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. J

    I David Deutsch (1985) attempt to solve the incoherence problem

    Can anyone elaborate on Deutsch's attempt to solve the incoherence problem? He postulates a continuously infinite set of universes, together with a preferred measure on that set. And so when a measurement occurs, the proportion of universes in the original branch that end up on a given branch...
  2. fluidistic

    Probability, genetic disorder related

    Note: not homework; thread moved to biology 1. Homework Statement It is not a textbook problem, but a real life scenario, as such I am not sure the solution exists. A child is born with a rare genetic condition (1 chance in a million). There is about 1/3 chance it comes from a mutation that...
  3. R

    Calculating Joint Probability Mass Function for Two Dice Toss

    Homework Statement Two dice are tossed. Let X be the smaller number of points. Let Y be the larger number of points. If both dice show the same number, say, z points, then X = Y = z. Homework EquationsThe Attempt at a Solution Find the joint probability mass function (X,Y) For (1,1) = 1/36, I...
  4. N

    Probability - Playing a Game with Two Fair Spinners

    Homework Statement Jack plays a game with two fair spinners, A and B. Spinner A can land on the number 2 or 3 or 5 or 7 Spinner B can land on the number 2 or 3 or 4 or 5 or 6 Jack spins both spinners. He wins the game if one spinner lands on an odd number and the other spinner lands on an...
  5. S

    MHB What is the probability that the second screw is also good?

    Question: All the screws in a machine come from the same factory but it is as likely to be from A as from factory B. The percentage of defective screws is 5% from A and 1% from B. Two screws are inspected and the first is found to be good. What is the probability that the second is also good...
  6. Zack K

    Importance of uncertainty and probability in research papers

    As someone who is interested in astrophysics, I always get emphasized on the importance of knowing statistics and error analysis in results of a calculation. However when I read about real physics papers, I never see any numerical solutions, just equations that demonstrate phenomena. I know that...
  7. iVenky

    Power of noise after passing through a system h(t)

    **Reposting this again, as I was asked to post this on a homework forum** 1. Homework Statement Hi, I am trying to solve this math equation (that I found on a paper) on finding the variance of a noise after passing through an LTI system whose impulse response is h(t) X(t) is the input noise...
  8. iVenky

    I White noise & 1/f noise after a system h(t)

    Hi, I am trying to solve this math equation on finding the variance of a noise after passing through a system whose impulse response is h(t) X is the input noise of the system and Y is the output noise after system h(t) if let's say variance of noise Y is σy2=∫∫Rxx(u,v)h(u)h(v)dudv where...
  9. R

    Birthday problem (probability)

    Homework Statement a) Consider a class with 30 students. Compute the probability that at least two of them have their birthdays on the same day. (For simplicity, ignore the leap year). b) How many students should be in class in order to have this probability above 0.5? Homework EquationsThe...
  10. F

    B Are Algebra, Geometry, Probability innate or cultural?

    Are the concepts of the rules of Algebra, Geometry and Probability things that all humans have some instinctive grasp at some level, or things that we basically learn from others, therefore cultural? Let me explain. I once saw an experiment with a mommy rat. She had 4 puppies, and someone put a...
  11. chwala

    Probability generating function when x is even

    Homework Statement [/B] A random variable x has a probability function ##G(t)##. Show that the probability that ##x## takes an even value is ## \frac 1 2 ( 1+G(-1))##Homework EquationsThe Attempt at a Solution [/B] ##G(t)= \sum_{k=0}^\infty p_k t^k ##... ## 1=P(X=even)+ P(X=odd)##...1 ##G(-1)=...
  12. Z

    Thermodynamics with use of Zusammenstand and probability

    Homework Statement Three-state system. The nucleus of the nitrogen isotope 14N acts, in some ways, like a spinning, oblate sphere of positive charge. The nucleus has a spin of lft and an equatorial bulge; the latter produces an electric quadrupole moment. Consider such a nucleus to be spatially...
  13. R

    Probability that both are defective

    Homework Statement Out of six computer chips, two are defective. If two chips are randomly chosen for testing (without replacement), compute the probability that both of them are defective. List all the outcomes in the sample space. Homework EquationsThe Attempt at a Solution I want to know...
  14. Demystifier

    Insights The Sum of Geometric Series from Probability Theory - Comments

    Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.
  15. J

    MHB Probability of Order Accuracy from A, C, or Not Accurate

    This is a problem I got for a review in probability and statistics The layout of the data is as follows: A B C D Order Accurate- 315 277 234 120 Order not - 34 50 35 18 accurate Like I said previously this is an or...
  16. J

