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. Addez123

    Probability of all runners finishing within 100 minutes

    The chance of everyone finishing should be $$Fx(100)^8 = (100 * 125/450 - 100^2/900)^8 = (50/3)^8$$ What am I doing wrong?
  2. H

    MHB What is the probability that at least 5 was shown for 5 or more times?

    A dice was thrown for 6 times.What is the probability that at least 5 was shown for 5 or more times?
  3. Leo Liu

    Understanding Probability Density Functions and Their Properties

    My questions are as follows: 1. How do we find them and why do we need them? 2. What are the meanings of the mean and the median of a PDF? Are the formulae below correct? $$\int_{a}^{median} f(x) \mathrm{d}x = \int_{median}^{b} f(x) \mathrm{d}x$$ $$\int_{a}^{mean} f(x) \cdot x \mathrm{d}x =...
  4. scottdave

    Probability of a 4 character code without repetition

    The lessons show using Permutations and Combinations for other problems in this section, but we were having difficulty what to choose to put in those formulas. I used this approach: We have 10 numeric and 6 alphabetic = 16 total possible. Let's look at the first digit numeric, and the next 3...
  5. J

    I Bernoulli Trials Homework Problem

    this is the answer Is this right?
  6. I

    MHB Probability distribution: Estimate the size of the fox population

    In a park , 200 foxes are tagged. In 100 sighting, 14 were tagged. Estimate the size of the fox population ? This is how approached . 200 tagged : Population 14 tagged : 100 (200x100/14)= Estimated Population = 1,429 I wonder if this is right !
  7. I

    MHB Expected Value of Cup of Coffee in Flip a Lid Contest

    In its flip a lid contest, a coffee chain offers prizes of 50,000 free coffees, each worth \$1.50; two new TVs, each worth \$1200; a snowmobile worth \$15 000; and sports car worth \$35 000. A total of 1 000 000 promotional coffee cups have been printed for contest. Coffee sells for \$1.50 per...
  8. karush

    MHB -find probability it will rain both days

    ok I still don't know where the (0.2) comes from
  9. Addez123

    Probability of receiving a binary one

    $$P(A|B) = P(A \cap B) / P(A)$$ $$P(A) = \text{Chance of 1 being received} = .4 * .98 + .6 * .01 = .398$$ $$P(A \cap B) = \text{Chance 1 being sent and 1 being received} = .4 * .98 = .392$$ $$P(A|B) = P(A \cap B) / P(A) = .392 / .398 = .985$$ The correct answer is 147/148 ~= .9932 What am I...
  10. Entertainment Unit

    B The Probability that N and E bridge hands have the same number of spades?

    My answer is as follows: Let ##S## be the set of all outcomes of dealing four labelled 13-card hands from a standard 52-card deck. Let ##A## be event "N and E have exactly the same number of spades." Let ##A_i## be event "N and E have exactly ##i## spades each." Note that when ##i > 6##...
  11. Addez123

    Conditional probability of dying from eating a poison fruit

    Summary:: There's 11 fruits, 3 of which is poisionous. A guy eats 4 of them, a girl eats 6 and a dog gets the last one. What is the conditional probability of both the girl and guy dying IF the dog made it? One fruit is enough to kill you. $$P(dog lives) = 8/11$$ $$P(allPeopleDie | dog...
  12. T

    I Probability current density of a stationary state

    I have written a finite difference program to solve 1D time-independent Schrodinger equation. It seems to work correctly for harmonic oscillator, particle in a box, etc. But I can't figure out how to calculate the probability current density. It should be constant, but what is it? The program...
  13. Addez123

    B Probability of drawing 5 spades, 3 hearts....

