Probability Definition and 1000 Threads

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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.)

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

Hello! A friend shared a problem he recently solved. It goes as follows: Given: Each dagger strike deals either normal damage = 20 or critical damage = 80. After each strike, the probability of dealing normal and critical damage changes (initial probabilities are 90% and 10% respectively). The...

Boltzmann's brain, entropy reduction, Poincaré's recursion theorem, the probability of oxygen molecules in a room gathering in one place, the probability of quantum tunneling of macroscopic objects, etc. are theoretically possible. But the probability of these events is very low. Additionally...

I have attempted the solution above and I am fairly sure that it is correct. My question is the following: What am I calculating if I multiply 4/52 x 3/51 x 4/50 x 3/49 = 144/ 6,497,400 = 0.002%. I got the correct answer by following the principles of combination probabilities, but intuitively...

1) At first my answer was ##n! \begin{pmatrix} n+r-1 \\ r - 1 \end{pmatrix} ## But I think that's not correct because let say first group consists of person A and B, by multiplying with n!, I also consider first group to be B and A which is just the same as A and B so there is double counting...

I know this problem can be done as follows. P(1 boy and two girls) = (C(2,1)*C(5,2))/C(7,3) = 20/35 My question can this be written as probility fractions? Meaning Lets say if that (2/5*3/7+1/2*1/7)/(3/7) But that doesn't give same result? what am I doing wrong?

At the 2005 World Bridge Championships in Estoril a hand was dealt with QT8642 in two different suits and with the remaining card even. The odds of this happening by pure chance are one in 531 trillion. We can say that AQT864 and KJ9753 would form an equivalent class of "hands spaced by two"...

[Mentor Note: Two similar thread starts merged] The questions are from MIT OCW. First of all, I cannot understand what is the meaning of the measurement outcome being 0. How can an eigenvalue be 0? I tried doing the problems guessing that by 0 they mean the posterior state will be |0>. The only...

Hi! I'm getting confused by these two things. If I have two uncorrelated probabilistic events, and I want to know the probability of seeing them both land beyond 3.3 sigma (for example), do I multiply the probabilities .001*.001 or do I do sum of the squares sqrt(.001^2 + .001^2). I assume it is...

The following problem is sometimes called “The Monty Hall Game Show Problem.” You are a contestant on a game show and have won a shot at the grand prize. Before you are three closed doors. Behind one door is a brand new car. Behind the other two doors are consolation prizes. The location of the...

Post-grad, my background is in mathematical physics, probability/statistics, and information theory. I am here for discussion and collaboration on things I find interesting from time to time.

Welcome to the reinstatement of the monthly math challenge threads! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by...

Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$ If we know that A wins, his score is specified by B's score; he...

In a board game, I need to reach a certain place in a board divided into boxes, I move by throwing 2 six-faced dices. If my goal is 4 boxes away I need to obtain at least a combination of numbers that sums to 4, but any higher number is also a favorable outcome. I want to calculate the...

TL;DR Summary: Chance of picking 2 named people when randomly choosing 3 from a group of 30. For my daughter's homework question: There is a group of 12 girls and 18 boys. Two of them are twins (girl and boy). If I select three at random, what is the chance that the twins will be chosen? I...

Let E be a finite nonempty set and let ## \Omega := E^{\mathbb{N}}##be the set of all E-valued sequences ##\omega = (\omega_n)_{n\in \mathbb{N}}F##or any ## \omega_1, \dots,\omega_n \in E ## Let ##[\omega_1, \dots,\omega_n]= \{\omega^, \in \Omega : \omega^,_i = \omega_i \forall i =1,\dots,n...

Hello all! I hope I have come to the right place, and I appreciate any help! My first disclaimer is that I am not a math professional of any sort, I am not bad at it but I just wanted to start with that so if I ask a stupid question, it's because I am ignorant mostly. What I work in is risk...

There are a number n of runners in a race. We know their expected times from start to finish μ(i) and the corresponding standard deviations σ(i). The probability of runner 0 to finish first is given by this integral: It's from here: https://www.untruth.org/~josh/math/normal-min.pdf The 0 is...

I started my research in statistical digital signal processing two years ago, so I need to familiarize myself with all the notations people use in probability and statistics. I come from a deterministic science background. I name my variables based on what they mean. A velocity is a v , a...

Probability of any random n points on a line being within a given distance Hi, I am a software engineer trying to solve the following problem analytically given a line segment in cm and n random points on it what is the probability that the distance between any 2 consecutive points on the...

The problem is stated like this : There are k people in a room. Assume each person’s birthday is equally likely to be any of the 365 days of the year (we exclude February 29), and that people’s birthdays are independent (we assume there are no twins in the room). What is the probability that two...

The question is as seen below: My attempt (note that my questions are in bold below) is below. Please note that I am self-studying AM: (a) By the independence of any interval ##dt## of time and time symmetry, we expect these two answers are the same (Is there any way to make this rigorous?)...

Hello, I am solid on the following concepts but less certain on the correct understanding of what a random variable is... Random Experiment: an experiment that has an uncertain outcome. Trials: how many times we sequentially repeat a random experiment. Sample space ##S##: the set of ALL...

I have a question that is bothering me. It is commonly accepted that when playing heads or tails with a fair coin and a large number of tosses are made, the probabilities of getting heads or tails are equal to 50% for each toss. However, the principle of determinism, which states that under the...

Hi, Given a spin in the state ##|z + \rangle##, i.e., pointing up along the z-axis what are the probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##? My problem is that I'm not sure to understand the statement. It seems like I have to find the probabilities of measuring an eigenvalue...

Will the probability to provide a negative value of a Sz 1/2 spin system always be 0? If lambda 1 = hbar/2 and lambda 2 = -h bar/2 ?

Abstract: If a laser shoots photons at a pinhole with a screen behind it, we get a circular non-interference pattern on the screen. Is this distribution Guassian, and if not, what would its wave function be? ===================== Assume a double-slit like experiment, but instead of double...

I would like to know how one can use the tree diagram...hence my post... otherwise, i was able to solve problem as follows, a. ##P(A∩B)= \dfrac{3}{4} ×\dfrac{1}{5}=\dfrac{3}{20}## b. ## P(B/A')=\dfrac{P(B)-\dfrac{3}{20}}{P(A')}## ##\dfrac{3}{7}=\dfrac{P(B)-\dfrac{3}{20}}{\dfrac{1}{4}}## ...

My interest is on part ##b## only. We know that ##A## and ##B## are independent and not mutually exclusive events therefore, ##P(C)=0.7×0.6=0.42## ##P(C|A')=\dfrac{P(C)-P(A∩C)}{P(A')}=\dfrac{0.42-(0.3×0.42)}{0.7}=\dfrac{0.294}{0.7}=0.42## which is wrong according to textbook solution. Where...