    MHB Probability of 4 Adults Having Health Insurance

    66% of adults have health insurance, what is the probability that 4 adults selected at random have health insurance?
  17. Boltzman Oscillation

    B Help with determining the order

    Hello guys, Ive been struggling on determining whether something is ordered or unordered. The example i get is something like, "a burger with pickles, tomato, beef will be the same regardless of the way you make the order." Then in this case it would be ordered right? So when something is...
  18. S

    B Expectation of probability density function

    E(X) of probability density function f(x) is \int x f(x) dx E(X2) of probability density function f(x) is \int x^2 f(x) dx Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ? I tried to find E(5 + 10X) from pdf. I did two ways: 1. I found E(X) then using...
  19. M

    A Can Probit or Logit Models Be Used on Panel Data with 50 Subjects?

    Is it possible to use probit or logit models on panel data? Mons
  20. J

    B Probability that 1000 coin flips results in >600 tails

    What is the formula for this? Throwing a flipping coin N times, what is the probability that the number of tails results is higher than M?
  21. S

    MHB Probability of Missed Bullseye 3+ Times in 4 Shots

    1.Mary has two friends, Ann and Sarah. Mary will visit Ann this evening if the 19A bus arrives before 6PM. Otherwise, She will visit Sarah. The probability of the 19A bus arriving before 6pm is 40%. If she visits Ann, the probability that Ann will be at home is 10% and if she visits Sarah, the...
  22. J

    MHB Calculating the Probability of Winning a Masters Degree H-1B Visa Lottery

    I am fairly new to statistics and probability, I'm currently studying for a career change. I was asked this question but I don't know how to navigate through and get to any kind of an answer. Previously, the government conducted a lottery to award visas to 20,000 advanced-degree holders first...
  23. W

    I Are There Any Theorems Relating Joint Distributions to Marginals?

    Hi all, I was wondering if there exist any theorems that allow one to relate any joint distribution to its marginals in the form of an inequality, whether or not ##X,Y## are independent. For example, is it possible to make a general statement like this? $$f_{XY}(x,y) \geq f_X (x) f_Y(y)$$...
  24. R

    Probability of at least two happening

    Homework Statement A computer program is tested by 5 independent tests. If there is an error, these tests will discover it with probabilities 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. Suppose that the program contains an error. What is the probability that it will be found by at least two...
  25. CivilSigma

    Bayes' Theorem and a car starting probability

    Homework Statement A man owns two old cars, A and B, and has trouble starting them on cold mornings. The probability both will start is 0.1; the probability B starts and A does not is 0.1; the probability that neither starts is 0.4 a) Find the probability that car A will start. b) Find the...
  26. R

    Probability Example: A vs B | Competing Software Companies

    Homework Statement I was reading this example in a book... " Two competing software companies are after an important contract. Company A is twice as likely to win this competition as company B. Hence, the probability to win the contract equals 2/3 for A and 1/3 for B." Homework EquationsThe...
  27. F

    I Why does information need probability?

    I have only recently started studying information theory and I am stuck even trying to understand the basic problem and concept of information theory. My question is why does information need a probability? I have read some text that starts by introducing the idea that the amount of information...
  28. WMDhamnekar

    MHB Joint probability distribution of functions of random variables

    If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation. Hint:The joint density function can be obtained by differentiating the...
  29. C

    Two teams, A and B, are playing a series of games

    My attempt I used negative binomial to solve the problem, however I'm left with a polynomial that is difficult to solve? Is there any other way to approach this problem? I used the inequality because I'm trying to find the range of p. Since the probability of winning the series for team...
  30. Another

    I Please help explain the probability density function

    ##E = \frac{1}{2}(kx^2+m \dot{x}^2)## ## \frac{2E - kx^2}{m}=\dot{x}^2## ##\frac{dx}{dt} = \sqrt{\frac{2E - kx^2}{m}}## or ## dt = \sqrt{\frac{m}{2E - kx^2}}dx ## ⇒##= \frac{1}{\sqrt{\frac{2E - kx^2}{m}}}dx## My Question please help me. 1. I know ##T = 2\pi\sqrt{\frac{m}{k}} .## but i don't...
  31. F

    I Probability-To-Exceed (PTE) and Chi^2 distribution

    I would like to know the difference between the ##\chi^{2}## distribution and the PTE (Probability-To-Exceed) ? I must compare 2 data sets A and B and in the article I am reading, they talk about this PTE : For the moment, I only know the ##\chi^{2}## distribution with ##k=2## degrees of...
  32. H

