Probability Definition and 1000 Threads

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. Agent Smith

    B Probability of hitting a target

    Say an archer has a probability p of hitting the target. Given n shots at the target, the number of hits = np The standard deviation of hits = ##\sqrt{np(1-p)}## Say p = 0.7 Given 100 shots, my expected/average number of hits = ##100 \times 0.7 = 70## The standard deviation for the number of...
  2. M

    Reflection probability in Quantum Mechanics

    For this, The solution is I have doubt where they got the reflection probability formula from. Someone may know how to find it. I think that ##R + T = 1##. But I'm not sure where the transmission probability formula comes from either. Kind wishes
  3. rcktbr

    Getting the probability distribution of a random variable

    X and Y are discrete random variables with the following joint distribution: a) Calculate the probability distribution, mean, and variance of Y. My attempt: I have calculated the probability for different values of Y and X using the following equation: ##\text{Pr(Y = y)}## = ##\sum_{i=1}^l##...
  4. P

    I A controversial application of Bayesian reasoning

    The gain in odds that aliens are visiting Earth (A) due to ##n## independent reports of close encounters (C) is given by: $$\frac{\rm Odds(A|C)}{\rm Odds(A)}=\left[\frac{\rm Prob(C|A)}{\rm Prob(C|\bar A)}\right]^n.$$ Let us assume that we have good cases such that an alien explanation (##a##) is...
  5. hongseok

    B Questions about state vectors

    ∣r⟩,∣l⟩,∣i⟩, and ∣o⟩ can all be expressed as expressions for ∣u⟩ and ∣d⟩. So, given the state vector ∣ψ⟩ = α∣u⟩ + β∣d⟩, is it possible to know not only the probability of ∣u⟩ but also the probability of ∣r⟩ and ∣i⟩? ∣ψ⟩ can be expressed as an expression for ∣r⟩, ∣l⟩ or ∣i⟩, ∣o⟩.
  6. rcktbr

    Searching for knowledge about probability and statistics

    I am currently studying Economics at undergraduate level and want to enhance my knowledge about probability and statistics in order to better understand econometrics.
  7. P

    Two cards are drawn from a deck without replacement

    Let ##X## be the event that the first draw is a picture card Let ##Y## by the event that the second is a picture card Then the probability that at least one of the cards is a picture card is the probability of ##X## union ##Y## and has the formula Equation 1 ## P\left(X \cup Y \right) = P...
  8. badr

    A Research about chained functions

    I am looking for some academical concept to work on 3 parts : 1) Real and imaginary analysis of two functions describing 2 events 2) If the first event's function is imaginary and the second is real , how can we analyse the intersection that show how the imaginary function turned out...
  9. Agent Smith

    B Relative And Absolute Probability (Probability Of Picking A Red Ball)

    A bag contains ##4## red balls, ##3## green balls and ##2## blue balls. A random ball is selected from this bag. P(ball is red) = ##P(R) = \frac{4}{9}## P(ball is green) = ##P(G) = \frac{3}{9}## P(ball is blue) = ##P(B) = \frac{2}{9}## P(ball is not red) = ##P(\neg R) = \frac{3}{9} +...
  10. Agent Smith

    B Quantum Superposition And A Coin

    Confetior ... I'm a layman with minimal physics background, but the most happening place in physics, going by media articles, seems to be Einstein's relativity and Max Planck et al's quantum physics. I did a little reading on Schrodinger's cat and what I could gather is unless an observation is...
  11. G

    Probability distributions for Maxwell-Boltzmann, B-E, F-D

    I don't even understand what question is being posed here. The answers given by the author are as follows: These are numbers, potentially very large ones.
  12. A

    I How do I express that a 100% occurrence in a small sample is low "confidence"?

    In experiment A: I observe an event 2 times in 2 trials. In experiment B: I observe an event 100 times in 100 trials. In both cases, I calculate a frequency of 100% In both cases, I calculate a 95% confidence interval of (1, 1). But intuitively the result of experiment B is "stronger" than...
  13. mozzie789

    I Total Winning Saddlecloth Numbers

    There is a race meeting with a number of races. Each race has horses entered and each horse has a (saddlecloth) number, 1….x, with x being the number of horses entered in each race. Each horse has a known probability of winning a race (reflected in its price). I’d like to set a market where...
  14. badr

    I Probability of passing a disease down to a child or a group of siblings?

