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. V

    Probability that A will win given a condition

    I tried to solve this problem using the chart given below. But I get a different answer of ##\frac {2}{3}## rather than ##\frac {3}{4}##. Maybe the answer given is incorrect? I determine from the chart the number of ways in which A could win given that A has already won 2 of first 3 points...
  2. C

    B Is the probability of a quantum outcome ever zero such as with....

    Greetings, Given an infinite universe or an infinite number of universes? - Regarding the location of an electron around an atom, is there a tiny volume in which finding the electron 100%? Or is there a possibility, no matter how remote, it might be found a meter away or a kilometer away? -...
  3. D

    I Primes -- Probability that the sum of two random integers is Prime

    im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000. note that I am looking for a novel proof, not just some brute force calculation. (this isn't homework, I am just curious.)
  4. E

    A Probability flux integrated over all space is mean momentum?

    In Sakurai Modern Quantum Mechanics, I came across a statement which says probabiliy flux integrated over all space is just the mean momentum (eq 2.192 below). I was wondering if anybody can help me explain how this is obtained. I can see that ##i\hbar\nabla## is taken as the ##\mathbf{p}##...
  5. WMDhamnekar

    MHB What Is the Probability of Distributing Remaining Trump Cards in Bridge?

    North and south have ten trumps between them ( trumps being cards of specified suit). (a) Find the probability that all three remaining trumps are in the same hand. (that is either east or west has no trumps). (b) If it is known that king of trumps is included among the three, what is the...
  6. shivajikobardan

    Comp Sci Poker test: probability of Full House (confusion)

    I'm studying this for poker test. This should not be memorized as this has 3,4 and 5 digit versions. Memorizing all of them isn't possible. So I need a way to calculate them. I'm trying to learn through this example. I'm not getting the process(I know math behind it ie permutations...
  7. shivajikobardan

    Comp Sci Markov model to find the probability of a Pepsi drinker buying a Coke?

    Design the markov model and transition matrix for the given data. Answer the following questions based on the mode. a) If a person purchase coke now the probability of purchase of coke next time is 80%. b) If a person purchases pepsi now the probability of purchasing pepsi next time is 70%...
  8. PhanthomJay

    Probability of Rain in weather forecasts

    I am trying to settle a debate over two definitions of the 'probability of rain' in a weather forecast area. Definition 1 states that for example there is a 50% averaged probability of rain at some point in the forecast area over a given duration of time, that is, there is a 50-50 chance that I...
  9. NewPhysi

    A Probability of U-235 fission happening

    Where can I find information about the ~82% of U-235 nuclear fission happening and ~18% not happening?
  10. I

    MHB Calculating probability that 3 events occur 1 after other

    Let's say we have 3 events that all have a certain chance of occurring. Each latter event occurring depends on if the prior event occurred based on the chance associated with it. For example, if Event #1 does not happen, Event #2 cannot happen. As such, if Event #2 doesn't happen, Event #3...
  11. WMDhamnekar

    Using Multiple integrals to compute expected value

    I want to know how did author derive the red underlined term in the following Example?
  12. WMDhamnekar

    MHB Probability, Expected value, joint P.D.F. and order statistics

    I want to know how did author derive the red underlined term in the below given Example? Would any member of Math help board enlighten me in this regard? Any math help will be accepted.
  13. Killtech

    I The interpretation of probability

    I am looking for a way to compare the handling of probability in QT with how it's done in classic PT (probability theory) - and their interpretations. QT does have it's own formalism that works, so there isn't much motivation to bring it into a usual representation which makes it hard to find...
  14. S

    Probability of getting the smallest value of cards

    There are 3 cases of getting 2 as smallest values: (1) Taking one card (n = 1) → Probability = 1/4 (2) Taking two cards (n = 2) → Probability = 1/4 x 2/3 x 2! = 1/3 (3) Taking three cards (n = 3) → Probability = 1/4 x 1/3 x 1/2 x 3! = 1/4 Total probability = 1/4 + 1/3 + 1/4 = 5/6 But the...
  15. M

