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. LCSphysicist

    Probability and shoes

    > 20 shoes, from 10 pairs of shoes, are lined randomly. What is the > probability that there is a set of 10 consecutive shoes with 5 left shoes > and 5 right shoes? I thought that would be a good idea to imagine 10 shoes as one unique object, as follow: Instead enumerate 20 objects, let's...
  2. M

    I Probability: why can we use the Dirac delta function for a conditional pdf?

    Hi, I have a quick question about something which I have read regarding the use of dirac delta functions to represent conditional pdfs. I have heard the word 'mask' thrown around, but I am not sure whether that is related or not. The source I am reading from states: p(x) = \lim_{\sigma \to...
  3. L

    A Probability and entropy in an exponentially increasing sample space

    Hi, I'm new to PF and not really sure which forum may be the most appropriate to find people with an interest in probability and entropy. But the title of this forum looks promising. If you share an interest in this topic would be delighted to hear from you.
  4. S

    I Probability models and entropy

    Thinking of the common language notion of "entropy" as "uncertainty", how can running a simulation based on a probability model implement entropy increasing? After all, the simulation picks definite outcomes to happen, so (intuitively) there is less uncertainty about the future as definite...
  5. Addez123

    What is the Probability of Engine Failure for a Plane with Four Engines?

    Given we only have one number I assume we are to use Poisson distribution. Probability for a plane with two engines to fail require both engines to fail: $$P_2 = P_o(2) =p^2/{2!} * e^{-p}$$ Probability of a four engine plane to fail requires 3 or 4 engines to fail: $$P_4 = P_o(3) + P_o(4) =...
  6. chwala

    Probability distribution for discrete data

    this is a textbook problem shared on a whattsap group by a colleague... i have no problem in finding the value of ##k=0.08##, i have a problem with part (ii) of the problem. I have attached the solution here; how did they arrive at the probability distribution of ##y##? attached below is...
  7. iVenky

    I What are the statistics of probability of dying today vs age?

    I don't intend to sound macabre, but I was having this thought if I have to quantify the probability of someone dying given his age (in days) how would I go about quantifying that with a minimal accuracy (ok if it's not accurate but I just need some number with days). Has anyone ever worked out...
  8. Kaguro

    Probability dependence on potential

    I only know that if E>V, then the frequency would be higher where E-V is higher. But what does that have anything to do with probability?
  9. Another

    Problem about dot product in probability density problem

    I don't understand why ? ## \Psi ∇ ⋅ (A \Psi^ *) + \Psi ^* ∇ ⋅ (A \Psi ) = 2 ∇ ⋅(A \Psi ^* \Psi) ## form ## ∇ ⋅ (fg) = ∇f ⋅ g + f(∇ ⋅ g) ## Attempt at a Solution ## \Psi ∇ ⋅ (A \Psi^ *) + \Psi ^* ∇ ⋅ (A \Psi ) = 2 ∇ ⋅ (A \Psi ^* \Psi) - ∇\Psi ^* ⋅ A\Psi - ∇\Psi ⋅ (A\Psi ^*) ##
  10. Dale

    Insights Exploring Frequentist Probability vs Bayesian Probability

    Continue reading...
  11. nomadreid

    I Why P(A), and not P(A)(1-P(A))

    The summary says it all: why is the probability of an event not calculated by the probability that it is the event AND that it is not any other? Sounds silly, and I am certain the explanation is simple, but I don't see it.
  12. lelouch_v1

    Probability Density of ##x## (Wiener Process)

    Suppose that W(t) is just a Wiener process (i.e. a Gaussian in general). I want to know what the probability density for x, P(x), is. I started off by just assuming that I want to measure the expectation value of an observable f(x), so ##<f(x)>=\int_{W=0}^{W=t}{P(W)f(g(W))dW} \ \ ,\ \ x=g(W) ##...
  13. K

    Probability density in statistical Mechanics

    First of all, I've calculated the partition function:Z=1h3∫e−βH(q,p)d3pd3q=1h3∫e−β(p22m−12mrω2)d3prdrdθdz=2πL(2mπh2β)3/2e12βmω2R2−1ω2mβThe probability of being of one particle in radius $r_0$ is: p(r=r0)=1Z∫e−βHd3pd3q=∫1Z2πL(2mπh2β)3/2eβmrω22rdr So I've thought that because, by definition, the...
  14. L

    MHB COVID-19 in a Small Town: Probability of Infection

    Kindly assist with these questions: Data showed that 22% of people in a small town was infected with the COVID-19 virus. A random sample of six residents from this town was selected. 1) What is the probability that exactly two of these residents was infected? 2) What is the probability that at...
  15. M

    B Probability of n events over a time period

    Let's say you have a leaking tab, and the probability of a droplet in any given second is 1%, regardless of whether there was a drop previously. How would you calculate the probability of n drops in a minute? No drops in a second is 0.99, so no drops over a minute is 0.99^60. Hence one or more...
  16. entropy1

    I Probability in MWI and the Copenhagen interpretation

    Suppose we have an operator with three eigenvectors/eigenvalues ##e_1##, ##e_2## and ##e_3##. The operator measures wavefunction ##\psi##. Could we say that we find outcome ##e_x## with probability ##P(\psi,e_x)##, and could we extend this to an infinite dimensional operator as a spectrum of...
  17. bob012345

    I Is Probability in Quantum States Proportional to Energy Levels?

