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. jk7297

    A BCS energy and Cooper pair probability amplitude

    This question is for those familiar with the BCS theory of superconductivity or familiar with R.D. Mattuck's book “A Guide to Feynman Diagrams in the Many-Body Problem.” I am working my way through the book, and I am stumped by some of the problems at the end of chapter 15 (superconductivity)...
  2. SchroedingersLion

    A Equilibrium, entropy, probability - Release of constraints

    Hi everyone, I have a fundamental question to the first part of Swendsen's Intro to StatMech and Thermodynamics (book). Suppose we have two isolated systems of volumes ##V_1## and ##V_2##. We distribute ##N## ideal gas particles across the two systems with total energy ##E##. Suppose we bring...
  3. Falgun

    Prob/Stats Looking for a probability and statistics textbook

    I want to learn some probability & statistics on my own. I am well versed in Calc 1-3 , elementary ODEs and very little linear algebra. I want a comprehensive , introductory textbook which is NOT COOKBOOK STYLE. I might be self studying AP statistics next term so if the book covers everything I...
  4. YouAreAwesome

    How many combinations? (High school math problem)

    Summary:: Year 11 Extension 1 Math problem (Australia) How many combinations can be made from a 4 digit pin code if we can only use two numbers to form our pin code, and we MUST use 2 distinct numbers. E.g. 1112, 4334, 9944, 3232. But NOT 1111, 2113, 0992 etc. We're using the numbers 0-9 and...
  5. T

    Probability of Receiving 2-4-6-7-Queen-King Sequence from 40 Cards

    A quick translation of the question: We have a deck of 40 cards, containing four suits (hearts, diamonds, spades, clubs), in which each suit has an ace, a queen, a jack, a king and six number cards (2 to 7). From the deck, six cards are distributed randomly and successively to a player who picks...
  6. L

    Statistics: Verifying a Probability Proof

  7. L

    How to Start a Problem I'm Struggling With

    I really don't know what to do for this problem. I looked at similar threads but couldn't seem to grasp the idea of it. I would like help on how to start.
  8. S

    Probability related to Normal Distribution

    a) Let X = distance walked on Friday and Y = distance walked on Saturday X ~ N (12, 0.192) and Y ~ N (10, 0.52) Let A = Y - X → A ~ N (-2 , 0.2861) P(Y > X) = P(Y - X > 0) = P(A > 0) = 9.2 x 10-5 But the answer key is 0.026 Where is my mistake? Thanks
  9. M

    Probability: posterior predictive probability

    Hi, This is a another question from the same MIT OCW problem in my last post. Nevertheless, I will try explain the previous parts such that the question makes sense. I know I am usually supposed to make an 'attempt', but I already have the method, but just don't understand it. Questions: 1...
  10. M

    Probability: Prior Distribution for a Continuous Variable

    Hi, I was attempting this problem from the MIT OCW website probability and statistics course. Context: When Jane started class, she warned Jon that she tends to run late. Not just late, but uniformly late. That is, Jane will be late to a random class by ##X## hours, uniformly distributed over...
  11. S

    MHB Calculating probability from linear regression parameters

    I'm a bit stuck on how to calculate the probability in part b from the linear regression parameters. I tried plugging the parameter values into the linear regression model: Y =β0+β1X+ε, ε∼N(0,σ) So P(Y=y| X=40) = 2.85 + 0.07 * 40 + 1^2 P(Y=y|X=40) = 5.65 But I don't think this is the...
  12. P

    I Find P(X+Y>1/2) for given joint density function

    Hey everybody, :smile: I have a joint density of the random variables ##X## and ##Y## given and want to find out ##P(X+Y>1/2)##. The joint density is as follows: $$f_{XY}(x,y) = \begin{cases}\frac{1}{y}, &0<x<y,0<y<1 \\ 0, &else \end{cases}$$ To get a view of this I created a plot: As...
  13. Buzz Bloom

    Polling for estimates of probability our universe is infinite

    I have noticed that many PF participants seem to favor a cosmological model that our universe is infinite (and flat), but some (like myself) favor a finite universe. I have become curious about the subjective probability distribution, so I am hoping that many PF participants will post their...
  14. R

    I Probability of finding a particle outside its light cone

    Say we just created a particle (high probability of one-particle state), is the probability of a very far away detector getting triggered at the time of creation (probability of finding a particle outside of its light cone) zero according to QFT? Since we can detect particles and make...
  15. karush

    MHB Aux24 probability student range

    The frequency chart below shows the cumulative number of Ms. Hernandez's science students whose test scores fell within certain score ranges All test scores are whole numbers. Cumulative number Score range of students $\begin{array}{lll} &65-70& 12\\ &65-80& 13\\ &65-90& 19\\ &65-100& 21...
  16. Marioweee