    Hello. So I got a question about heredity . Let's say the probability of inheriting schizophrenia is 6 % if one parent is affected. So i know that for 6 % probability, there is 1.2 kid out of 5 who will inherit that illness . So is it better not to have kids in this case ?
  15. badr

    I What branch of electrical engineering uses a lot of probability?

    Hello everybody. I have used probability in troubleshooting electronic pcbs , but the complexity of the designs forced me to apply it in other branches of electrical engineering. What branch in electrical other than maintenance uses probability theory ?
  16. P

    I The set of nonzero values of pmf is at most countable

    The claim is Proof. If it was ##P(X = x ) > 0## for uncountably many ##x##, then there would exist an ##N\in\mathbb N## such that ##P(X = x) > \frac{1}{N}## for uncountably many ##x## (1). This in turn would imply that there exist countably infinite many ##x## such that ##P(X=x)>\frac{1}{N}##...
  17. alima

    Comp Sci Which is the probability of JohnCalls given Burglary? Why?

    Questions: P (JohnCalls|Burglary) ? Why? Source of the image: Artificial Intelligence: A Modern Approach - Third Edition, by Stuart Russell and Peter Norvig. My attempt at solving: using Bayes' Theorem = P (A|B) = ( P(B|A) * P(A) ) / P(B) P(JohnCalls|Burglary) = P(J|B) = ( P(B|J) * P(J) /...
  18. Lucchini

    Changing the minimum value of a probability matrix

    I am doing a study of the possibility of transition between 12 different events. I have a dataframe with these key events (listed from 1 to 12) over a period of time. I constructed a transition probability matrix between these events (photo of the matrix is attached below). As I don't have a...
  19. Lotto

    B Probability - two possible points of view?

    I know it is a simple problem, but I am confused by the fact that we can look at the problem from two points of view. In general, we differentiate between the balls of the same colour, I mean we could mark them B1, B2, ... and B6, the same way R1, R2, R3 and R4. Then we could say that the...
  20. Username34

    I Airplane crash probability fallacy

    https://anxieties.com/self-help-resources/fear-of-flying/how-safe-is-flying/ "In fact, based on this incredible safety record, if you did fly every day of your life, probability indicates that it would take you nineteen thousand years before you would succumb to a fatal accident." The...
  21. Username34

    If theres a 15% probability each month of getting a woman pregnant...

    How many times do I need to have sex each month to ensure it will happen one of those months? Does it statistically have to happen sooner or later? Say I live for 250 years and copulate absurd amount of times.
  22. ergospherical

    I Calculating the inverse of a function involving the error function

    I have a probability distribution over the interval ##[0, \infty)## given by $$f(x) = \frac{x^2}{2\sqrt{\pi} a^3} \exp\left(- \frac{x^2}{4a^2} \right)$$From this I want to derive a formula for the inverse cumulative density function, ##F^{-1}##. The cumulative density function is a slightly...
  23. physicsdick

    Higher Chance to get Lightning Strike by Large Power Consumption?

    Assume there are two tall building with same height, and the materials are the same, having same distance away from the storm cloud. Will the one building with extreme high consumption of electricity cause higher chance to get struck by lightning than the one building without ? Thanks!
  24. Juanda

    I Adding random numbers: Tolerance analysis

    I am not a fan of random and statistics. I know it is extremely useful and probably the mathematical branch more applicable to real life to understand the world around us but I am a Calculus and Vectors boy. This problem though I find interesting. I would like to find a generalized solution for...
  25. T

    A combinatorial probability question

    I've a small molecule ABCD made of building blocks A,B,C,D. The molecule can get cleaved at any 'bond' between the building block one bond at a time. Researchers have compiled probabilities from dissociation studies of large number of molecules( made up of many other building blocks) and the...
  26. R

