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. K

    I Probability in statistical mechanics

    Suppose we've an isolated box having ##N## classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical. Its said that the probability of having the configuration of ##n## particles in the left side is given as...
  2. J

    Can double masking increase efficacy by 126% against particles?

    Would the two masks together be 126% effective?
  3. M

    MHB Probability to get a chocolate snowman or a chocolate reindeer

    Hey! :giggle: $N\in \mathbb{N}$ christmas presents will be distributed. In every gift there is additional either a chocolate snowman or a chocolate reindeer. It will be independent of each other and with the same probability of giving out gifts with a chocolate snowman or chocolate reindeer...
  4. docnet

    Probability problem -- The joint PDF of X and Y is uniform on this rectangle....

    we haven't learned howw to do parts d-f yest. could you please give me a hand? (a) $$E(X)=E(Y)=\int_0^3\int_0^3\frac{1}{9}dxdy=\frac{3}{2}$$ (b)its typoe suppoesed to be W=Y-2 $$E(Z)=E(X-2)=E(X)-E(2)=-\frac{1}{2}$$ $$E(W)=E(Y-2)=E(Y)-E(2)=-\frac{1}{2}$$ (c) i guess joint pdf from ##E(Z)## and...
  5. L

    MHB Probability game - Simulation

    Two numbers are bring chosen by random from the set {1,2,3,4,5}. If the sum of the two numbers is even, you win 100 dollars, otherwise you win nothing. In order to participate in the game, you pay $80. What is the expected value and variance of the profit after 17 games ? I solved this one...
  6. issue

    A complex probability question

    There is a box with 2324784 bullets. Balls are divided into 5 groop and there is a definition for each groop and how many balls and what is the probability of getting each groop. The game with the return of the balls List the probability and quantity groop 1 906192 0.389796213 groop 2 1006880...
  7. A

    B Calculating the coprime probability of two integers in a different way

    The probability that two randomly chosen integers to be coprimes is known to be equal to ## \prod_{2}^{\infty}(1-\frac{1}{p^2})=6/\pi^2## I tried to conceptualize the problem in the following way but got different results. Suppose that we pick up an integer at random which could be either prime...
  8. docnet

    The joint probability - two coins

    (a) The probability mass function of X and Y: $$p_{X,Y}(1,1)= \frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}$$ $$p_{X,Y}(1,0)=\frac{1}{2}\cdot \frac{1}{4}=\frac{1}{8}$$ $$p_{X,Y}(0,1)= \frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}$$ $$p_{X,Y}(0,0)= \frac{1}{2}\cdot \frac{1}{4}=\frac{1}{8}$$ or $$...
  9. A

    MHB Want to use probability algorithm for my web app

    I'm a freelance web developer and right now I'm stuck into a mathematical(probability) issue where I'm developing an answer filtering tool. This tool works as follows: 1) There are six questions that the user will be asked 2) Based on the answer of first question, my server will decide which...
  10. R

    MHB Probability regarding balls

    Hi guys, I have a maths question about Probability, hope somebody can help me out, thank you very much. There are 5 red, 4 blue and 3 green balls in a bag. Two balls are taken from the bag, one after the other. If the first ball is not replaced, and it was not green. What is the probability...
  11. evceteri

    Neutron scattering probability distribution

    Hi, I'm reading Chapter 2-II of of Duderstadt & Hamilton's "Nuclear Reactor analysis". In the section "Differential scattering cross sections with upscattering" it is discussed the situation in which neutrons suffers elastic scattering collisions in a hydrogen gas at finite temperature T and the...
  12. hackedagainanda

    Intro Math Introductory Probability without Calculus

    Are there any good introductory texts aimed at students with only a working knowledge of High School Algebra? I currently have: Probability: An Introduction by Samuel Goldberg I do plan on eventually learning Calculus, but I would like to start learning probability sooner.
  13. M

    MHB How can we find probability space and events

    Hey! :giggle: Give a probability space and events $A$, $B$ and $C$ such that $P [C \mid A]>P [C \mid A \cup B]$.So we want that $$\frac{P[C\cap A]}{P[A]}>\frac{P[C\cap (A\cup B)]}{P[A\cup B]}=\frac{P[(C\cap A)\cup (C\cap B)]}{P[A\cup B]}$$ But how can we find these events ? :unsure:
  14. M

    MHB Probability that a randomly selected chicken has the genetic modification

    Hey! :giggle: In the case of the reproduction of the chicken of the breed Krueper, in some cases it can be that they are extremely short-legged, which has as a result that they already die in the egg. A group of researchers now considers that they have found two main genetic modifications...
  15. M

    MHB How can we calculate the probability?