    Probability of measuring an eigenstate of the operator L ^ 2

    Calculate, with a relevant digit, the probability that the measure of the angular momentum $L ^2$ of a particle whose normalized wave function is \begin{equation} \Psi(r,\theta,\varphi)=sin^2(\theta)e^{-i\varphi}f(r) \end{equation} is strictly greater than ##12(\hbar)^2##...
  17. A

    MHB Probability of Rolling Sum > 3 with Two Dice

    Two fair dice are rolled. What is the probability of rolling a sum that exceeds 3?
  18. Q

    MHB Probability of O & AB Blood Types

    The blood groups of 200 people are distributed as follows: 40 have type A blood, 75 have type B blood, 60 have type O blood, and 25 have type AB blood. If a person from any of the group is selected at random, what is the probability that this person has an O blood type? How about the AB blood...
  19. S

    MHB What is the probability of exceeding maximum weight with a normal distribution?

    The weight of goats at a farm is normally distributed with a mean of 60 kg and a standard deviation of 10 kg. A truck used to transport goats can only accommodate not more than 650 kg. If 10 goats are selected at random from the population, what is the probability that the total weight exceeds...
  20. M

    Probability: Multivariate distribution change of variables

    Hi, I was attempting the problem above and got stuck along the way. Problem: Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf: f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ##...
  21. chwala

    Find P(B) and P(A/B) by considering the given probability question

    i managed to find the values...i am seeking alternative approach to the problem. See my attempt below.
  22. chwala

    Solve the conditional probability question

    My question is on part ##c## of the problem. Kindly see attached question,...is the second approach correct?
  23. E

    Probability a particle is in a certain region

    I think I made an error somewhere. In ##[0,a]## I let ##\varphi(x) = \varphi_1(x) := p\sin{kx}## whilst in ##(\xi, \infty)## I let ##\varphi(x) = \varphi_2 (x) := re^{-\gamma x}##, and the constraints at ##x=\xi## are \begin{align*} \varphi_1'(\xi) = \varphi_2'(\xi) &\implies pk\cos{k\xi} =...
  24. A

    Solving QM Problem: Fermi's Golden Rule & Transitional Probability

    Hello all, I would like some guidance on how to approach/solve the following QM problem. My thinking is that Fermi's Golden Rule would be used to find the transitional probability. I write down that the time-dependent wavefunction for the free particle is...
  25. Purpleshinyrock

    Probability of girls on a team getting shirts with pair numbers

    Summary:: probability, Probability and Combinatorics Hello, I need someone to check If I correctly analized the probability of the following event: A group of younglings formed by 5 girls and 5 boys, Is going to divide in two teams of 5 elements each to play a game. a) suposing that the...
  26. Frabjous

    I Probability fun time: Proof that 1/3=1/2=1/4

    Forgive me, I am not a probability guy, so I am unsure how well known this is. I was trying to figure something out and found this. I found it cool. Here's the explanation. The first solution is a fraction (damn scanner!) Oops! From Kendall Geometrical Probability (1963)
  27. Y

    A What is the Probability of Sampling a Set with a Certain Size?

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  28. Leo Liu

    Answer to Probability Question: Finding Outcomes Under Constraint

    The answer is ##4/7##. I know the denominator which is number of unconstrained outcomes is $$\dfrac{\dbinom{8}{2}\dbinom{6}{2}\dbinom{4}{2}}{4!}$$But I am not sure how to find the number of outcomes under the constraint. Could some give me a hint? Thanks.
  29. H

    How to Calculate Absolute Gamma Probability from Relative Intensities?

    Hi All, I want to ask how to calculate the absolute gamma probability from relative intensities ( found on the tables of nucliedes) following alpha or beta decay. I mean the probabilities that all add to 1. Many thanks.
  30. R

    I Understanding Measurement Probability in Electron Spin: Explained by Susskind

    Susskind explains how if you prepare an electron along any axis n (with an electromagnet) and then measure it along any other axis m, the probability of finding the electron with spin up or spin down is given by the angle between the axis. I have left out the linear algebra, because my question...
  31. S

    B Value of t for Probability Generating Function

    My questions: 1) What about if t = 2? Is there a certain meaning to ##G_X (2)## ? 2) PGF for uniform distribution is ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and for t = 1 ##G_X (1)## is undefined so ##G_X (1) =1## is not true for all cases? Thanks
  32. I

