey everyone. Question! If the probability of A is .3 and the probability of B is .8, can A and. B be mutually exclusive? This is confusing me!
  14. S

    Determine probabilities involving exponential distribution

    Homework Statement Problem(s): Suppose that X has an exponential distribution with mean equal to 10. Determine the following: (a) P(X > 10) (b) P(X > 20) (c) P(X < 30) (d) Find the value of x such that P(X < x) = 0.95. Correct answers: (a) 0.3679 (b) 0.1353 (c) 0.9502 (d) 29.96 Homework...
  15. D

    Combinatorial argument probabilities

    Hello 1. Homework Statement Consider fk N*→N, k≥0 fk(n)=Σ jk n>0 We're looking to establish these identities, using a combinatorial argument fk(n)=n if k=0 fk(n)= 1/(k+1) [ nk+1 + Σ ( i , k+1) (-1)k+1-ifi(n) ] if k>0 the sum is about i from i=0 to i=k-1 and (i, k+1) is the combination...
  16. D

    How do probabilities balance the odds?

    Let's consider a simple roulette game, where one may either get red or black. Since the probabiblity P of an event A, P(A) is defined as the relative frequency at which the event occurs, if we get red , say 3 times in a row, it is very likely that the next random pick will be black. My question...
  17. O

    Calculating Transition Probabilities & Expected Values of a Markov Bus Chain

    Homework Statement -A bus is moving along an infinite route and has stops at n = 0, 1, 2, 3, ... -The bus can hold up to B people -The number of people waiting at stop n, Yn, is distributed Poisson(10) and is independent from the number waiting at all the other stops. -At any given stop each...
  18. S

    "A computer system uses passwords [..]": Probabilities=?

    Homework Statement PROBLEM(S): A computer system uses passwords that contain exactly eight characters, and each character is one of 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote the set of all possible passwords, and let A and B denote the events...
  19. jk22

    Limited probabilities : a nonsense ?

    Suppose i have an experiment which can give result 0,4 but that the probability p(4)<=1/sqrt(2). Does this make sense in a frequentist approach since if i do the real experiment once and got 4 then the probability (statistics a posteriori) for 4 is 1 which is a dumb counterexample.
  20. R

    Determine the state |n> given results and probabilities [QM]

    Homework Statement In a spin-\frac{1}{2} system all particles are in the state |\psi\rangle. 3 experiments performed and are separate, the results are as follows: Particle in state |\psi\rangle, measured S_z = \frac{\hbar}{2}, with P=1/4 Particle in state |\psi\rangle, measured S_x =...
  21. M

    Calculating Probabilities of Particle in a L-box After Wall Movement

    Homework Statement A particle is in the ground state of a L-box. At t = 0 the wall at x = L is suddenly moved to 2L. (a) If an energy measurement is made after the wall is moved, what is the probability of measuring the energy to be that of the ground state of the new box? (given solution...
  22. L

    Probability of Visiting a Physical Therapist: Analysis and Solution

    Homework Statement Among a large group of patients recovering from shoulder injuries, it is found that 22% visit both a physical therapist and a chiropractor, whereas 12% visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.14 the probability that a patient...
  23. M

    Modified Harmonic Oscillator probabilities

    Homework Statement The e-functions for n=0,1,2 e-energies are given as psi_0 = 1/(pi^1/4 * x0^1/2)*e^(x^2/(2*x0^2) psi_1 =... psi_2 =... The factor x0 is instantaneously changed to y= x0/2. This means the initial wavefunction does not change. Find the expansions coefficients of the...
  24. C

    Positive probabilities for neg sums of uniformly distributed variables

    I've been thinking about the Central Limit Theorem and by my understanding it states that the sum of randomly distributed variables follows approximately a normal distribution. My question is if you have, say, 100 uniformly distributed variables that range from 0 to 10, their sum has to be...
  25. B

    Transition probabilities subject to Lloyd's finite information limit?