    Hey! :giggle: The distribution of a random variable $X:\Omega\rightarrow \mathbb{R}$ is given by the probability function $p_X$ as follows: (i) Calculate the distribution function $F_X$ of $X$. (ii) Let $Y(\omega)=|X(\omega)|$ for all $\omega\in \Omega$. Determine $p_Y$. I have done the...
  16. M

    I Using a probability argument I got the sum. Can it be done directly?

    ##\sum\limits_{k=0}^{n-m} \frac{\binom{n-m}{k}}{\binom{n}{k}}\frac{m}{n-k}=1##. Can be derived from question. For ##n\ge m##, pick ##m## marbled out of a set size ##n## labeled from ##1## to ##n##, what is probability distribution of minimum of the number labels on the marbles? The terms is...
  17. M

    Probability of getting 5 consecutive heads before 2 consecutive tails

    Hi, I was just having a go at this question, and I suppose I have two sub-questions: 1. Does my approach look correct? 2. If my approach is correct, is there a better approach than what I have done? Original question: Suppose we are flipping a fair coin. What is the probability that we observe...
  18. chwala

    Solve the probability without replacement problem

    My interest is solely on question ##6.V## only. This is the question paper; This is my working to solution; I was able to use tree diagram in this question. This is the mark scheme: I would appreciate other ways of tackling the problem. Cheers! :cool:
  19. L

    I The Prisoner Paradox: Examining the Relativity of Probability in Everyday Life

    For example, If I have a constant probability of 50% to death per day (sorry for the macabre example but I find it good); Every day I wake up, I can say:"Today, at the end of the day, I have a probability of 0.5 to be live( or to be dead)!". Now, one can calculate the probability that after...
  20. chwala

    Solve the probability distribution and expectation problem

    This is the problem; Find my working to solution below; find mark scheme solution below; I seek any other approach ( shorter way of doing it) will be appreciated...
  21. J

    Calculating the Probability of Particles Traveling Through Different Size Pores

    Homework Statement:: Not homework, but I want to calculate the probability of different size spheres traveling through different size pores. Relevant Equations:: V=4/3πr^3 volume of sphere. Volume of 60nm diameter particle = 1.13x10^5 nm^3. Volume of 70nm particle = 1.8x10^5 nm^3 Hi all, this...
  22. J

    A Probability distributions in an ordered set of extracted elements

    Hello, I tried looking for an existing solution for the following problem: "Assume that S is a set of d elements, and R is a total order relation on S. Assume that n elements are randomly extracted from S, and then they are ordered according to R. Which is the probability that in the i-th...
  23. C

    Prob/Stats Books on Combinatorics, Permutations and Probability

    Hello! I am looking for textbooks to relearn Combinatorics, Permutations Combinations and Probability and also Matrix algebra( decomposition, etc). I had done these many years ago and the course/books provided to me at that time weren't that great. So I want to relearn this with a more...
  24. AngleWyrm

    I Confidence as used in Probability

    Example problem A casino offers you a gamble with a 1% chance of winning a try. How many tries will it take to win at least once? Solution For this example, I chose 95% confidence, a willingness to be wrong once in twenty...
  25. PainterGuy

    Random variable and probability density function

    Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...
  26. T

    B Probability in a roll of dice

    I was trying to calculate the probability of throwing only one six when throwing a pair of dice. Using the formula for non-mutually exclusive events : P(A)+P(B)-P(A,B) I get 1/6+1/6-1/36=11/36 but when I count all the 36 possibilities on paper I get 10/36 ways of getting only one 6. What am...
  27. benorin

    I Juicy game probability calculation

    This Q was asked in a chat room: Find the number of pulls (draws from the distribution) required to have a 50% chance of getting a full set of event 5*s (*'s denote rarity in the game) from the Magic Tower Summon (of the Empires & Puzzles mobile game) which has 5x event 5*s each at a 0.2% drop...
  28. M

    Probability: Dice Game between 20- and 12-sided dice with re-rolls

    Hi, I was reading around and found this problem. I have seen some discussion about the solution (but nothing verified) with some disagreement. Problem: One person has a 12 sided die and the other has a 20 sided die. They each get two rolls and they can each chose to stop rolling on either one...
  29. Amitkumarr

    I Finding bias of the coin from noise corrupted signals

    Suppose there are two persons A and B such that both have a personal communication system which can transmit and receive bits. B has a biased coin whose bias is not known. A asks B to toss the coin 2000 times, send a 0 when a tail comes up and a 1 when a head comes up. It is known that whatever...
  30. D

    Probability of 1 Letter E in 4 Randomly Selected Letters from 'ENCYCLOPAEDIA

    Four letters are randomly selected from the word ENCYCLOPAEDIA. Find the probability that one letter E will occur in the selection of 4 letters. My calculation was 2C1 x 11C3/13C4 = 0.46 However, the answer is 2/33. I think the repeated letters might need to be accounted for. I divided the...
  31. D

    Combinatorics: calculating Oz Lotto odds for divisions

    In Oz Lotto, balls are numbered 1 to 45. Nine are selected, seven of which are winning numbers and two being supplementary numbers. Players select seven numbers. The odds of winning can be found here: https://www.lottoland.com.au/magazine/oz-lotto-everything-there-is-to-know.html I tried...
  32. S

    I Does this defense lawyer's probability argument sound like BS?

