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. M

    Other How Can I Maintain Interest and Improve My Understanding of Probability?

    I want to have a deep understanding of probability. I've tried William Feller's first book on Probability, and E.T Jaynes' Probability theory - the logic of science (which is very different from most probability books.) But, neither books could hold my interest for long. The first was too...
  2. Y

    MHB Calculating the Probability of Winning $25 in a Bag Game

    Hello all, In a bag there are 18 paper notes. On five of them there is the digit 2, on seven the digit 3, and on six the digit 5. A man takes 3 notes by random. If the multiplicity of the notes is even, he wins 25 dollars. If for each game he pays 6 dollars, what is the average of profit he has...
  3. A

    Find probability of certain event, total probability theorem

    Homework Statement Suppose you're at a college campus. 3/4 of the people on the campus are students or professors from that college, and the rest 1/4 aren't. When asked a question, students and professors from that college will give you a correct answer every time, and those that aren't from...
  4. L

    Probability math confusion sample space (I think)

    Homework Statement high school has 417 students total 186 of total are athletes (play sports) 136 of total are musicians (play music) 74 of total are musicians and athletes. (play music and play sports) a) at which probability does randomly chosen athlete also play music (i.e. be a musician)...
  5. S

    MHB Calculation of odds: probability and deviation

    *WARNING* My understanding of maths is pretty limited and i don't know if i am being really stupid or not. Ive been looking into data analysis of football matches, working out proabilities of certain situations happening and how steady the averages are. i come across a website which as really...
  6. H

    I How to find the inverse of an integral transform?

    I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral...
  7. M

    MHB Probability of Combinations and Permutations

    Hi, I'm a mom trying to help my son understand why he got answers wrong on his online math program. He is taking Geometry, but the last unit in the class is an introduction to Probability and Statistics. After re-reviewing the lesson and re-working the problems he got wrong, we were able to...
  8. W

    B Simple probability with high frequency

    I'm trying to remember simple probability form high school. I'd like to know how the probability of anyone event changes with frequency. So for example the probability of getting one ace of spades is 1/52; what is the probability of getting at least one ace of spades if the card is put back and...
  9. micromass

    Challenge Micromass' big probability challenge

    Probability theory is very nice. It contains many questions which are very easy to state, but not so easily solved. Let's see if you can solve these questions. For an answer to count, not only the answer must be given but also a detailed explanation. Any use of outside sources is allowed, but...
  10. Danielm

    Calculating Winning Probability for a Chess Master in a Competition

    Homework Statement A chess master plays n games at a chess competition, each against a different opponent. Based on their past performances, we can estimate the probabilities of each opponent beating the master: p1, . . . , pn ∈ [0, 1]. Describe an algorithm, which given integer k returns the...
  11. NatFex

    I Sum of Probability Density Function > 1?

    I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
  12. bapowell

    Insights Scientific Inference P3: Balancing predictive success with falsifiability - Comments

    bapowell submitted a new PF Insights post Scientific Inference P3: Balancing predictive success with falsifiability Continue reading the Original PF Insights Post.
  13. EhrEnFeest

    What is the physical dimension/unit of Probability current?

    Homework Statement Question: What is the physical dimension of Probability Current for a particle in 1 dimension? (Quantum Mechanics) Homework Equations Quantum mechanical Probability Current: The Attempt at a Solution I know the physical dimension of mass, that is kg. If I know every...
  14. CynicusRex

    I Poisson process approximation error

    X = # of cars that pass in one hour E(X) = λ = n * p λ cars/1hour = 60min/hour * (λ/60) cars/min In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?" "We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1...
  15. S

    I Why is the standard deviation the error on the singular meas

    I'm a beginner with the study in data analysis in Physics. I'm trying to understand the meaning, in the field of experimental Physics, of the standard deviation ##\sigma## of a series of data. There is one fundamental thing about ##\sigma## that I read but I could not understand. >In a series...
  16. S

    A Normalized Differential Scattering Probability

    The normalized differential quantum-field-theoretic probability ##dP## of scattering is given by ##dP=\frac{|\langle f |S|i\rangle|^{2}}{\langle f|f\rangle\langle i|i\rangle}d\Pi,## where ##|i\rangle## is the initial state, ##|f\rangle## is the final state, ##\langle f|S|i\rangle## are the...
  17. mikebrisingr

    Probability coin and die Question

    I started by looking at the probability of any given situation and maybe making a tree but my professor hasn't explained things very well and most the notes I see online are overwhelming A Game of Coin and Die. This game is played with a fair coin and a die. First player flips a coin. If it...
  18. T

    I Relationship between E-field and Probability Amplitude of Waves

    Electromagnetic waves can be classically described by Maxwell's equations. Photons can be described by probability waves.In this case, what is the relationship between the electric field and the probability amplitude? Are they directly proportional to each other? What about the fact that one...
  19. G

    I Probability involving n dice

    I'm studying probability and am currently stuck on this question: Let's say we have n distinct dice, each of which is fair and 6-sided. If all of these dice are rolled, what is the probability that there is at least one pair that sums up to 7? I interpreted the above as being equivalent to the...
  20. Smalde

    QM: Time development of the probability of an Eigenvalue

    The problem is actually of an introductory leven in Quantum Mechanics. I am doing a course on atomic and molecular physics and they wanted us to practice again some of the basics. I want to know where I went conceptually wrong because my answer doesn't give a total probability of one, which of...
  21. R

    I Can Conditional Probability Be Solved Generally with PDFs of Variables?

