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. red65

    B Concerning the birthday problem in probability

    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...
  2. E

    Ensemble vs. time averages and Ashcroft and Mermin Problem 1.1

    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?)...
  3. F

    I Sample space, outcome, event, random variable, probability...

    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...
  4. N

    B Head or Tails: The Question of Determinism and Probability

    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...
  5. H

    Probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?

    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...
  6. E

    Probability of Negative Value in Sz 1/2 Spin System w/ Lambda 1 & 2

    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 ?
  7. Z

    I Free particle probability distribution

    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...
  8. chwala

    Solve the given problem that involves probability

    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}}## ...
  9. chwala

    Calculate ##P(C|A')## in the given probability problem

    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...
  10. chwala

    Solve the given problem that involves Probability

    I may seek an alternative approach; actually i had thought that this would take a few minutes of my time..but just realized that it just takes a minute; My interest is only on highlighted part. Text solution My take; ##P(\text{at least one of the first three days is wet})=1-P(ddd)##...
  11. J

    B Probability of seeing peak noise in a given time window

    Hi! Say I have a electric signal that has an RMS noise value of 10uV, I would calculate peak noise by multiplying by 6.6, so 66uV. I am looking for an equation that describes the probability of seeing a noise voltage that reaches 66uV in a given viewing time window. For example if I look at the...
  12. V

    Probability of not getting a prize

    Let W be the event that a prize is received. Then ##p(W) + p(not ~ W) = 1##. We need to find ##p(not W)## and so let's try to find ##p(W)## and then we can subtract it from 1 to get ##p(not ~ W)##. The experiment is buying 2 tickets. So, $$p(W) = \frac { {}^{10}C_2} { {}^{10000}C_2}$$ Thus...
  13. N

    I What is the probability of life on exopanet LHS475b?

    Recently the James Webb Telescope discovered an exoplanet 99% the size of Earth. Its name is LHS 475 b. What is the probability that there is life on this planet?
  14. V

    Are both sample spaces the same or do they mean different things?

    I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up. $$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$ $$ S = \{ (i,j) : i...
  15. red65

    B The law of total probability with extra conditioning

    Hello, I am studying probability and came across this theorem, it's the law of total probability with extra conditioning, I tried to work out a proof but couldn't ,does anyone know the proof for this : thanks!
  16. I

    I QED/Quantum Mechanics: Probability or Spatial Function?

    In "QED, The Strange Theory of Light And Matter" Richard Feynman describes the probability path of a single photon emitted from a source, reflected from a mirror surface, and finally reflected to a probe detector. This path is the least time path relative to the probe/viewer, determined by the...
  17. red65

    B Where is the error in my reasoning about palindromes?

    Hello everyone, I found this problem online about probability, for me, I think that to have a 2 letter palindrome is less likely because we need to have the same letter in the 2 places which gives us 26 possibilities (aa , bb, cc ....) however for words with 3 letters we have 26 possibilities...
  18. red65

    B About the naive definition of probability

    hello, I took an introductory course about statistics, we viewed the naive definition of probability which says "it requires equally likely outcomes and can't handle an infinite sample space ", I understood that it requires finite sample space but I didn't understand "equally likely outcomes "...
  19. James1238765

    I What is the probability of an electron emitting a photon?

    I have seen many tutorials that provide steps how to transcribe a Feynman diagram into algebra, for instance [here]: However, I have never seen the final line of the calculation converted into a real number. What are the steps to get from the algebra equations transcribed using the Feynman...
  20. dubeypuja

    Expectation of Product of three RVs

    We have three Random variable or vector A,B,C. Condition is A & B are independent as well as B & C are independent RVs . But A & C are the same random variable with same distribution . So How can determine E{ABC}. Can I write this E{ABC}= E{AE{B}C}?
  21. G

    Probability density of a 1-D Tonk Gas

    It is a 1D Tonk gas consisting of ##N## particles lined up on the interval ##L##. The particles themselves have the length ##a##. Between two particles there is a gap of length ##y_i##. ##L_f## is the free length, i.e. ##L_f=L-Na##. I have now received the following tip: Determine the...
  22. A

    Probability involving Gaussian random sequences

    How do I approach the following problem while only knowing the PSD of a Gaussian random sequence (i.e. I don't know the exact distribution of $V_k$)? Or am I missing something obvious? Problem statement: Thoughts: I know with the PSD given, the autocorrelation function are delta functions due...
  23. S

