Probabilities Definition and 398 Threads

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. keyzan

    Probability of the outcomes of ##J^2## and ##J_{z}##?

    Hi guys, I have a problem with point 2 of this exercise: The electron of a hydrogen atom is initially found in the state: having considered the quantum numbers n,l,m and epsilon related to the operators H, L^2, Lz and Sz. I am asked: determine the possible outcomes of a measurement of J^2...
  2. alima

    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) /...
  3. PeroK

    I Tennis Probabilities Challenge

    Let's assume that a player has a probability ##p## of winning a point on the opponent's serve. 1) What is the probability (##P##) of a break of serve - i.e. winning the game? 2) What is the expected number of break points per service game? PS I had a mistake in my original calculation for 2)...
  4. Memo

    How Many Type A Pigs Have 3 DYL Blood in a Herd of 1,000?

    Mentor note: Thread moved from technical section to here, so is missing the homework template. TL;DR Summary: The weight of DYL 3-blood hybrid pigs after correction of a farm is a random quantity with a normal distribution. Knowing that the probability of a pig weighing over 20 kg is 0.1587 and...
  5. J

    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...
  6. C

    A Multiparticipant events and pairwise probabilities

    Is there a formula for this ? Consider the following simple looking problem. We have three contestants A, B and C, there is a competition between them and the best wins. For example a race, discus throw, javelin throw ... We know that A beats B with a probability 0.6, A beats C with a...
  7. S

    I Many worlds interpretation with unequal probabilities

    Let's pay a visit to one of Schrodinger's cats. In the classical statement of the case, we have to decide if the cat is alive or dead when the probability of the radio-active decay mechanism has a 50/50 chance of releasing the cyanide, most often posed as 60 minutes. If I understand the MW...
  8. 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...
  9. 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...
  10. F

    Looking for a particular function

    TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...
  11. B

    I Roulette system -- which optimization is better?

    Hey, gotta do some explanation first: I assume you know how roulette works. (if you dont: ball is thrown into a pit and it can either land on red, black or zero, each having a certain likeliness to land there. you can bet on where the ball will land) let's assume unrealistically you have the...
  12. 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!
  13. 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 "...
  14. SrivastavaHarsh

    A A game of seemingly mutually independent events

    There are 4 players numbered 1 to 4. There is a room with an entrance door to one side and exit door to opposite side. Inside the room, there are 4 boxes numbered 1 to 4. Inside each box, there is a chit containing a number (with equal probability) from 1 to 4 (inclusive). No two chits can have...
  15. 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...
  16. Feynstein100

    I Is there a way to calculate expected value from probabilistic data?

    So I ran a python simulation of 1,000 games of toss (50/50 odds) where each game consists of 100,000 consecutive flips. The result was this: 1000 is our starting balance and as expected, there's a nice normal distribution around it. I also calculated the average value after all the games and it...
  17. G

    B Understanding Waves, Particles and Probabilities

    In the ongoing quantum interpretations and foundations thread vanahees71 explained to me that the wave particle duality has been explained by the model where the position of a particle is calculated according to a probability distribution traveling in space. Am I understanding this...
  18. jjson775

    I Understanding Wave Function Probabilities for a Free Particle

    The textbook I am self studying says that the wave function for a free particle with a known momentum, on the x axis, can be given as Asin(kx) and that the particle has an equal probability of being at any point along the x axis. I understand the square of the wave function to be the probability...
  19. shivajikobardan

    MHB Calculating Probabilities in Poker Hand: An Example

    An example of how probabilities are calculated in poker hand. https://lh5.googleusercontent.com/pCynfBFNqfR00y8rEKWoXYkbSCGR310FpejMJ_iGWlwD7ttkCZjunp-TLKFMmU0A94CDsR4Bb-X8i6ai8RxiLLPdWlf1j9g6BZdjq1ppPZzp0JZOBjCVqwvKCK9XmGfg7Ks7VnN4IoWZIY3gqWvmKw Probability and Statistics with Applications: A...
  20. murshid_islam

    I How to calculate the probabilities here? (3-person card game)