    Is it possible to solve something like this generally or does it depend on the pdf's of the variables? P(x < f(y) | x > -f(y))
  22. lucytranxx

    MHB Bernoulli Trials and Probability

    Let X be a random variable defined as the sum of 5 independent Bernoulli trials in which the probability of each Bernoulli taking the value 1 is given by r. Suppose that prior to the 5 Bernoulli trials, r is chosen to take one of three possible values with the following probabilities: R=r...
  23. S

    A Using probability to predict my sales amount

    Suppose I'm trying to answer the following question: How much more insurance premium will I get, by raising rates by 10%. for this exercise, I"m assuming price is the only factor, just to simplify things. Lets say I give 20,000 quotes per month. Obviously, not all will purchase my policy...
  24. Amine_prince

    Probability question on buckets and balls

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  25. thegreengineer

    Binomial distribution problem

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  26. T

    MHB What is the Probability of Getting Three Heads in 30 Coin Tosses?

    Conduct 30 trials and calculate emperical probability of the outcomes. What is the probability of tossing 1 coin three times and getting three heads? Need a flowchart of some sort to show all possible outcomes. This math is new to me and has little understanding of this math lol so any help...
  27. Rampart

    Conditional Probability exercise with dice

    Hey there community, I have a question on an exercise. Actually it is a general question based on it. Here is the exercise: We throw 3 dice. If we know that the sum of these 3 is 10, then what is the probability of at least one of them being 3? Well now, this exercise is very simple. I mean I...
  28. S

    A Discrete Multivariate Probability Distribution

    Homework Statement A fair coin has a ##1## painted upon one side and a ##2## painted upon the other side. The coin is tossed ##3## times. Write down a sample space for this experiment. Let ##X_1## be the sum of the numbers obtained on the first ##2## tosses and ##X_2## be the sum of the numbers...
  29. R

    I Sphere Probability: Understanding Joint Distribution

    Hi If I'm using this method to generate points inside a sphere with radius K: X = S^(1/3)*sqrt(1-V^2)*sin(O) Y = S^(1/3)*V Z = -S^(1/3)*sqrt(1-V^2)*cos(O) where (0 < s < K^3), (-1 < v < 1) and (0 < o < 2*pi), i guess that: S_PDF(s) = 1/(K^3) V_PDF(v) = 1/2 O_PDF(o) = 1/(2*pi) How come the...
  30. E

    MHB Discrete Probability Distribution

    Okay, my online class has posed another word problem and I cannot seem to understand this week's material or how to formulate a solution. Here it is: Imagine you are in a game show, a money give-away! There are 4 prizes hidden on a game board with 16 spaces. One prize is worth \$4000...
  31. amjad-sh

    I Transition probability and superposition

    Hello suppose that we are dealing with a particle moving in an infinite potential well(a box of length L). Let the wavefunction of the particle be \psi(x,t)=c1\psi_{1}(x,t)+...+cn\psi_{n}(x,t) suppose that after measurement we find the particle at the energy eigenstate \psi_{1}(x,t). Now let's...
  32. kenyanchemist

    Probability distribution curve for an electron in 2s and 2p

    hi, so my lecturer decides to give me manic depression by sending me on a wild goose chase. what is the general form of a plot of Ψ, Ψr2 and r2Ψ versus r for both Ψ2s and Ψ2p orbital... am not even sure i said it right So far I have only gotten the Ψ2r2 versus r
  33. J

    I Probability current in positive finite square potential

    Hello! I want to prove that the probability current is a continuous entity at the boundaries of the square for the situation of 0< E< Vo in the problem where V is zero except a finite region in space where it is +Vo and we consider an incoming particle from the left(for example). I thought that...
  34. kenyanchemist

    I Probability distribution plots

    hi, so my lecturer decides to give me manic depression by sending me on a wild goose chase. what is the general form of a plot of Ψ, Ψr2 and r2Ψ versus r for both Ψ2s and Ψ2p orbital... am not even sure i said it right
  35. DavidReishi

    I Square of absolute value of amplitude for a single photon

    I understand that this determines a probability, but of what exactly for a single photon? The probability that the photon will be detected on a surface where the photon is pumped, e.g. where on the surface the laser is aimed?
  36. P

    MHB What is the Probability of Knee Problems Requiring Full Knee Replacement?