    The odds are $$\frac {13C5 \times 2 \times 13C3 \times 13C2} {52C13}$$ This is correct according to my book. The follow up question then becomes, what if you can pick 5 of any suit, 3 of any other suit, another 3 from the remaining 2 suits and 2 from the last untouched suit? The solution...
  14. michaelwright

    B Fun with (im)probabilities

    Hi folks - I need some help with a tricky probability. Here's the situation: Let's say there are 4M internet users in Age Group A. (The total set) Of those 4M, there are 1,000 users who play a specific sport. Those 1,000 are spread evenly over 125 teams, so 8 players each. 1. What's the...
  15. C

    MHB What is the probability that exactly 8 of them are over the age of 65

    2) The Agency for Healthcare Research and Quality reported that 53% of people who had coronary bypass surgery in 2008 were over the age of 65. Fifteen coronary bypass patients are sampled. a) What is the probability that exactly 8 of them are over the age of 65? b) P (less than 10 are over...
  16. C

    MHB Find the probability of 1 tire with low air pressure P (1)

    1) Let X represent the number of tires with low air pressure on a randomly chosen car. The probability of distribution of X is as follows: X 0 1 2 3 4 P(X) 0.1 0.2 0.4 0.2 0,1 a) Find the probability...
  17. E

    B Mapping between rules of classical vs QM probability?

    Apologies in advance for my ignorance, I don't really have a reference to consult and Google hasn't been too helpful! In standard probability theory there are a few common useful formulae, e.g. for two events ##S## and ##T## $$P(S\cup T) = P(S) + P(T) - P(S\cap T)$$ $$P(S \cap T) = P(S) \times...
  18. Physics lover

    Tricky problem from Probability

    I decided to take cases.For example-:A gets one 1 duck and B gets 2,3,4,...,51.So i can write this as 50C1(1/2)5051C2(1/2)51+50C1(1/2)5051C3(1/2)51+... But i was unable to solve it further. please help.
  19. coolfish21

    B Probability IB Math HL: Is it a difficult question to solve?

    A person picks 8 cards without replacement from a bag containing cards numbered from 1 to 6 (117 in total). What is the probability that the sum of those 8 cards is 28? Given that P(1)=8/117, P(2)=14/117, P(3)=34/117, P(4)=39/117. P(5)=14/117 and P(6)=8/117.
  20. domingoleung

    B Poisson Distribution - Selecting cookies that are indistinguishable

    Here's the problem: A chef made 500 cookies randomly mixed with 1000 nuts including 600 almonds and 400 hazelnuts in which each nut is the same size. Suppose the number of pieces of nuts in a piece of cookie follows a Poisson distribution. (a) Suppose cookies are randomly selected one-by-one...
  21. E

    How Do Measurements Affect GHZ States?

    Here's what I think I understand: First off, the GHZ state ##|GHZ \rangle = \frac {|000\rangle+|111\rangle} {\sqrt 2}##, and ##\sigma_x## and ##\sigma_y## are the usual Pauli matrices, so the four operators are easy to calculate in Matlab. I'm thinking the expectation values of each operator...
  22. Addez123

    B Probability of picking one black and one white marble

    My approach is the amount of successfull options / total amount of options. I can first pick white in 3 different ways. Then black in 4 different ways 3 * 4 But I can also pick black first then white 4 * 3 Total amount of ways to pick marbles are 7 *6 So the probability is: (3*4 + 4 * 3) / (7...
  23. G

    Calculating Φ - Wrong Answer Found

    I calculated <+|Φ> to be (1-√2i)/4√2 + (3-2i)/4√2. When I squared this I for 16/32 but this is no the right answer.
  24. Z

    I Is the calculation related to average or probability?