    B Are both of these the same thing? (logical NOT and the complement rule)

    Hi everyone, This is an example of binary variable called as logical NOT https://www.fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/chap2s1.html?scroll=sseclognot ...and this is the complement rule of probability...
  27. D

    B Roulette wheel physics and probability

    Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is...
  28. Hennessy

    Help Calculating probability between 2 limits for the ground state

    TL;DR Summary: Looking for help on a Intro to QM Problem Hi All, THIS IS A GRADED PIECE OF WORK AT MY UNIVERSITY PLEASE DO NOT JUST GIVE ME THE ANSWER , I have made this post to see if what i've calculated seems reasonable, it sounds unlikely as 0.4 - 0.5L is in the middle of the well. The...
  29. R

    B Why did it suddenly become subtractive? (Example of Bayes’ Theorem)

    From https://corporatefinanceinstitute.com/resources/data-science/bayes-theorem/#:~:text=Formula for Bayes' Theorem&text=P(A|B) –,given event A has occurred Example of Bayes’ Theorem Imagine you are a financial analyst at an investment bank. According to your research of publicly-traded...
  30. T

    I What Is the Counterintuitive Probability in the Bag of Balls Problem?

    In Aubrey Clayton's book" Bernoulli's Fallacy" which documents the conflict between frequentists and Bayesian interpretations of probability, he describes a problem that was proposed in the 19th century that gives a counterintuitive result. The Problem: "Infer the state of a bag of 3 balls...
  31. K

    I Are different interpretations of probability equivalent?

    There appear to be at least three concepts of probability. In my words frequencies in a history (theretical and measured) reasonable expectation propensity to outcomes There may be more. I am wondering whether these are actually different meanings, which could affect how probability is used in...
  32. K

    I Why is there no consensus about the meaning of probability in MWI?

    As there appears to be no consensus about the meaning of probability in a deterministic model, I am asking what the sticking point is? That's all really.
  33. MAXIM LI

    Limit of probabilities of a large sample

    My first thought as well but I think the problem is deeper than that. I think that as the n tends towards infinity the probability of the the sample mean converging to the population mean is 1. Looking at proving this. By the Central Limit Theorem the sample mean distribution can be approximated...
  34. J

    I Probability, observers and the multiverse

    Not sure if I'm putting this in the right place! I have a question about probability and conscious observers. Aside from other arguments for and against a multiverse, does the idea that a multiverse could contain a vast number of consicous observes make it more likely, given that we find...
  35. WMDhamnekar

    I Computing the expectation of the minimum difference between the 0th i.i.d.r.v. and ith i.i.d.r.v.s where 1 ≤ i ≤ n

    Problem :Let ##X_0,X_1,\dots,X_n## be independent random variables, each distributed uniformly on [0,1].Find ## E\left[ \min_{1\leq i\leq n}\vert X_0 -X_i\vert \right] ##. Would any member of Physics Forum take efforts to explain with all details the following author's solution to this...
  36. chwala

    Solve the given problem involving conditional probability

    Phew! took time to figure this out...i guess there may be a way to use combinations or markov process i do not know... anyway, it was pretty straightforward, we have the ##P_r(w) = \dfrac{n-3}{n}## from box ##X## and this will result in ##P_r(w) = \dfrac{4}{n+1}## in box ##Y##. Together i...
  37. MichPod

    I A probability of field amplitude in QFT

    Per quantized scalar field (quantized Klein-Gordon equation), suppose we act on a vacuum state |0> with some set of creation operators to have some particles. How then can we calculate a probability density for the field to have a particular value ##\psi_0## (upon measurement) at a specific...
  38. B

    B Entanglement & Superposition Probabilities

    I cannot find a clear answer on the following beginner’s question on some QM fundamentals: Suppose we have two particles, A and B. Let’s say we generated these as (or otherwise entangled them as) an entangled pair with opposite/orthogonal states. Perhaps horizontally and vertically polarized...
  39. M

    Multi-event probability puzzle - is my answer correct?