    For b and c I have no idea where to go. I know for part b I need the probability of the full knee replacement, but that's all I know. I have no idea what needs to be done for part c. If someone could help that would be greatly apprecieted. I think part a is right. If not could it be explained...
  37. a255c

    Show that a sample space is valid by verifying properties

    Homework Statement http://puu.sh/nYQqE/2b0eaf2720.png Homework Equations http://puu.sh/nYSjQ/e48cad3a8b.png The Attempt at a Solution http://puu.sh/nYYjW/174ad8267c.png My main issue is with part b) and part d). I think that part b) is mostly right, but part d) is definitely wrong and...
  38. Kingyou123

    What is the Probability of a Loaded Die?

    Homework Statement (10 points) Consider a six-sided die that is loaded so that numbers 2, 3, 4, 5 and 6 are all equally likely to appear, and 1 is three times as likely to appear as 2. (a) What is the probability of rolling a 2? (b) What is the probability of rolling a 1? (c) What is the...
  39. Alanay

    I How do I find out the exact probability of this....

    I am making a game, there are 1-100 numbers presented on a 10 by 10 table. To make your guess you choose a range from 1-100. So I can choose 45-55. Then the server generates a range from 1-100 and as long as my guesses range is inside of that range I win. How do I find the exact probability I...
  40. E

    What is the Relationship Between Probabilities of Independent Events?

    Homework Statement Of the three independent events E1 , E2 and E3, the probability that only E1 occurs is α, only E2 occurs is β and only E3 occurs is γ. Let the probability p that none of the events E1 , E2 and E3 occurs satisfy the equations ## (α - 2β) p = αβ ## and ## (β - 3γ) p = 2βγ ##...
  41. E

    Probability Question: At Least One of A, B & C Occurring

    Homework Statement For the three events A, B and C, P (exactly one of the events A or B occurs) = P (exactly one of the events B or C occurs) = P (exactly one of the events C or A occurs) = p , and P (all the three events occur simultaneously) = p2 , where 0 < p < ½ . Then the probability of...
  42. fluidistic

    I Place pieces on a chessboard randomly until 2 are adjacent

    I've written a program that simulates placing pieces randomly on an nxn chessboard, one by one until 2 are adjacent. When 2 are adjacent I stop, and I report the number of pieces it took. I've plotted a histogram of the number of pieces it took vs number of occurences just to see what kind of...
  43. P

    I Supremum inside and outside a probability

    I'm trying to deal with the supremum concept in a specific situation, but I think I'm getting the concept wrong. A step of a proof I'm going through states: P\ [\sup\limits_{x}\ |f(x)\ -\ f'(x)|\ >\ y\ |\ z]\ \ \leq\ \ \sum_{i=1}^M\ P\ [\ |f(x)\ -\ f'(x)|\ >\ y\ |\ z]\ \ \leq\ \ M\times\...
  44. Lenus

    I Two-player card game, combined probability

    There is a card game where one player gets three cards and only uses two (the third one is discarded without showing to an opponenet), and the second player gets also three cards and uses only two, discarding the third card in the similar way - without showing. I am trying to enumerate all...
  45. D

    MHB Question regarding a probability mass function of a random variable

    Thank you for your time, I really appreciate it I have no idea where to even begin
  46. D

    MHB Consider the following probability mass function of a random variable x

    $p(x) ={p}^{x}*{(1-p)}^{1-x}$ for $x=0.1$ $0$ otherwise Where $p$ is such that $0<=p<=1$ Question: Find the mean and variance of $X
  47. I

    I Probability density and kinetic

    energy. Consider a particle in a box of the form ##V(x)= \begin{cases}0 \; \; \; -L < x < 0\\ V_0 \; \; \; 0<x<L\\ \infty \; \; \; \text{ elsewhere}.\end{cases}## One can show that the probability density ##P(x) = \Psi^* \Psi## is greater in the region of lower kinetic energy (that is at higher...
  48. SlowThinker

    B Boy/girl riddle (conditional probability)

    I've found this video about conditional probability: All steps look correctly, but the result does not make any sense. I'm ok with the part about frogs, but not so with the boy/girl computation. To sum it up: 1) I have two children and at least one of them is a boy. What is a probability I...
  49. K

    B Do probabilities for entangled polarization states add up to one?

    Two photons are moving in opposite directions along the y-axis are in the entangled polarization state |Ψ> = (1/√2)(|VV>+|HH>) V is vertical polarization relative to the z axis and measured from an axis n1 and is defined as θ=θ1 and H is horizontal polarization to the z axis measured from an...
  50. F

    I Does the probability collapse theory of QM imply backward in time

    In the thought experiment known as Schrodinger's cat a cat is placed in a sealed box, and its life or death is tied to an uncertain quantum event such as radioactive decay. If the radioactive particle decays, the cat dies. If not, nothing happens. According to probability collapse QM, as...
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