    Hi, I am trying to understand the calculation in the following research paper: http://cramton.umd.edu/market-design/abdulkadiroglu-sonmez-house-allocation.pdf
  25. Z

    Request to Improve the Complexity of a Probability Question for students

    Hi, I want to frame the above Probability question for computer science students. I have stated my idea above but I want to refine it so that it becomes a more comprehensive real world problem. Zulfi.
  26. S

    Ehrenfest chain transition probability

    Here is the Ehrenfest Chain that the question is talking about: I was able to solve parts 1 and 2 as shown in the image below. But I'm not really sure how I'd prove part three. Any help would be appreciated, thanks!
  27. Glype11

    B Odds of Drawing Sequence 4457: Calculating Probability

    Suppose we have a code: 4457 If these numbers are randomly drawn what are the odds of drawing those numbers? Here's my approach: The available numbers range from 0123456789 and there are four columns, minus the numbers which were drawn gives: 1/(10^4-4)~0.01% Now what about the odds of...
  28. S

    Understanding Recurrence in Probability: Solving for hN(1) and cNcN

    I set hN(1)hN(1) equal to cNcN, but I'm confused on how I'd be able to solve it and because of that I was not able to conclude that 0 is recurrent when qx/px = infinity
  29. J

    MHB Probability of Having a Totally Dominating Set (Probabilistic Methods)

    Problem: A vertex set $S$ in a graph $G$ is said to be totally t-dominating, for a positive integer t, if $|N(v) \cap S| \geq t$ for all $v \in V (G)$. Suppose that r, t, n are positive integers such that $r > 2t$ and $t \geq \frac{14}{3}\cdot ln(2n)$, and let $G$ be an r-regular n-vertex graph...
  30. Killtech

    I Probability density expansion

    This is probably a stupid question but i don't want to make a stupid mistake here, so i thought better ask: I'm starting with the simple free Schrödinger Equation ##V(x)=0## (can be 1 dim) and an initial condition where the wave function is somehow constrained to be entirely localized around a...
  31. Killtech

    I Hydrodynamic interpretation and consistency classic probability

    So back in the other thread I asked about compatibility of classical probability theory (PT) and QM – and it turns out there is no inherent reason why they need to be incompatible. Therefore I was looking for something that makes them compatible, which wasn’t easy to search for. But there seems...
  32. Addez123

    B Probability of rolling a "6" k times out of 20 rolls of a die

    The probability should be ## (1/6)^k * (5/6)^{20-k} ## But the book says the answer is : ## \begin{pmatrix} 20 \\ k \\ \end{pmatrix} * (1/6)^k * (5/6)^{20-k} ## Because there are 20 over k different sequences, but the order doesn't matter? I just don't understand why the 20 over k is there...
  33. Saptarshi Sarkar

    Probability of finding a particle in a region

    I know how to calculate the probability of finding the particle in a region by integrating the mod square of the wave function within that region. But in this question only the operator is provided but not the wave function. I am not sure how am I supposed to proceed with this problem.
  34. I

    What is the Relationship Between Probability Amplitudes and the Sum of Terms?

    I am not sure what I can do with the equation. I realize that ## \vert c_1 \vert ^2 = \vert c_2 \vert ^2 = \frac{1}{2} ## does not mean that ## c_1 ^2 = c_2 ^2 = \frac{1}{2} ## or that ## c_1 = c_2 ##, so I don't know how to use it. I think ideally I might have something like ##P = \vert c_1...
  35. S

    Probability of X: Geo Distribution with 2/3

    a. P(X = 3) = (4/6) x (4/6) x (4/6) x (2/6) = 8/81 b. X ~ Geo (2/3) Is this correct? Thanks
  36. I

    A Building a better ranking system (probability)

    Suppose we have four games and the probability that a player will win the game are as follows: Game 1: 71% Game 2: 55% Game 3: 58% Game 4: 16% Suppose player b won these games with the following percentages of time: Game 1: 100% Game 2: 96% Game 3: 87% Game 4: 67% In other words, he's a very...
  37. Saptarshi Sarkar

    Meaning of thermodynamic probability

    I was studying statistical mechanics when I came to know about the Boltzmann's entropy relation, ##S = k_B\ln Ω##. The book mentions ##Ω## as the 'thermodynamic probability'. But, even after reading, I can't understand what it means. I know that in a set of ##Ω_0## different accessible states...
  38. A

    On the width of the kinetic energy distribution of a gas

    In these lecture notes about statistical mechanics, page ##10##, we can see the graph below. It represents the distribution (probability density function) of the kinetic energy ##E## (a random variable) of all the gas particles (i.e., ##E=\sum_{i}^{N} E_{i}##, where ##E_{i}## (also a random...
  39. fRod57

    How do I predict rogue waves using simple probability models ?