    This is a question about The Computational Capacity of the Universe by Seth Lloyd. It seems to me that arbitrary real numbers cannot be part of the state of the universe, since they carry an infinite amount of information. There are transition probabilities from the current state of the...
  26. W

    QM:Expectation values and calculating probabilities

    Homework Statement An operator \mathbf{A}, corresponding to a physical quantity \alpha , has two normalized eigenfunctions \psi_1(x)\quad \text{and}\quad \psi_2(x), with eigenvalues a_1 \quad\text{and}\quad a_2. An operator \mathbf{B}, corresponding to another physical quantity \beta, has...
  27. M

    Markov equilibrium probabilities

    Homework Statement A Markov process with 4 states evolves in unit time steps with a transition probability matrix given by P = |0.5 0 0.3 0.2| |0.1 0.4 0.2 0.3| | 0 0.4 0.4 0.2| |0.1 0.1 0.1 0.7| Find the equilibrium probabilities for this system. What is the most likely state...
  28. jk22

    Sum of the probabilities equals 3 in bipartite covariance ?

    If we consider a bipartite system as in EPRB experiment we get the probabilities : p(++)=p(--)=1/4*(1-cos(theta)) p(+-)=p(-+)=1/4*(1+cos(theta)) p(+A)=p(+B)=p(-A)=p(-B)=1/2 Thus the sum of all the probabilities equals 3... How does that come ? Is it because in fact there are only...
  29. R

    HMM with continuous observation - PDFs to probabilities

    So I am working with a Hidden Markov Model with continuous observation, and something has been bothering me that I am hoping someone might be able to address. Going from a discrete-observation HMM to continuous-observation HMM is actually quite straightforward (for example see Rabiner's 1989...
  30. DaTario

    QM and a new way to calculate probabilities

    Hi All, Would it be reasonable to state that, together with the quantum mechanics appearance, there appeared also a new way to calculate probabilities? I have never heard any teacher pointing out that QM has brought a new probability method. They speak of amplitudes of probability but...
  31. A

    MHB Two probabilities question

    i got the answer to the following problem wrong: "there are four children in in the family. what is the probability that there are three girls, given that the youngest child is female?" my (updated) answer: the youngest is female, so three out of two children must be female. there are three...
  32. A

    System described by probabilities

    Imagine some kind of system, where you have at t=0 N single atoms (a gas). Now in a later instant dt there is a certain probability that 2 atoms will have collided and formed a 2-atom unit. Similarly dt after this event there is a certain probability that this 2-atom unit has either collided...
  33. W

    Calculating Probabilities for Quantum Spin Measurements

    Homework Statement The problem is a series of problems all relating to each other. I have gotten most of this done but I will add my answers and continue until the last part which is my main question: We choose a magnetic field in the z direction. Use your knowledge of 2-dimensional...
  34. Ackbach

    MHB Wackerly/Mendenhall/Schaeffer Problem 2.19: Assignment of Probabilities

    The Bureau of the Census reports that the median family income for all families in the United States during the year $1991$ was $\$35,353$. That is, half of all American families had incomes exceeding this amount and half had incomes equal to or below this amount (Wright 1992, p. 242). Suppose...
  35. U

    Possible measurement, eigenvalues of eigenfunctions and probabilities

    Homework Statement Suppose the angular wavefunction is ##\propto (\sqrt{2} cos(\theta) + sin (\theta) e^{-i\phi} - sin (\theta) e^{i\phi})##, find possible results of measurement of: (a) ##\hat {L^2}## (b)##\hat {L_z}## and their respective probabilities. Homework Equations...
  36. H

    C/C++ App that counts probabilities in c++

    Hey,I am making an app that counts probabilities in c++(win32){for those who knows}. I have found three formulas of equation,but I need some help for one. For example we have a die,and we hit it 6 times. Can we count with an formula,the probabilities to have the number "5",three times...
  37. J

    Do Past, Unrelated Events Affect Future Probabilities?