    Here is my attempt. Beginning state: Bag B : B, B, O Bag A : R, R, G, V, Y Final state: Bag B: B, B, O, + G/V/Y Bag A: remaining balls First possible exchange that would have exactly 3 different colors in each bag is: Move 1: P ( Arjun moves either the green, violet, or yellow ball to...
  40. R

    I Variation of the Liar's Paradox

    A variation of the Liar's Paradox occurred to me: "Statistics are wrong 90% of the time". This statement seems to refute itself, but does so in a less straightforward way. I would appreciate any insights! And what about, "Statistics are wrong 50% of the time"? (Even odds.)
  41. E

    I TOUGH Textual Criticism math statistics question

    Okay let me rephrase this math question and frame it. It is math dealing with ancient Biblical texts and textual criticism. . Codex 01 (350AD) agrees with the MT (mjority text) about 87% of the time. Codex 03 (350AD) Agrees with MT about 87% of the time. 01 and 03 agree with each other...
  42. C

    A Are these results possibly the outcome of a probabilistic calculation?

    Hello, I am seeking the opinion of specialists in statistics and probabilities to evaluate the results of a forecast on a univariate time series derived from an experiment at hazcard.com. They are presented as obtained from a logic that includes no probability calculation, no learning, no use of...
  43. A

    B Definition of a random variable in quantum mechanics?

    In a line of reasoning that involves measurement outcomes in quantum mechanics, such as spins, photons hitting a detection screen (with discrete positions, like in a CCD), atomic decays (like in a Geiger detector counting at discrete time intervals, etc.), I would like to define rigorously the...
  44. S

    Prob/Stats Probability book for basic understanding

    I want to learn topics related to combinatorics, probability theory, discrete and continuous random variables, joint pdf and cdf, limit theorems and point estimation, confidence intervals and hypothesis testing. Any recommendations for books to learn those topics? High school level or...
  45. S

    Finding probability related to Poisson and Exponential Distribution

    My attempt: (i) ##\lambda =3## (ii) (a) ##P(N_{2} \geq 1=1-P(N_{2} =0)=1-e^{-6} \frac{(-6)^0}{0!}=0.997## (b) ##P(N_{4} \geq 3)=1-P(N_{4} \leq 2)=0.999## (c) ##P(N_{1} \geq 2) = 1-P(N_{4} \leq 1)=0.8## Do I even understand the question correctly for part (i) and (ii)?(iii) The expectation of...
  46. Memo

    Combination, partial permutation

    a) p=(4C1*6C2)/(10C3)=0.5 b) p=(4C1*6C2)/(10C3) + (4C2*6C1)/(10C3) + (4C3*6C0)/(10C3)=0.83 Please check if my answer is correct. Thank you very much.
  47. Hamiltonian

    Drawing marked balls from an urn

    part (a) was straightforward ##\mathcal{P} = \frac{20}{200} = 0.1##. Instead of directly trying to find the probability of the 20th drawn ball being marked I decided to start with finding the probability of the second ball drawn being marked and then after figuring that out moving to the cases...
  48. S

    Probability related to cumulative distribution function

    I have tried to answer all the questions but I am not that sure with my answer. That's the graph of ##F_X (x)## (I think) (i) P (X ≤ i) = ##\frac{i^2}{N^2}## and P(X < i) = 0 All of these are based on the graph (ii) P(X = i) = P(X ≤ i) - P(X < i) = ##\frac{i^2}{N^2}## Are my answers...
  49. tworitdash

    I Parameters of a distribution of a physical variable

    Pardon me if this is a very silly question. Although my research involves a lot of probability distributions, I consider myself a fledgling statistician. When people assign a probability distribution to a variable in a physical process, is it inherently assumed that the parameters of this...
  50. WMDhamnekar

    I How to obtain moment bound from the importance sampling identity?

    Let ##X## be a non-negative random variable and let a > 0. We want to bound the probability ##P\{X \geq a\}## in terms of the moments of X. - Define a function ##h(x) = \mathbb{1}\{x \geq a\}##, where ##\mathbb{1}\{\cdot\}## is the indicator function that returns 1 if the argument is true and 0...
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