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. 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...
  2. A

    Expected value of a function of a random variable

    Homework Statement Let X be a random variable. It is not specified if it is continuous or discrete. Let g(x) alway positive and strictly increasing. Deduce this inequality: $$P(X\geqslant x) \leqslant \frac{Eg(X)}{g(x)} \: $$ where x is real. Homework Equations I know that if X is discrete...
  3. H

    Probability at a temperature T that a system has a particular energy

    Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a small system that has 2 normal modes of vibration, with natural frequencies $$\omega_1$$ and...
  4. volnei_cipriano

    B Probability density in physics

    Studying probabilistic density, I know that a function that is integrated between two limits presents a probability. But how should I think to solve a problem where I need to determine the probability of a particle being seen being that its moment liner is a constant value
  5. P

    MHB Calculation of probability with arithmetic mean of the sum of random variables

    Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...
  6. H

    Prove that there exists a graph with these points such that....

    Homework Statement Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and $$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$ Prove also the ##10## in the inequality can't be replaced...
  7. DaTario

    I Probability Amplitudes and the History of Science

    Hi All, I would like to know who was the first scientist to use probability amplitudes in solving either math or physics problems. Best wishes, DaTario
  8. R

    MHB Likelihood of things happening on with regularity depending on initial probability

    Hello All, Firstly, what a wonderful place for people to help one and other. I am glad that I have stumbled across : ) I have posted this query in advanced because to me there it is advanced but if it is basic to others then please excuse me. I can easily work out the probability of...
  9. Chris Miller

    B Probability of n consecutive tails in n coin tosses

    The probability P of k consecutive tails occurring in n coin tosses is 1 - (1 / F) where F is element n+2 in the k-step Fibonacci series divided by 2n. As n approaches infinity, P approaches 1 for any value of k. Is the above statement true? EDIT: sorry, mistyped heading question, can't edit...
  10. J

    MHB What is the probability that the ball was labeled

    A tank contains 50 balls. 10 are frosted (F) and labeled (L), 23 are mild (M) and unlabeled, 7 are frosted and not labelled (N), 10 are mild and labeled. 1. A ball is randomly selected from the tank and it was frosted. What is the probability that the ball was labeled? 2. The 1st ball was...
  11. Mathfan7

    B How to determine the wavefunction of photons?

    Hi, I'm sorry if this question has already been answered somewhere and I'm just too incompetent to find it, buuut: As the title already says, I really do not get that part of quantum physics (if you can even say I'm getting ANY part at all...). As I searched all Google for an answer I just...
  12. R

    MHB Find the probability that all 7 witnesses would pick the same person.

    In a homicide case, 7 different witnesses picked the same man from a lineup. The line up contained 5 men. If the identification were made by random guesses, find the probability that all 7 witnesses would pick the same person.
  13. R

    MHB Success Rate of College Grads: Probability of 1+ Job in 1 Year

    A study conducted at a certain college shows that 55% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that among 7 randomly selected graduates, at least one finds a job in his or her chosen field within a year of graduating
  14. R

    MHB What is the probability that she selects none of those containing errors

    A IRS auditor randomly selects 3 tax returns from 45 returns of which 15 contain errors. What is the probability that she selects none of those containing errors? Round to four decimal places.
  15. entropy1

    B Calculating Entropy for Independent Coin Toss Sequences with Varying Differences

    If we have two sequences s1 and s2, both of N coin tosses, is the entropy of getting two sequences that are exactly the same then lower than sequences of which can be said that they differ by x incident tosses? Is the entropy of getting sequences s1 and s2 that differ by N/2 tosses the highest...
  16. Flabbergast

    Standard deviation and probability for decay

    Homework Statement (b) A nuclear research reactor produces radiation for neutron scattering measurements. A safety procedure shuts the reactor down if a radiation level monitoring detector measures more than 3 counts per minute. In a test, 156 counts are recorded during a random 24 hour...
  17. W

    Probability Theory: Order statistics and triple integrals

    Homework Statement Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##. a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##. b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...
  18. 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...
  19. Eclair_de_XII

    If duelist A shoots first, what is the probability A wins?