    A Probability of resonant ionization

    Hello! I am trying to make some predictions for an experiment in which we have a first ##E_2## transition in an atom driven by a laser, and then we have a second laser that is ionizing the molecule only if the first laser was resonant (i.e. if the atom was excited). For the purpose of the...
  16. LCSphysicist

    Probability to hit a spherical area

    I was asked to derive the relation $$p = u/3$$ for photon gas. Now, i have used classical mechanics and symmetry considerations, but the book has solved it in a interisting way: I can follow the whole solution given, the only problem is the one about the probability to colide the sphere!. Where...
  17. chwala

    Determine the probability that Andy wins the match

    My interest is on the highlighted part only...the other questions are well understood. Find ms solution here; Even this is well understood...they made use of sum to infinity to arrive at the solution. I am interested on an alternative approach. Cheers guys.
  18. S

    I Derive the probability of spin at arbitrary angle is cos( )

    From Dr. Leonard Susskind's Stanford Lecture: Quantum Entanglement, Lecture 4, he sets up a "given particle is spin up along n (arbitrary direction) and discusses : what is probability we measure up along another arbitrary m directionHe does all of the setup, - calculates the eigenvectors and...
  19. M

    Given an integer, find the probability its cube ends in 111

    The last three digits of ##x^3## must be solely dependent on the last 3 digits of ##x##. So let ##x=a+10b+100c## for integers ##a,b,c##. Then ##x^3 = a^3 + 30 a^2 b + 300 a b^2 + 300 a^2 c +O(1000)## where of course ##O(1000)## don't affect the last 3 digits. Evidently ##a^3## is the only...
  20. M

    Probability that 𝑌>3𝑋 where 𝑋,𝑌 are 𝑁(0,1) random variables

    After plotting the above (not shown) I believe one way (the hard way) to solve this problem is to compute the following integral where ##f(x) = e^{-x^2/2}/\sqrt{2\pi}##: $$\frac{\int_0^\infty \int_{3X}^\infty f(X)f(Y)\, dydx + \int_{-\infty}^0 \int_0^\infty f(X)f(Y)\...
  21. S

    Probability of getting arithmetic sequence from 3 octahedron dice

    I try to list all the possible sequences: 1 2 3 1 3 5 1 4 7 2 3 4 2 4 6 2 5 8 3 4 5 3 5 7 4 5 6 4 6 8 5 6 7 6 7 8 I get 12 possible outcomes, so the probability is ##\frac{12 \times 3!}{8^3}=\frac{9}{64}## But the answer key is ##\frac{5}{32}## . Where is my mistake? Thanks
  22. murshid_islam

    Probability of Two Consecutive Same-Value Cards in a Shuffled Deck

    Not really a homework question. I was reading a book on card tricks and it said that it's almost certain that in a shuffled deck of cards, there will be at least two consecutive cards of the same value. I just wanted to know the actual probability of that. So, here's my question: in a standard...
  23. Samama Fahim

    Finding Probability of Two-Identical-Particle System in a Given State

    Problem: A system contains two identical spinless particles. The one particle states are spanned by an orthonormal system ##|\phi_k>##. Suppose that particle states are ##|\phi_i>## and ##|\phi_j>## (##i \neq j##). (a) Find the probability of finding the particle in the state ##|\xi>## and...
  24. WMDhamnekar

    MHB Probability of misprints on each page

    Hi, Each page of a book contains N symbols, possibly misprints. The book contains n =500 pages and r =50 misprints. Show that (a) the probability that pages number 1, 2, . . . , n contain, respectively , $r_1, r_2 , . . . , r_n $ misprints equals $$\frac{\binom{N}{r_1}\binom{N}{r_2}. ...
  25. T