    Given a particle in a 1D box with a finite number of states ##m##, is the probability a particle is in a certain state ##n## equal to the energy of that state divided by the sum of energies of all states? In other words, given $$ E_n = \dfrac{n^{2}h^{2}}{8mL^{2}}$$ is $$P_n=...
  18. LCSphysicist

    I Probability involving a needle

    My approach to this problem is a little laborious, it involves three coordinates, probably it is right, but tiring and extensive beyond what the question wanted. Be the origin in the rectangle middle. It would be like: imagine a rectangle with opposite sides L and R with length l, so to find...
  19. L

    I Two vectors and two perpendicular lines

    In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...
  20. L

    I Probability that two points are on opposite sides of a line

    I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $ [0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction...
  21. E

    Online course for probability and statistics with emphasis on python

    I have been looking for a way to learn probability and statistics online and have searched but found nothing yet. I am looking for a course on probability and statistics that will not only teach me the basics but all there is to know about the subject. I would love it if the visualizations are...
  22. thaiqi

    I The probability when the wave-function collapses

    Hello, everyone. The projection postulate says the wave-function collapses to one of its eigenstates under measurement, does it talk about each probability with which the wave-function collapses to those possible eigenstates?
  23. Zack K

    Spin probability of a particle state

    Starting with finding the probability of getting one of the states will make finding the other trivial, as the sum of their probabilities would be 1. Some confusion came because I never represented the states ##|\pm \textbf{z}\rangle## as a superposition of other states, but I guess you would...
  24. B

    I Proving Probability of Union with Indicator Variables in Three Events

    "Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product." I'm taking...
  25. H

    B Is there a probability in QM that an event happens at time t?

    Hi PF, A(x,t) is the probability amplitude at time t that a particle is at x. If it was emitted at (0,0) the propagator gives its value. I wonder if QM can give the amplitude of time probability B(y,t) that an impact will occur (for a given y) at any t. consider a screen behind the two slits, it...
  26. archaic

    Exponential distribution probability exercise

    1) Since I want at least ##6## flights to come within ##2## hours, then the time interval between each should be, at worse, ##2/6=1/3## hours, and the probability is ##P(X\leq1/3)=1-e^{-1/3}##. 2) The probability that at best 5 airplanes arrive at the airport is...
  27. entropy1

    B Simple probability question: Suppose P(B|A)=1. Does that mean that P(A|B)=1?

    Suppose P(B|A)=1. Does that mean that P(A|B)=1?
  28. V

    Finding the probability of making a measurement E1

    Summary:: How to solve this? See attached. This is from Griffiths 3rd edition quantum mechanics textbook, problem 2.7: This is the solution for an infinite square well. I am to find the probability of obtaining E1 where En = n^2pi^2hbar^2/2ma^2. E1 = hbar^2pi^2/2ma^2 When finding the...
  29. U

    I Conditional distribution of geometric series

    Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
  30. S

    Probability of a state given the partition function

    If my partition function is for a continuous distribution of energy, can I simply say that the probability of my ensemble being in a state with energy ##cU## is ##e^{-\beta cU} /Z##? I believe that isn't right as my energy distribution is continuous, and I need to be integrating over small...
  31. D

    Probability that X1=0 given that N=1

    Given ##X_1,X_2,X_3## three independent random variables that can assume values 1 ,0 ,-1 with probabilities ##\frac 1 4 ,\frac 1 2 ,\frac 1 4 ##. Let ##S=X_1,X_2,X_3## and let N be the number of##X_i## assuming value ##0##. Knowing that N=1 i have to find what os the probability of ##X_1=0##...
  32. D

    What is the probability that Ugo and Zoe will not meet at the party?

    So first i compute the number of manners i can choose a sett of 3 ppl among 6 ##\frac {something}{\binom {6}{3}}## but then i don't understand from where should i pick up the two friends, from the set of those going to the party or what? Sorry but have problems on understanding how to deal...
  33. D

    Calculating Probability: \binom{8}{2} \binom{6}{2} \binom{4}{2}\binom{2}{2}

    My reasoning is : first i evaluate in how many manners i can choose 2 guys from the group of 8. ##\binom{8}{2} \binom{6}{2} \binom{4}{2}\binom{2}{2}## then i consider the probability of choosing the seventh floor is 1/4. But now I don't now how to proceed, supposing up to here i am correct...
  34. AN630078

    Probability Questions: Union, Intersection and Combinations

    Question 1: a) T' is the complementary event of T Therefore, T'=1-T In set T = {3,6,9,12} P(T)=4/12 =1/3 P(T')=1-1/3=2/3 b) The addition rule states; P(A ∪ B)=P(A)+P(B)-P(A⋂B) Therefore, P(S ∪ E) = P(S)+P(E)-P(S⋂E) S={1,4,9} P(S)=3/12=1/4 E={2,4,6,8,10,12} P(E)=6/12=1/2 (S⋂E)={4} P(S⋂E)=1/12...
  35. DarkMattrHole

    B How does the electric field of an electron compare to its probability wave?