    Homework Statement "Mr. A and Mr. B fire at each other until one of them is hit. Each duelist has the same probability of hitting each other each time a shot is fired. Mr. A has probability ##a## of hitting, and Mr. B has probability ##b## of hitting. Given Mr. A shoots first, calculate the...
  20. Chris Miller

    Probability of randomly selecting 32 bytes in ascending order

    I chose the thread level for what I imagine to be the difficulty of this problem, not my own math ability. Obviously (?) there are 256^32 possible series. So knowing how (i.e., how to calculate) how many of these are in ascending order would give the answer. Is there some polynomial time algorithm?
  21. S

    Probability of Getting a Total of 15 or Higher in Three Dice Throws

    Homework Statement If two identical dice are thrown, what is the probability that the total of the numbers is 10 or higher? [Hint: list the combinations that can give a total of 10 or higher.] [2] Two dice have been thrown, giving a total of at least 10. What is the probability that the throw...
  22. Abdul Quader

    I Probability current density

    In order to check if a quantum particle crosses a barrier or not, isn't calculating ##|\psi(x)|^2## enough in that particular region ? Why do we need to calculate probability current density for that matter?
  23. S

    Calculating the expected value for a probability

    Homework Statement An ice-cream store makes 150 ice-cream balls every day. The cost of making each ice-cream ball is $3. The price of an ice-cream ball is $8. The demand distribution is as follows: 100 ice-cream balls with probability 25%, 150 ice-cream balls with probability 50%, and 200...
  24. N

    How Do You Solve These Continuous Probability Problems?

    Homework Statement f(x) = (3/4)(-x^2 + 6x - 8) for 2 < x < 4 (0 elsewhere) A) Find F(x) integral 2 to 4 ((3/4)(-x^2 + 6x - 8))dx B) Use F(x) to find P(3 < X < 3.5) integral 3 to 3.5 ((3/4)(-x^2 + 6x - 8))dx 11/32 C) Use F(x) to find P(X > 3.5) 1-( P(3 < X < 3.5)) = 21/31 D) Find E(X)...
  25. W

    Probability theory: Understanding some steps

    Homework Statement Hi all, I have some difficulty understanding the following problem, help is greatly appreciated! Let ##U_1, U_2, U_3## be independent random variables uniform on ##[0,1]##. Find the probability that the roots of the quadratic ##U_1 x^2 + U_2 x + U3## are real. Homework...
  26. B

    MHB Probability Density Function of Average Value of Log-Normal Trials

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  27. Krushnaraj Pandya

    Probability of decay of a nucleus

    Homework Statement I want to know why probability disintegration per second of a radioactive nucleus does not depend on time lived by it. Homework Equations N/N(initial)=e^(-λt) The Attempt at a Solution According to the above equation, the probability should increase with the passage of...
  28. D

    Approximating Probability for a Wave Function

    Homework Statement The wavefunction at t = 0 is given by $$\Psi = N*e^{-\frac{r}{a_0}}$$ where ##r = |\mathbf{x}|##. ##a_0## is a constant with units of length. The electron is in 3 dimensions. Find the approximate probability that the electron is found inside a tiny sphere centered at the...
  29. C

    B Rough probability that aliens would visit the Earth?

    According to a survey in 2017, nearly half of the Americans believe, that aliens visit earth.However, I think that a lot of conditions have to be met, so that aliens would visit earth. First of all, they have to exist. Meaning their home planet has to be a habitable planet (but only a very...
  30. W

    I Time-dependence in probability amplitudes

    Hi all, I am rather confused about the following concept. Assistance is greatly appreciated! A time-dependent probability amplitude can be written as $$\langle a_k| e^{-\frac{i}{\hbar}\hat{H}t} |\psi\rangle$$ where ##a_k## is an eigenvalue. Suppose I want the x-representation of the ket, I can...
  31. Eclair_de_XII

    What is the probability of getting r heads before s tails?

    Homework Statement "Toss a coin repeatedly. Denote the event of getting heads or tails on the ##i##-th try by ##H_i## and ##T_i## respectively, where ##P(H_i)=p## and ##P(T_i)=1-p##, for some ##0\leq p \leq 1##. Now denote by ##E## the event of getting ##r## consecutive heads before ##s##...
  32. N

    Probability of Opening a Pushbutton Lock

    Homework Statement A password has numbers 0-9 in it. The password is 5 digits, repeats are not allowed, and order doesn't matter (I just have to have the correct 5 digit buttons depressed). A. If I guess at the password, what is the probability that the box will open? B. If I have completely...
  33. S

    I Probability that X is less than a set

    Hi everyone, I am currently working through the textbook Statistical Inference by Casella and Berger. My question has to do with transformations. Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in...
  34. W

    Probability Theory: Simultaneous picks

    Homework Statement [/B] Hi all, I have an issue understanding the concepts pertaining to the following problem, assistance is greatly appreciated. I understand the "flow" of the problem; first find the probability of obtaining balls of the same colour, then use the geometric distribution...
  35. Sushmita

    A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x)....