    Finding a Mistake: Probability of Y with X and Spins

    I divide the question into three cases: 1) P (Y = 1 and X = 5 or 7) = 1/5 + 1/5 = 2/5 2) P(Y = 2 and 1st spin = even and 2nd spin = 7) = 2/5 x 1/5 = 2/25 3) P(Y = 3 and 1st and 2nd spin = even and 3rd spin = 7) = 2/5 x 2/5 x 1/5 = 2/125 Total probability = 2/5 + 2/25 + 2/125 = 62/125 But...
  24. Peter Morgan

    I The collapse of a quantum state as a joint probability construction

    The titular paper can be found here, https://doi.org/10.1088/1751-8121/ac6f2f, and on arXiv as https://arxiv.org/abs/2101.10931 (which is paginated differently, but the text and equation and section numbers are the same). Please see the abstract, but in part this 24 page paper argues that we...
  25. S

    Calculating Probability Using Bayes' Formula: Solving for P(urn III | silver)

    From my understanding of Bayes formula, it should look like something like this P(Silver| III) = \frac{P(III | silver) \times P(silver)}{P(III)} now we know that P(urn III) = 1/3 and the probability of P(silver) = Pr(silver|urn I) + P(silver|urn II) + P(silver|urn III) = 1/3 (0) + 1/3 (1/2)...
  26. J

    B How helpful is probability theory?

    I am interested in stomach acid and heat expansion, for instance the stomach will become heated due to an athelete competing. The heat causes atheletes to live shorter than people who don't have their body heated so often. I do a lot of differential equations and number theory, but I was...
  27. bsharvy

    B Understanding Probability of Bias in Coin and Dice Tosses

    I was thinking that the probability of a set of events not happening is the same as the probability of that the die/coin is biased. So, if I flip a coin 10 times and get heads every time, the probability the coin is biased is 1- (.5)^7. Roll a die 5 times, get "4" all times, probability of...
  28. C

    I Probability of White Ball in Box of 120 Balls: Solved!

    Problem: In a box there are ##120## balls with ## X ## of them being white and ## 120 - X ## being red for random variable ##X##. We know that ## E[ X] = 30 ##. We are taking out ## k ## balls randomly and with returning ( we return each ball we take out, so there is equal probability for each...
  29. earthling75

    Stuck calculating probability of measuring ##S_y## for spin 1 particle

    I know how to construct Sy for spin = 1 case from the raising and lowering operators. I get $$ S_y=\frac{i\hbar}{\sqrt{2}}\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & -1 \\ 0 & 1 & 0 \\ \end{pmatrix} $$ From what I have seen, the eigenspinor for $\hbar$ is found by solving $$...
  30. T

    I Probability paradox: P(X=x)=1/n >1

    I have a random variable X in range(0,n) where n<1, with a uniform distribution Then the probability of sample space S=n x P(X=x) x<=n which must be 1 Manipulating the equation P(X=x)=1/n >1 Then this violates the fundamental law of probability which says that any probability must be at most 1...
  31. B

    B Which probability calculation is "more correct" in Baccarat games?

    Hello, this question isnt really much about calculation but rather which view point is more correct. See, in a gambling game called Baccarat a game is played where a player A ("player") and a player B ("bank") draw cards according to a fixed ruleset from a given card pot. First A and B both...
  32. C

    I Prove that the tail of this distribution goes to zero

    Theorem: Let ## X ## be a random variable. Then ## \lim_{s \to \infty} P( |X| \geq s ) =0 ## Proof from teacher assistant's notes: We'll show first that ## \lim_{s \to \infty} P( X \geq s ) =0 ## and ## \lim_{s \to \infty} P( X \leq -s ) =0 ##: Let ## (s_n)_{n=1}^\infty ## be a...
  33. chwala

    Understanding the concept of Probability distribution

    Consider the attachment below; How did they arrive at ##F_X (u) = \dfrac{u-a}{b-a}## ? I think there is a mistake on the inequality, probably its supposed to be ##a≤u<b## and that will mean; $$F_X (u) =\dfrac{1}{b-a} \int_a^u du= \dfrac{1}{b-a} ⋅(u-a)$$ as required. Your thoughts...then i...
  34. chwala

    Solve this problem that involves a probability density function

    I am refreshing on this; ..after a long time... Note that i do not have the solution to this problem. I will start with part (a). ##f(u)= 3u-\dfrac{3u^2}{2k}## with limits ##0≤u≤k## it follows that, ##3k - \dfrac{3k}{2}=1## ##\dfrac{3k}{2}=1## ##k=\dfrac {2}{3}## For part (b)...
  35. S

    Probability of Hypokalemia w/ 1 or Multiple Measurements

    TL;DR Summary: Finding the probability with one measurement and multiple measurements on separate days. Question: Hypokalemia is diagnosed when blood potassium levels are low, below 3.5 mmol/L. Let’s assume we know a patient whose measured potassium levels vary daily according to N(µ = 3.8...
  36. J