    I need to predict an upcoming rogue wave or analyse old rogue wave events using simple probability models and real-time data for a physics high school project.
  40. M

    MHB Probability of the error of type 2

    Hey! :o We have data of a sample of $100$ people from a population with standard deviation $\sigma=20$. We consider the following test: \begin{align*}H_0 : \ \mu\leq 100 \\ H_1 : \ \mu>100\end{align*} The real mean is $\mu=102$ and the significance level is $\alpha=0.1$. I want to...
  41. Saptarshi Sarkar

    Total probability conservation

    I found out that the operator H is not a Hermitian operator but I didn't understand the second part of the question. What do I calculate the probability of?
  42. CaptainX

    B What is Conditional Probability and its Properties?

    1. Definition If E and F are two events associated with the same sample space of a random experment, the conditional probability of the event E given that F has occurred, i.e. P(E|F) is given by P(E|F) = (E∩F)/P(F) (P≠0) 2. Properties of conditional probability Let E and F be events of...
  43. G

    B Calculating Dice Roll Probability: Learn Basics with Frank

    Hello, I am trying to learn the basic concepts of calculating probability as it pertains to dice rolling. I have searched the internet and not been able to figure it out. If I have a regular 6 sided dice and I want to know the probability of rolling a 3, I know its 1/6 or 16.6%. This is...
  44. M

    Find the probability of a particle in the left half of an Infinite Square well

    Attempt: I'm sure I know how to do this the long way using the definition of stationary states(##\psi_n(x)=\sqrt{\frac {2} {a}} ~~ sin(\frac {n\pi x} {a})## and ##\int_0^{{a/2}} {\frac {2} {a}}(1/5)\left[~ \left(2sin(\frac {\pi x} {a})+i~ sin(\frac {3\pi x} {a})\right)\left( 2sin(\frac {\pi x}...
  45. U

    Probability generating function

    (a) I find the geometric distribution $$X~G0(3/8)$$ and I find p to be 0.375 since the mean 0.6 = p/q. So p.g.f of X is $$(5/8)/(1-(3s/8))$$. (b) Not sure how to find the p.g.f of Y once we know there are 6 customers?
  46. Killtech

    I Quantum physics vs Probability theory

    Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems...
  47. P

    Biology Genetics Probability (Autosomal Recessive)

    I think both Jane and Bill are carriers. I'm unsure if the answer is 1/4 (25%).
  48. snatchingthepi

    I Step unclear in Peebbles' Probability Book

    If anyone could help me understand how Peebles gets from line one of the autocorrelation to the second line, I'd be most grateful. I don't understand what identity or property is being used to go from a product in the expectation value to a sum in the expectation value.
  49. K

    A Martingale, calculation of probability - Markov chain needed?

    I am trying to estimate probability of loosing (probability of bankrupt ##Pb##) using Martingale system in betting. I will ilustrate my problem on the following example: Let: ##p## = probability of NOT getting a draw (in some match) We will use following system for betting: 1) We will bet only...
  50. S

    Average speed of a probability density wave and wave packets

    Show that ##v_{av}=\frac{\hbar k_2 + \hbar k_1}{2m}## is equal to ##v_{av}=\frac{\omega_2 - \omega_1}{k_2-k_1}##. Which of the identities listed above (if any) would make the sign change between ##k_2## and ##k_1##? One can attain a "wave packet" by superposing two or more sinusoidal waves...
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