    Homework Statement [/B] A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x) = (1/2) kx2 . If the spring constant of the oscillator is suddenly doubled, then the probability of finding the particle in ground state of new potential will be? (A)...
  36. Vital

    I Understanding part of the multinomial formula

    Hello. I am trying to decipher the formula, making sure I understand what exactly is going on in each part of the expression. I will be grateful for your guidance, corrections and help. Below I show the formula and the example for only 3 possible outcomes (in general it would be k) \(p =...
  37. Z

    Can Bayes Theorem Predict the Next Winner in a Team Matchup?

    Homework Statement Team 0 and Team 1 have played 1000 games and Team 0 has won 900 of them.[/B] When the two teams play next, knowing only this information, which team is more likely to win? Homework Equations P(X,Y) = P(YlX) x P(X) = P(XIY) x P(Y) (Not Sure) The Attempt at a Solution Hi, I...
  38. 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...
  39. E

    Calculations using Standard Deviation and Mean

    Homework Statement Homework Equations Chebyshev's Theorem: The percentage of observations that are within k standard deviations of the mean is at least 100(1 - (1/k2))% Chebyshev's Theorem is applicable to ANY data set, whether skewed or symmetrical. Empirical Rule: For a symmetrical...
  40. N

    What is the Probability of a Group of 4 Getting a Job Out of 12 Applicants?

    Homework Statement Assume a job has 12 applicants, and 4 job openings. I want me and my 3 friends to all get the job. What is this probability. Homework Equations ! Factorial and Permuation & combinatoins The Attempt at a Solution Number of possible solutions: 12C4 = 495 Possible ways Number...
  41. W

    Probability Theory: Need help understanding a step

    Homework Statement Discrete random variables ##X,Y,Z## are mutually independent if for all ##x_i, y_j, z_k##, $$P(X=x_i \wedge Y=y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_k )$$ I am trying to show (or trying to understand how someone has shown) that ##X,Y## are also independent as a result...
  42. W

    Probability Theory, work check

    Homework Statement Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance. Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?Homework Equations The Attempt...
  43. W

    I Independent events and variables

    Hi all, I have a few questions regarding the issue of independence. Many thanks in advance. ##\textbf{1}## If I find that some events ##A, B, C## obey the following formula $$P(A \cap B \cap C ) = P(A)P(B)P(C)$$ it does not necessarily mean that a) they are mutually independent and b) ##A##...
  44. W

    Probability Theory: Multinomial coefficients

    <Moderator's note: Moved from homework.> Hi all, I have an issue understanding a statement I read in my text. It first states the following Proposition (Let's call it Proposition A): The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n...
  45. R

    I Rayleigh distribution and the probability of a point

    Okay, so I just found out about the Rayleigh distribution being the radial distribution of a point composed of normal distributed cartesian components. And this is because of the area element, right? But how then can the joint density of the cartesian component's distributions equal that of the...
  46. Prez Cannady

    I Confused by this result for the tensor product of two vectors

    Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...
  47. P

    Probability per atom and per second for stimulated emission to occur

    Homework Statement We are investigating hydrogen in a plasma with the temperature 4500 ºC. Calculate the probability per atom and second for stimulated emission from 2p to 1s if the lifetime of 2p is 1.6 ns Homework Equations ##A=\frac{1}{\Sigma \tau}## $$A_{2,1} = \frac{8*\pi *h *...
  48. Aleoa

    I Probability density functions for velocity and position

    In the first volume of his lectures (cap. 6-5) Feynman asserts that these 2 can be the PDF of velocity and position of a particle. Under which conditions it's possible to model velocity and position of a particle using these particular PDFs ? ps: Is the "Heisenberg uncertainty principle"...
  49. malawi_glenn

    I Probability distribution of random events

    Hi Imagine we have a lottery, with chance of winning 1 in 1000 (1/1000). I have made computer simulations in order to find confidence levels for winning. At 1000 bought lottery tickets, the confidence of winning is 64.1% and 2000 bought lottery tickets the confidence of winning is 87.1% By...
  50. C

    MHB Find Probability of two polygons

    There are five hexagons. The edges of each hexagon have been colored with one of three colors randomly. If you pick two hexagons randomly without replacement, what is the probability that they are the same? (Rotation is okay). The total space or denominator is 3^(2×6), therefore we have...
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