# 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. ### 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...
2. ### 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 ?
3. ### 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 ?
4. ### 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}##...
5. ### 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) /...
6. ### 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...
7. ### 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...
8. ### 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...
9. ### 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.
10. ### 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...
11. ### 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!
12. ### 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...
13. ### 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...
14. ### 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...
15. ### 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...
16. ### 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...
17. ### 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...
18. ### 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...
19. ### 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...
20. ### 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.
21. ### 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...
22. ### 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...
23. ### 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...
24. ### 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...
25. ### 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...
26. ### 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...
27. ### 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...
28. ### 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.)
29. ### 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...
30. ### 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...
31. ### 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...
32. ### 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...
33. ### 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...
34. ### 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.
35. ### 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...
36. ### 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...
37. ### 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...
38. ### 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...
39. ### A Calculation of average damage using analytical methods

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...
40. ### I Impossible probability between theoretical possibility and reality

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...
41. ### Probability using combinations or alternatives

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...
42. ### Number of ways to partition n persons and probability to form n groups

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...
43. ### Basic probability question (1 boy + 2 girls of a group of 7)

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?
44. ### B An Observed Extreme Probability Event

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"...
45. ### Questions from Quantum Measurements

[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...
46. ### B Multiply Probabilities vs. Sum of the Squares

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...
47. ### Comp Sci Monty Hall Game Show Problem in Javascript

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...
48. ### How Can Mathematical Physics and Information Theory Enhance Collaboration?

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.
49. ### Challenge Math Challenge - June 2023

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...
50. ### I Write probability in terms of shape parameters of beta distribution

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...