    A The Probability Distribution of a Bosonic Field when Emitted

    If a bosonic field is probabalistic, and if it can be emitted (suddenly coming into existence), what determines its probability distribution when it is emitted from a fermion? In other words, one thinks (or at least I think) of a fermion field as already being in existence and already having...
  37. Ahmed1029

    I Spin expectation value for one particle vs actual measurement

    When the expectation value of spin in the z direction for one particle is zero and I make measurements for an even number of particles in the same state, do I get exactly half to be spin up and half to be spin down along the z direction? More generally, what does spin expectation value for one...
  38. MathMan2022

    Conditional probability problem

    A) P(A and B) = 0.45 * 5/10 B P(Not B) = 1 - ( 0.45 * 5/10) Is it like this?
  39. K

    Conservation of probability issue when solving ODE in Mathematica

    I am trying to solve this two level (Schrodinger) equation as a function of time:$$i\begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & iW+dE_0sin(\omega t)\\ -iW+dE_0sin(\omega t) & \Delta \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$ (I can go into more details about...
  40. Ahmed1029

    I Can I find a particle in two states simultaneously?

    If I want to get the spin angular momentum of a particle using the Stem-Gerlach experiment, I think I will find the spin 1/2 particle either spin up or spin down, but not both. I however want to ask this : Is there a non-zero probability that a particle which is spin-up in the z direction to be...
  41. WMDhamnekar

    B Probability of firing exactly one shot in each annular zone

    My attempt to answer this question: With the radii in the ratio ## 1: \frac12: \frac13 ##, the area of the corresponding circles will be in the ratio of ##1: \frac14: \frac19 ##. The areas of the three rings will be in the ratio of ## \frac34 : \frac{5}{36}: \frac19 ## So, if three shots are...
  42. WMDhamnekar

    I Probability Spaces | What You Need to Know

  43. Euge

    POTW Convergence in Probability

    Prove that if ##\{X_n\}_{n = 1}^\infty## is a sequence of real random variables on probability space ##(\Omega, \mathscr{F},\mathbb{P})## such that ##\lim_n \mathbb{E}[X_n] = \mu## and ##\lim_n \operatorname{Var}[X_n] = 0##, then ##X_n## converges to ##\mu## in probability.
  44. chwala

    Solve the problem involving probability density function

    This is the question: This is the ms solution- from Further Maths paper. My question is referenced to the highlighted part. I can see they substituted for the lower limit i.e ##x=1## to get: ##F(x)=\dfrac{x^3-1}{63}## supposing our limits were; ##2≤x≤4## would the same approach apply? Anything...
  45. S

    Find the probability that the shorter piece of a 20cm string is at least 8cm when the string is cut randomly

    I want to ask about part (c). This is what I did: the length of the shorter piece should be 8 ≤ X < 10 so P (8 ≤ X < 10) = 2 . (1/20) = 1/10 But my teacher said the correct answer is 2/10. Where is my mistake? Thanks
  46. Drakkith

    B Calculating Probability of Event After Y Attempts

    Hey all. I've got next to no education in probability and I was wondering how to figure out the chance of some event occurring after Y number of attempts given X chance of happening per attempt. For example, if event A has a 0.15% chance of occurring each attempt, and you make, say, 1000...
  47. A

    Probability: pair of random variables

    Hello all, I would like to check my understanding and get some assistance with last part of the following question, please. For part (d), would I use f(x | y) = f(x, y) / f(y) ? Problem statement: My attempt at a solution, not too confident in my set-up for part (d). I drew a sketch of the...
  48. A

    Probability/Random variables question

    Hello all, I am wondering if my approach is coreect for the following probability question? I believe the joint PDF would be 1 given that the point is chosen from the unit square. To me, this question can be reduced down to finding the area of 1/4 of a circle with radius 1. Any help is appreciated!
  49. chwala

    Solve the problem involving toss of a biased coin- Probability

    This is the problem; My thinking on this is based on Von Neumann Strategy i.e ##e=pf+(1-p)((f+e)## where ##e##= Expected value, ##p##= Probability and ##f## = number of tosses ...in our case ##f=1## ##e=\frac{f}{p}=\frac{1}{p}## This is clear (as indicated on the left hand side of the ms...
  50. V

    Probability of selecting two white balls from two bags

    My attempt for part (a) is as given below. I will attempt part (b) after getting part (a) correct. (a) Based on what is asked, we can identify 3 independent events as follows: (i) select any 2 bags followed by (ii) select a ball from one bag followed by (iii) select a ball from the other bag...
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