    Suppose I have a shuffled pack of cards, out of which 4 red cards have been lost or removed. So now I have 22 red cards and 26 black cards in random order. I'm dealing cards in pairs. If a pair has 2 red cards, they are dealt to Mr. Red, if the pair has 2 black cards, they're dealt to Mr. Black...
  21. E

    MHB Probabilities of certain events in a lucky wheel game

    Given information: A wheel of fortune with ten equal sectors is used for a candidate game. Five of these sectors are labelled only with the number 1, three only with the number 2 and two only with the number 3. The game for a pair of candidates is as follows: The two candidates �K1 and K2...
  22. Neo Tran

    The occupation probabilities of electrons in different states

    occup is proportional to [gi x exp(-Ei/kT)] where gi is the numver of states at energy Ei
  23. 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...
  24. H

    A Probabilities of couples of impacts - Bob and Alice and Young experiment

    Let O be where pairs of EPR particles are emitted to Alice on the righr and Bob on the left. the pairs have a constant total energy, a null total momentun and a null total angular momentum. Alice at a distance D on the right of O have a Young double slits device with a screen at D+ L . Bob has...
  25. 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...
  26. 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...
  27. kfulton

    Winning and losing distributions

    I started by trying to write a normal distribution function for losing and not sure if that is possible.
  28. K

    Possible Results and Probabilities of a Measurement of Operator Q

    I first Normalise the wavefunction: $$ \Psi_N = A*\Psi, \textrm{ where } A = (\frac{1}{\sum {|a_n^{'}|^{2}}})^{1/2} $$ $$ \Psi_N = \frac{2}{7}\phi_1^Q+\frac{3}{7}\phi_2^Q+\frac{6}{7}\phi_3^Q $$ The Eigenstate Equation is: $$\hat{Q}\phi_n=q_n\phi_n$$ The eigenvalues are the set of possible...
  29. C

    A Beta/gamma decay probabilities

    Hi all, I have received this following question which I can't really figure out all the way to the end: Consider the beta decay of 212Pb: * What is the probability that the decay leads to the second excited state of 212Bi at 238.6 keV? This is straight forward - from nndc NuDat, it seems...
  30. M

    MHB How to Show Equality of Probabilities?

    Hey! :giggle: Let $(\Omega, p)$ be a discrete probability room with induced probability measure $P$ and let $A, B\subseteq \Omega$ be two events. I want to show that $P(A\cap B)-P(A)P(B)=P(A^c)P(B)-P(A^c\cap B)$. For that do we write to what for example $P(A^c\cap B)$ is equal to simplify the...
  31. pinball1970

    B Combining probabilities / results

    I had a LFT test this week which gives a 75% accuracy result.Negative.I also wanted to stick in some prior knowledge if I can, 20% of the UK currently has Covid say.Vanadium 50 illustrated this to me a while back so I hope I don’t mess it up 20 False pos No Covid in UK 80% 60 Neg...
  32. PainterGuy

    I Conditional and joint probabilities of statistically dependent events

    Hi, If the events A and B are statically dependent then the following formulas are used to calculate conditional probability and joint probability but there is a problem. As I see it both formulas are dependent upon each other. One cannot calculate conditional probability without first...
  33. S

    B Do eigenstate probabilities change with time?

    To my understanding any quantum system can be describes as a linear combination of eigenstates or eigevectors of any hermetian operator, and that the eigen values represent the observable properties. But how does the system change with time? I suppose big systems with many particles change with...
  34. R

    Expected value question -- Probabilities when tossing two coins

    Summary:: checking an expected error Given the question: "If a person tosses two coins and gets two heads, the person wins $10. How much should the person pay if the game is to be fair?" The book gives the answer as $2.5 while I calculate $3.333... E(X) = 0 = $10(1/4) - a(3/4) => a =...
  35. H

    A Heisenberg lens and probabilities

    Hi Pf in her experiment Birgit Dopfer uses an https://www.researchgate.net/figure/color-online-The-Dopfer-experiment-of-the-Zeilinger-Group-Innsbruck-If-detector-D2-is_fig7_265787833 the distance between the source and the lens is 2f and the detector may be at the distance f or 2f behind the...
  36. A

    A What are the probabilities of a die with an offset center of gravity?