    For a long time now I've been thinking about something I find odd about probabilities. For example, let's say that I win the lottery, something that is extremely unlikely. What is the probability that I will be struck by lightning? The odds of someone winning the lottery and getting struck by...
  38. K

    MHB Properties of probabilities

    Let $(\Omega,\mathcal F,P)$ be a space of probabilities and $(A_n)_n\subseteq \mathcal F.$ Show that a) if the sequence satisfies $\inf\{P(A_n)_n:n\ge1\}=\alpha,$ with $\alpha\ge0,$ then $P\left( \bigcap\limits_{n=1}^{\infty }{\bigcup\limits_{k=n}^{\infty }{{{A}_{k}}}} \right)\ge \alpha .$ b)...
  39. F

    Calculating entropy, microstate/macrostate probabilities

    Hi all, could somebody look over my answer please. I'm pulled the equation I used off the internet but can't remember where so I'm not sure what it's called. I took a picture of my answer as I thought it would be easier to read than fiddling with symbols here. QUESTION ANSWER ATTEMPT...
  40. W

    Probabilities on Non-Standard Models.

    Hi, I think I read here; maybe not, that , within a non-standard model of the Reals, it is possible to have probabilities , say over an interval, so that each point has non-zero probability. AFAIK, the transfer principle ( a.k.a elementary equivalence of models) does not disallow having a...
  41. beyondlight

    How to solve poisson process probabilities

    Homework Statement Let X(t) and Y(t) be independent Poisson processes, both with rates. Define Z(t)=X(t)+Y(t). Find E[X(1)|Z(2)=2]. 2. The attempt at a solution...
  42. M

    Need a refresher on finding probabilities of a wave function.

    After a few months off (yay summer/internships), I'm 'back in the saddle' and I'm trying to catch up with my Q-mech. I have a wave function which is given as a particle sliding freely on a circular wire: \Psi = A(1 + 4cos\phi) I need to find the corresponding probabilities. So I know that I...
  43. G

    Conditional Probabilities in Tennis

    Homework Statement This isn't really homework, just reviewing for a test. This is problem 3.17 in 'A modern introduction to probability and statistics: understanding why and how' Dekking. But since it can be seen as a HW problem, might as well post here. Question: You and I play a...
  44. S

    Effect of changing probabilities on interference

    Let's say we are doing interference experiment via say a Mach-Zehnder http://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer In the Mach-Zehnder two of the mirrors are half silvered. The photon has a 50% chance of passing through the half-silvered mirror and 50% chance of...
  45. M

    What is the Probability of Selling 8 Listings Out of 10?

    Homework Statement You are a real estate salesperson and you currently have 10 listings. Past experience has shown that you will sell approximately 70% of your listings. If sales are independent: What is the probability that you make exactly 8 sales? Homework Equations The Attempt...
  46. F

    CDF: Prove a>0 for F(x) = 1 - e^(-ax) - axe^(-ax)

    Homework Statement Given the function F(x) = 1 - e^(-ax) - axe^(-ax) for x>=0 and 0 elsewhere, for which values of 'a' does the function constitute a CDF? Homework Equations The Attempt at a Solution I started with saying for that to occur, provided x1>x2 -> F(x1) > F(x2)...
  47. M

    How Do You Calculate and Combine Probabilities in a Two-Card Poker Game?

    Let's say we have a very simple game of two-card poker, where the deck consists of only 4 aces and 4 kings. There are 28 possible combinations: Six pairs of aces, six pairs of kings, and 16 ace-king hands. In play against a single opponent, both players are dealt two cards, face down. The...
  48. M

    What is the probability of males liking or being neutral towards the commercial?

    Finding probabilities through general addition rule Homework Statement http://postimg.org/image/q9ztktpot/ Homework Equations P(A or B)=P(A)+P(B)-P(A and B) P(A and B)=P(A)P(B)The Attempt at a Solution The question is for part b). It said it was only focused on the mens so i ignored all the...
  49. S

    Intuition behind superposition probabilities?

    Hi, This is silly, but I'm confused as to how we use the unit circle as a representation of particle states. I've been given the formula (probability distribution)=α|0> + β|1> (or in radians sin∅|0> + cos∅|1> or something), where the probability of a particle being in a certain state is the...
  50. A

    In RBW there are no probabilities per se.

    continuing from another thread. or rewriting physics ? and epistemology and ontology :rolleyes: another acausal model. "The causaloid framework" http://arxiv.org/abs/gr-qc/0509120v1 "Quantum theory is a probabilistic theory with fixed causal structure. General relativity is a deterministic...
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