    Looking for a book on Statistics and probability

    Hey There! I've finished reading The Introduction to Probability and Statistics book. Now I'm looking for another book as awesome as that was. Can anyone suggest me a book? Thanks in advance.
  33. K

    I Questions about probability

    Hi. What is the probability of throwing three dice at the same time while we are getting at least one of the particular number? I mean, for example, the probability of getting at least one number 2 among the three dice thrown Is it 1/6+1/6+1/6 = 3/6 = 1/2 ? How to explain to my daughter that...
  34. chwala

    Probability: 4 Coin Toss Cumulative Dist Table & Median

    Homework Statement (i) Construct the cumulative distribution table for the number of heads when the four coins are tossed. Coins are fair. (ii) Find the Median.Homework EquationsThe Attempt at a Solution (i) x 0 1 2 3...
  35. R

    I Polarization of light and probability

    I Dont understand how light polarized along a certain axis in a prepared state can then pass through a second polarizer that is at an angle to the first. The experiment I’m trying to understand starts with unpolarized light which then passes through a first filter which vertically polarizes the...
  36. R

    I Probability of getting a dice value

    If I have ten dice and pick one at random, given that I don't know anything about them the probability of getting a dice of value 4 would be 1/6? Given that I know that there is two dice of value 4, the probability of getting a 4 would be 2/10 = 1/5? Which would be correct in repeated trials?
  37. A

    MHB Best procedure to determine Lambda to calculate Poisson probability

    What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10. The question is, should I take the previous 10 days? Or calculate...
  38. S

    MHB Given probability density function find its cumulative distribution function

    Hi :) Here's my problem along with what I've done. Here is the problem: That is the p.d.f. of a random variable X. I have to find the cdf. I don't know which I should do so I tried it two ways. First: $\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...
  39. JackLee

    How to read a joint discrete table?

    Homework Statement [/B] Given a group of 100 married couples, let X1 be the number of sons and X2 the number of daughters the couple has. P(X1 = 0, X2 = 2) = f(0, 2) = 8 /100 = 0.08 2. Homework Equations The Attempt at a Solution I tried to look for a similar example online, I found this...
  40. B

    MHB Calculating theoretical probability

    Hello All, I have the following question in one of my tutorials. I need some help in resolving this. Background: A manufacturing company developed 40000 new drugs and they need to be tested. Question The QA checks on the previous batches of drugs found that — it is 4 times more likely that a...
  41. A

    Question regarding Ray Optics and Probability

    Let there be a sphere whose inner surface consists of a perfectly reflecting surface. It has a hole on it which allows a ray of light to enter. Give the angle made with the normal of the hole when the ray of light enters such that the probability that the ray comes out is the least?Assuming the...
  42. Hiero

    Probability of exactly n cars passing in time T

    Homework Statement In a short time dt, the probability that a car passes an observer is dt/τ. In a random time interval of length T, what is the probability P(n) that exactly n cars pass the observer? The Attempt at a Solution I can only see how to find P(0) the probability that no cars pass...
  43. pawlo392

    Calculating Expected Value and Variance of Coin Toss Results

    A coin had tossed three times. Let ##X##- number of tails and ##Y##- number of heads. Find the expected value and variance ##Z=XY##. My solution: We know, that ##Y=3-X##, so ##Z=(3-X)X## for ##X=0,1,2,3##. ##Z=2## for ##X=1,2## and ##Z=0## for ##X=3,0## So, ##E(Z)=E((3-X)X))= 2 \cdot ⅜ +2 \cdot...
  44. TytoAlba95

    What is the probability that the child is affected?

    Homework Statement The following pedigree shows that the trait is autosomal recessive. The probability of carriers in the population is 1/3. What is the probability that the child of III-3 and 4 is affected? 2. Relevant info The answer provided is 1/36. The Attempt at a Solution The...
  45. D

    A Probability of finding a k-subset in a d-dimensional random

    For a somwehat simple version of the problem, imagine a group of ten people (n=10), for which we observe two binary attributes (d= 2). One of the attributes tells us if any given person wears glasses (d_1) the other if any given person wears a hat (d_2). We know that in total 5 people wear hats...
  46. R

    MHB Probability of Student Seating Arrangement in Height Order

    a class has 9 students who are to be assigned seating by lot. What is the probability that students will be arranged in order from shortest to tallest? (assume that no two stusents are the same height.)
  47. 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...
  48. 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...
  49. 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...
  50. 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
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