    With a pure die, all odds are equal. With a pure die, the center of gravity is exactly in the middle of the die. But what if the center of gravity is not in the center? How are the odds then. For example, how do the odds become if the center of gravity is exactly on the line that runs through...
  37. AndreasC

    I Exceptions to the postulate of equal a priori probabilities?

    Not sure if this is the appropriate forum for this, hopefully if it isn't someone can move it to a more appropriate place. The fundamental postulate of equal a priori probabilities in statistical physics asserts that all accessible microstates states in an ensemble happen with equal...
  38. S

    Prob/Stats Material on complex random variables and exotic probabilities

    I am looking for books that have sections or even chapters devoted to complex random variables, or random variables that can take on complex values (NOT probabilities that are valued in the complex range, in this regard). On the other hand, if someone does know any books that contain material on...
  39. A

    A Question about a property of a matrix of transition probabilities

    In a 2012 article published in the Mathematical Gazette, in the game of golf hole score probability distributions were derived for a par three, four and five based on Hardy's ideas of how an hole score comes about. Hardy (1945) assumed that there are three types of strokes: a good (##G##)...
  40. J

    I Origin of Probabilities in Quantum Mechanics?

    The non-normalized wavefunction of a general qubit is given by: $$|\psi\rangle=A|0\rangle+B|1\rangle.$$ The complex amplitudes ##A## and ##B## can be represented by two arrows in the complex plane: Now the wavefunction can be multiplied by any complex number ##R## without changing the...
  41. kshitij

    What is the ideal number of guesses to maximize your marks in MCQs?

    To approach this, I first assumed the case when the students attempts all the remaining questions. Probability that they gain 4 marks for a guess = ##\frac 1 4## Probability that they lose 1 for a guess = ##\frac 3 4## Now let us say the number of correct guesses = ##r## Now we should have at...
  42. K

    I Odds of Winning Lottery: Expected Reward & Profit

    Hello! My friend got me a lottery ticket (which I didn't win) and I decided to check the odds of winning for that particular game. The prizes for this game are: 5, 10, 15, 20, 50,100, 500,1000, 5000,1000000 ($) and the probability for each of the prizes is 1 over: 10, 10, 150, 50, 150, 131.63...
  43. hdp12

    Bernoulli and Bayesian probabilities

    Summary:: Hello there, I'm a mechanical engineer pursuing my graduate degree and I'm taking a class on machine learning. Coding is a skill of mine, but statistics is not... anyway, I have a homework problem on Bernoulli and Bayesian probabilities. I believe I've done the first few parts...
  44. L

    I Probability that two points are on opposite sides of a line

    I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $ [0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction...
  45. J

    B Are proofs needed for definitions? Conditional probabilities

    My probability class has me wondering about pure math questions now. We started with the axioms and are slowly building up the theory. Everything was fine but then a definition of Conditional Probability P[A|B] = \frac{P[AB]}{P} appeared and it's just not sitting right with me. I know that...
  46. L

    MHB Q - Learning : Computing q(s,a) given transition probabilities

    Hi, I'm learning Reinforcement Learning, and computing q values is challenging. I'm not sure if the question wants me to follow this formula since I'm not given the learning rate \( \alpha \), and also because Q- Learning doesn't need a transition matrix. I'm really not sure where to begin...
  47. P

    I Prove that the norm squared of a superposition of two states is +ve

    This is what I have so far: $$ |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + \alpha^*\beta\Psi_1^*\Psi_2 + \alpha\beta^*\Psi_1\Psi_2^* $$ $$=> |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + 2Re(\alpha^*\beta\Psi_1^*\Psi_2) $$ I am...
  48. wolfy

    B Quantum Tunneling different individual transmission probabilities

    I have gone through all the videos on Youtube about Quantum Tunneling and became interested in it, so any helpful feedback would be appreciated. Do all the different individual transmission probabilities of electrons, protons, and such remain constant? May I ask what is the "formula" or...
  49. J

    Markov chain on state {1, 2, 3, 4, 5, 6 , 7}

    I need this for a programming project. Could you help?
  50. 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...
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