    A single electron sitting in a void has an electric field that spreads out evenly in all directions as far as there is open empty space to allow it, is this roughly a correct statement? Let's say we now introduce a singe proton into the void, 100 miles from the electron - it will also have an...
  36. T

    I How do you calculate the probability without using the complement?

    Let's say I'd like to calculate the probability of getting at least one 4 when rolling two dice. That's 1 minus the probability of not getting any 4, i.e 1 minus the complement, 1-(5/6)^2. But how would I calculate without using the complement?
  37. Addez123

    15 people flip coins, find the probability distribution

    Say I make it so that the 2 coin flips count as a single number 1,2,3,4 representing head-head, head-tails, tails-head, tails-tails. Then what do I do? I'm just lost as to how I would even approach this problem.
  38. jisbon

    Conditional Probability + Poisson Distribution

    Confused and not sure if it is correct, but please do correct my steps. We let event B be that there are at least 3 customers entering in 5 minutes. Hence P(B) = 1- P(X=0)- P(X=1) - P(X=2) = ##1- \dfrac{e^{-5}5^{0}}{0!}-\dfrac{e^{-5}5^{1}}{1!}-\dfrac{e^{-5}5^{2}}{2!} ## = 0.8753... Now we let...
  39. O

    I Proving the product rule using probability

    I thought this was kind of a cool proof of the product rule. Let ##F(x)## and ##G(x)## be cumulative distribution functions for independent random variables ##A## and ##B## respectively with probability density functions ##f(x)=F'(x)##, ##g(x)=G'(x)##. Consider the random variable...
  40. Addez123

    Probability that the weight of the carts exceeds 255 tons

    Summary:: When filling up carts with iron the real weight deviates from the nominal value 10 ton. The standard deviation is .5. What's the probability that 25 carts exceed 255 ton? The arithmetic median value is: $$X \in N(25 * 10, 0.5 / \sqrt{25}) = N(250, .1)$$ $$P(x > 255) = 1 - P(x < 255)...
  41. tanaygupta2000

    Position and Momentum probability for +x direction

    For the region where V = 0, solving the schrodinger equation leads to the above value of wave function, psi = sqrt(2/L) sin(pi x/L) Since in the qus. it is not stated about the 'direction of movement' only restricted to +x direction, I think that the probability will be 1/2. And finding the...
  42. B

    A The meaningfulness item on math probability

    Hello, In probability math,because of math's nature that is merely quantitative and not a qualitative, for any case,it give you just a number; so, I think for every cases, there should be a boundary probability number that is " meaningfulness " just for that specified case and out of that...
  43. archaic

    Prob/Stats Introductory textbook for Probability and Statistics

    Hello! I'll be taking a probability and statistics course this semester. Does anyone know of any good textbook? I have access to an extensive catalogue of books on springer, so it would be extremely preferable for me if you could recommend something from there. Thanks.
  44. nomadreid

    I Probability of neutrino switching: more massive, more probable?

    In Scientific American, July 2020, the article "The Darkest Particle" by Louis and Van de Water, page 46, discussing the hypothetical sterile neutrino, there is the sentence: "Because sterile neutrinos are likely to be more massive than the regular flavors, however, particles could make the...
  45. J

    Question about an Eqn. in Shankar - wave function probability

    I don't see why it is not ##P(\omega)\propto |\langle \psi | \mathbb{P}_{\omega}|\psi\rangle |^2.## After all, the wavefunction ends up collapsing from ##|\psi\rangle## to ##\mathbb{P}_{\omega}|\psi\rangle.##
  46. entropy1

    I Compatibility of MWI with probability of outcomes

    Can MWI account for the probabilities of outcomes? If MWI says all outcomes are realized, is the probability that an outcome occurs then not 100%? How is this explained with the entanglement of the measured object and the measurement apparatus?
  47. A

    A Identification of a probability density function

    I have the following probability density function: $$f(x) =...
  48. marialovesphysics

    Data Management - Probability of Cards

    Here is my work so far: 52-13=39 There are 39 decks of cards left since the spades were removed. a) Then there 13 hearts therefore, (13/39 ) * (13/39 ) that would be two hearts but I am not sure what to do next. But I am sure that it would be 39 cards and 13 hearts on top (maybe) cus it is...
  49. Philip Koeck

    A Do bosons contradict basic probability laws?

    This questions was brought to my attention by Kazu Okayasu. According to probability theory the probabilities of mutually exclusive events add upp. As an example we can distribute 2 balls in two boxes with two compartments each. So there's a box on the left with a lower and an upper compartment...
  50. A

    A How to transform a probability density function?

    I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...
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