    B Probability of eating one egg with salmonella

    ln Will Kurt's Kurt's book "Bayesian Statistics The Fun Way" he gives a problem at the end of a chapter " Raw eggs have a 1/20,000 probability of having salmonella. If you eat two raw eggs what is the probability that you ate a raw egg with salmonella." The online answer he gives: "For this...
  26. guyvsdcsniper

    Probability of electron in hydrogen nucleus for 1s and 2s wave-functions

    For this problem, Is it as simple as using the probability density function, P = Ψ2 and plugging in the radius value given to me? So essentially I am just squaring the wave function and plugging in?
  27. chwala

    Determine the probability of ##P(x>4)##

    Find the solution here; Ok my interest is on part (b) and (c) only. Let's start with (b), My take is, $$\int_4^5 \dfrac{2}{75}x\,dx=\left.[\frac{x^2}{75}]\right|_4^5$$ $$=(0.33333333-0.21333333)+\frac{2}{15}×5$$ $$=0.12+0.6666666666=0.78666666$$ note that at ##f(x)##=##\dfrac{2}{15}##, the...
  28. D

    I Probability of Sum of 2 Random Ints Being Prime

    if I select two integers at random between 1 and 1,000, what is the probability that their sum will be prime?
  29. J

    The infinite limits of the probability transition matrix for Markov chain

    Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix P given below: Now, I have already figured out the solutions for parts a,b and c. However, I don't know how to go about solving part d? I mean the question says we can't use higher powers of matrices to justify our...
  30. chwala

    Problem involving Probability density function

    I just want to be certain, i think the inequality indicated is not correct...ought to be less than. Kindly confirm...This is a textbook literature.
  31. L

    [Bayes' Theorem] Finding the probability of guessing correctly

    I calculated the probability of box 1/2/3 given white. P(Box 1 | white) = P(Box 1)*P(White | Box 1)/(P(Box 1)*P(White | Box 1) + P(Not Box 1)*P(White | Not Box 1)) = ((1/3)*(1/2)) / (((1/3)*(1/2) + (2/3)*(5/7) = 7/27 P(Box 2 | white) = P(Box 2)*P(White | Box 2)/(P(Box 2)*P(White | Box 2) +...
  32. chwala

    Find the probability of choosing exactly 4 red cards

    Find the question and solution here ( sorry its a bit blurred) ... given using the hypergeometric method...i wanted to understand what is the clear distinction between hypergeometric and binomial distribution? Could it be in in reference to no replacement and replacement...? My approach on this...
  33. C

    I Taking socks out of drawers, conditional probability

    Problem: In a dresser there are 3 drawers. In one drawer there are two black socks and one white sock, in the second drawer there are two white socks, and in the third drawer there is a black and white sock. Suppose I chose a drawer randomly ( meaning, in a uniform distribution ) and I took a...
  34. Yan Campo

    I Probability when measuring a local observable

    I have information that $$\rho_{ab}=\sum_{j}p_{j}\ket{\Psi_{j}^{ab}}\bra{\Psi_{j}^{ab}}$$ and $$Pr(o_{j}^{(a)}|\Psi_{ab})=Tr_{ab}(\ket{\Psi_{ab}}\bra{\Psi_{ab}}(\ket{o_{j}^{(a)}}\bra{o_{j}^{(a)}}\otimes \mathbb{I}_{2})) \text{.}$$ I started by representing the density operator for pure states...
  35. S

    Conditional probability of a test records this positive result

    My attempt: $$P(\text{B is positive}|\text{A is positive})=\frac{P(\text{B is positive} \cap \text{A is positive})}{P(\text{A is positive})}$$ $$=\frac{P(\text{B is positive})\times P(\text{A is positive})}{P(\text{A is positive})}$$ $$=P(\text{B is positive})$$ $$=0.01 \times 0.99 + 0.99 \times...
  36. K

    I Probability in a small interval is ##P. dx##

    Reif says " ... variable ##u## which can assume any value in the continuous range ##a_{1}<u<a_{2}##. To give a probability description of such a situation, one can focus attention on any infinitesimal range of the variable between ##u## and ##u+d u## and ask for the probability that the variable...
  37. Lapse

    I How to Combine These Two Probabilities?