    Probability distributions using 4 dice

    Let X denote the largest number shown on the four dice. P(X ≤ x) = (x/6)4 , for x = 1,2,3,4,5,6. Complete the following table: x 1 2 3 4 5 6 P(X=x) 1/1296 15/1296 65/1296 175/1296 369/1296 671/1296 The values in red are the answers, I don't understand how the answers were found. Thanks.
  33. Monoxdifly

    MHB [ASK] Probability with Factors

    In a bag there are m white balls and n red balls with mn = 200 and there are more white balls than red balls. If two balls are taken randomly at once and the probability of taking two different colored balls is \frac{40}{87} then the value of 2m + 3n is ... A. 30 B. 45 C. 50 D. 70 E. 80 Okay...
  34. M

    MHB Probability that the total annual profit is not more than 800000

    Hey! An insurance office has $2500$ contracts with mean annual profit (per contract) $\mu=330$ and standard deviation $\sigma=540$. Calculate the probability that the total annual profit is not more than $800000$. I have one the following: The annual total profit should be not more than...
  35. L

    B Probability Distribution with a resettable count to win

    Hi hi, I was thinking about this, all of this starts playing a game, I'll show a simplification: We ca win several times. We have a count ##n##, where is the max number of rolls until you win, let's say we can win a ##m## amount. In every roll we can win ##m## with a probability of ##p##. If...
  36. redtree

    I Separation of variables for Named Probability Density Distributions

    Given a probability density distribution ##P(\vec{x})##, for what named distributions is the following true: \begin{equation} \begin{split} P(\vec{x}) &= P_1(x_1) P_2(x_2) ... P_n(x_n) \end{split} \end{equation}
  37. W

    I Probability of ending up with Rhode island

    Today the youtube algorithm directed me to a video of someone playing geoguessr in the US until he got all of the 50 states at least once. If all states are equally likely this would take about 50*ln(50) tries. 50*(1+1/2+1/3+ ... 1/50), so about 200. Of course all states aren't equally...
  38. M

    Probability notation: question about joint and conditional probability

    Hi, Just a quick question about conditional and marginal probabilities notation. Question: What does ## p(a|b, c) ## mean? Does it mean: 1) The probability of A, given (B and C) - i.e. ## p[A | (B \cap C)] ## OR 2) The probability of (A given B) and C - i.e. ## p[(A | B) \cap C] ## I was...
  39. anemone

    MHB Roots of equation and probability

    Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$. Find the probability that $\sqrt{2+\sqrt{3}}\le |v+w|$.
  40. Armine

    Proof of a formula with two geometric random variables

    The image above is the problem and the image below is the solution I have tried but failed.
  41. M

    MHB Probability to get the correct message

    Hey! :giggle: One of the techniques we are using at the digital communications to improve the reliability of a noisy communication channel, is to repeat a symbol many times. For example, we can send each symbol $0$ or $1$ say three times. More precisely, applying the rule of majority, a $0$...
  42. H

    If the wave function is normalized, what is the probability density at x?

    The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value? (A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3 Ans : D
  43. hdp12

    Bernoulli and Bayesian probabilities

    Summary:: Hello there, I'm a mechanical engineer pursuing my graduate degree and I'm taking a class on machine learning. Coding is a skill of mine, but statistics is not... anyway, I have a homework problem on Bernoulli and Bayesian probabilities. I believe I've done the first few parts...
  44. M

    Probability Density Functions: Transformation of Variables

    Hi, I have a question about probability transformations when the transformation function is a many-to-one function over the defined domain. Question: How do we transform the variables when the transformation function is not a one-to-one function over the domain defined? If we have ## p(x) =...
  45. M

    Finding marginal distribution of 2d of probability density function

    Hi, I have question about finding marginal distributions from 2d marginal pdfs that lead to the probabilities being greater than 1. Question: If we have the joint probability distribution ## f(x, y) = k \text{ for} |x| \leq 0.5 , |y| \leq 0.5 ## and 0 otherwise. I have tried to define a square...
  46. S

    MHB Probability of Surviving 10 Years for A, B, and C

    The probability that at least 2 of 3 people A, B, and C will survive for 10years is 247/315. The probability that A alone will survive for 10 years is 4/105 and the probability that C alone will die within 10 years is 2/21. Assuming that the events of t.he survival of A, Band C can be regarded...
  47. F

    I What stands for probability in QFT?

    When we apply creation operator in vacuum we certainly have one particle,similarly for annihilation operator.Then what is stand for chance(probability) in QFT?
  48. N

    Find the probability that this number is in set A but not in set B

    Find the probability that this number is in set A but not in set B. (please see attachments) Is the answer to this simple maths secondary school homework 3/14. If not please help?
  49. 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...
  50. 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...
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