    Either that, or the author is a typical "pop scientist" author that doesn't understand probability too well. https://nautil.us/issue/4/the-unlikely/the-odds-of-innocence
  33. M

    MHB Probability - Amount of money in pocket

    Hey! :giggle: The amount of money a student in the Accounting Department has in his pocket is a random variable that follows the normal distribution, with an average price of $30$ euros and a variance of $100$. a) What is the probability that a student has $25$ to $35$ euros in his pocket...
  34. M

    MHB Probability : minutes that customers get served

    Hey! :giggle: The average time of customer service at the cash registers of a department store is $30$ people per hour. If a new customer arrives at the checkout, then calculate the probabilities: a) to be served in less than 15 minutes b) to need to be served from 6 to 8 minutes a) Do...
  35. K

    How to calculate the probability of winning top prize of lottery?

    Just take a simple example, Among 1-36 numbers, 6 balls are randomly drawn without putting back the ball. Only if you match the six numbers exactly you win the top prize, order doesn't matter. I think it should be 1/36 * 1/35 * 1/34 * 1/33 * 1/32 * 1/31, is it? [Moderator's note: moved from a...
  36. Moara

    Conditional probability and criminal DNA analysis

    We know that ##P(A-) = (95\% \cdot 0.5\% + 5\% \cdot 98.5\% )## and ##P(guilty \ and \ A-) = (95\% \cdot 0.5\%)##, so letter a) is just ##P(guilty \ and \ A-)/P(A-)##. What I tried to do in letter b) was again using the conditional probability theorem. First calculating the probability that...
  37. Lynch101

    I The Probability Distribution and 'Elements of Reality'

    The below comment by @vanhees71 is an interesting one and I would be interested in exploring its implications. I am inclined to think that we can draw certain inferences about nature based on how we interpret the probability function and what it tells us about the elements of reality of the...
  38. M

    Probability of Getting 1 Jack when taking 5 cards from a deck

    Hi, I was looking through probability questions and attempting it. Question: (a) What is the probability of picking up ONLY one jack when taking 5 cards from a 52 card deck (b) Considering that in the first 3 cards there is a jack what is the probability that the jack was the first card to be...
  39. M

    Variance of a point chosen at random on the circumference of a circle

    Hi, I was looking at this problem and just having a go at it. Question: Let us randomly generate points ##(x,y)## on the circumference of a circle (two dimensions). (a) What is ##\text{Var}(x)##? (b) What if you randomly generate points on the surface of a sphere instead? Attempt: In terms of...
  40. karush

    MHB Gaussian Probability: 1 to 2 in tikz

    \begin{tikzpicture}[scale=0.6] %preamble \usepackage{pgfplots} \newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma \begin{axis}[every axis plot post/.append style={ mark=none,samples=50,smooth}, % All plots: 50 samples, smooth, no marks...
  41. karush

    MHB Probability of z > 1.28: Visualizing the Gaussian Curve

    ok I was able to get the graph of P(z>1.28) \begin{tikzpicture} %preamble \usepackage{pgfplots} \newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma \begin{axis}[every axis plot post/.append style={ mark=none,samples=50,smooth}, % All...
  42. N

    A Probability Density Function: Converting Experimental Observations to PDF

    Hi All I am currently doing Master in data science. I came across the function PDF probability density function which is used to find cumulative probability(range) of a continuous random variable. The PDF probability density function is plotted against probability density in y-axis and...
  43. gentzen

    A Probability via Expectation and Callen's criterion

    In the thermal interpretation, the collection of all q-expectations (and q-correlations) is the state of a system. The interpretation of q-expectations is used only to provide an ontology, the apparent randomness is analysed and explained separately. This may be non-intuitive. Callen's criterion...
  44. C

    I Desorption probability calculation

    Hi all! I would like your assistance with wrapping up my thoughts regarding the following problem. Say I have Am-241 nuclide, which emits alpha particle in every decay (for the sake of this discussion, let's assume that 100% of the decays lead to a daughter nuclide, Np-237 + an alpha...
  45. A

    A The name of this probability density function

    Does anyone knows the name of the probability density function f(y) =aeby-cy2
  46. entropy1

    I Why are there no probability amplitudes in MWI?

    If I consider the MWI, one of the notions for what happens during measurement is that the initial wavefunction, if I use Dirac notation and two dimensions, ##|A\rangle+|B\rangle## undergoes the transformation ##(|A\rangle+|B\rangle)|E_{before}\rangle \rightarrow...
  47. 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)...
  48. 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...
  49. 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...
  50. 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...
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