    I am trying to determine the likelihood of a driver winning a race based on an associated rating as well as the team he drives for. The probability that Driver A beats Driver B = .8504 The probability that Team A beats Team B = .7576 How do I combine these two probabilities, where the outcome...
  38. F

    Prob/Stats Which Probability books are good for the first course?

    Please tell me some good books on Probability for first course.
  39. shivajikobardan

    Comp Sci What is the probability that there is a burglary given John and Mary calls?

    this is the question Here is a tutorial video but his steps are very confusing to me. I personally know bayes theorem and have already studied probability and got good marks in it(It may not be a metric for being quality in it given that it is nepal we are talking about.)...
  40. Buzz Bloom

    I An idea for calculating the probability that the Universe is flat

    References (1) https://www.physicsforums.com/threads/what-is-the-probability-that-the-universe-is-absolutely-flat.971984/ (2) https://www.physicsforums.com/threads/calculating-the-probability-that-the-universe-is-finite.1011826/ Suppose the Friedmann equation is used to analyze two models. (1)...
  41. P

    B Decision for conditional probability instead of intersection of events

    Hello, I have a question about the following sentence and would appreciate if someone could explain how to read out the conditional probability here. "Each microwave produced at factory A is defective with probability 0.05". I understand the sentence as the intersection ##P(Defect \cap...
  42. K

    Using Poisson random variables to calculate this probability

    I calculated the mean which is 78.4 And the Standard deviation is 5.6 I thought the answer would be (90^(-78.4)/78.4!)*e^-90 But looking back having a decimal factorial doesn't make sense I have the numerical answers for c)= 0.019226 and d)=0.022750 but I my solution was wrong. Any help on...
  43. Buzz Bloom

    I Calculating the probability that the Universe is finite

    Reference https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L06.pdf I note that the use of Gaussian probabilities is mentioned many times in the reference. However in many discussions via posts in many threads, there seems to be a consensus that the distribution is...
  44. karush

    MHB Pb.20 What is the probability that Hiroko....will be chosen

    The 35 member History Club is meeting to choose a student government representative. \item The members decide that the representative, who will be chosen at random, CANNOT be any of the 3 officers of the club. What is the probability that Hiroko, who is a member of the club but NOT an officer...
  45. A

    A Solving an Integral involving a probability density function

    In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows: "For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...
  46. Grinkle

    B Can probability waves be "focused"?

    I really cannot ask this question well. I can only hope its not simply a waste of the readers time. I won't finish every sentance with "maybe I'm wrong", just assume its in my mind every time I hit the period key. An electron on a screen leaves a pixel spot, this pixel spot is a measurement...
  47. Mr_Allod

    Probability of finding a pion in a small volume of pionic hydrogen

    Hello, I am trying to figure out the right way to approach this. First of all, other than the different Bohr radius value, does the change to a negative pion make any other difference to calculating the probability? Also what would be the correct way to apply the "small volume"? What I'm...
  48. A

    I The probability density function for the double-slit experiment

    I am desperate. I've scoured the web for the formula for the probability density function for the interference pattern obtained in the double slit experiment with both slits open. So I want to know the probability density function and not the intensity function. I prefer not to have references...
  49. Fobi

    I Probability of 10 consecutive tails with 30 coin flips

    Hi, i was doing a programming exercise that asked me to simulate te flip of coins until it finds 10 consecutive tails. The program usually needs to flips like 6000/8000 coins before finding 10 tails consecutively, but suddenly i found 10 tails with only 30 coin flips, i think that what happened...
  50. L

    Find the probability of different scenarios

    Summary:: Bag A contains 1 white straw, 2 red straws and 2 green straws. Bag B contains 2 white straws, 2 red straws and 1 green straw. One straw is drawn at random from each bag. Find the probabilities that (a) the two straws drawn are of the same colour; (b) one straw is red and the other...
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