Probability Definition and 320 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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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}?
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. WMDhamnekar

    I Probability spaces

  11. 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...
  12. 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!
  13. 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...
  14. 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...
  15. WMDhamnekar

    Using Multiple integrals to compute expected value

    I want to know how did author derive the red underlined term in the following Example?
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. J

    Double Masking Efficacy

    Would the two masks together be 126% effective?
  24. C

    Prob/Stats Books on Combinatorics, Permutations and Probability

    Hello! I am looking for textbooks to relearn Combinatorics, Permutations Combinations and Probability and also Matrix algebra( decomposition, etc). I had done these many years ago and the course/books provided to me at that time weren't that great. So I want to relearn this with a more...
  25. Amitkumarr

    I Finding bias of the coin from noise corrupted signals

    Suppose there are two persons A and B such that both have a personal communication system which can transmit and receive bits. B has a biased coin whose bias is not known. A asks B to toss the coin 2000 times, send a 0 when a tail comes up and a 1 when a head comes up. It is known that whatever...
  26. D

    Combinatorics: calculating Oz Lotto odds for divisions

    In Oz Lotto, balls are numbered 1 to 45. Nine are selected, seven of which are winning numbers and two being supplementary numbers. Players select seven numbers. The odds of winning can be found here: I tried...
  27. M

    Variance of a point chosen at random on the circumference of a circle

    Hi, I was looking at this problem and just having a go at it. Question: Let us randomly generate points ##(x,y)## on the circumference of a circle (two dimensions). (a) What is ##\text{Var}(x)##? (b) What if you randomly generate points on the surface of a sphere instead? Attempt: In terms of...
  28. Falgun

    Prob/Stats Looking for a probability and statistics textbook

    I want to learn some probability & statistics on my own. I am well versed in Calc 1-3 , elementary ODEs and very little linear algebra. I want a comprehensive , introductory textbook which is NOT COOKBOOK STYLE. I might be self studying AP statistics next term so if the book covers everything I...
  29. Y

    How many combinations? (High school math problem)

    Summary:: Year 11 Extension 1 Math problem (Australia) How many combinations can be made from a 4 digit pin code if we can only use two numbers to form our pin code, and we MUST use 2 distinct numbers. E.g. 1112, 4334, 9944, 3232. But NOT 1111, 2113, 0992 etc. We're using the numbers 0-9 and...
  30. L

    Statistics: Verifying a Probability Proof

  31. L

    Statistics: Prove following theorem by expressing all the binomial coefficients in terms of factorials

    I really don't know what to do for this problem. I looked at similar threads but couldn't seem to grasp the idea of it. I would like help on how to start.
  32. P

    I Find P(X+Y>1/2) for given joint density function

    Hey everybody, :smile: I have a joint density of the random variables ##X## and ##Y## given and want to find out ##P(X+Y>1/2)##. The joint density is as follows: $$f_{XY}(x,y) = \begin{cases}\frac{1}{y}, &0<x<y,0<y<1 \\ 0, &else \end{cases}$$ To get a view of this I created a plot: As...
  33. Armine

    Proof of a formula with two geometric random variables

    The image above is the problem and the image below is the solution I have tried but failed.
  34. 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...
  35. nomadreid

    I Why P(A), and not P(A)(1-P(A))

    The summary says it all: why is the probability of an event not calculated by the probability that it is the event AND that it is not any other? Sounds silly, and I am certain the explanation is simple, but I don't see it.
  36. L

    I Two vectors and two perpendicular lines

    In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...
  37. U

    I Conditional distribution of geometric series

    Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
  38. entropy1

    I Compatibility of MWI with probability of outcomes

    Can MWI account for the probabilities of outcomes? If MWI says all outcomes are realized, is the probability that an outcome occurs then not 100%? How is this explained with the entanglement of the measured object and the measurement apparatus?
  39. J

    I Bernoulli Trials Homework Problem

    this is the answer Is this right?
  40. 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...
  41. 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.
  42. 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...
  43. E

    Measurements of GHZ state

    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...
  44. S

    Transition Probability

    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
  45. 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...
  46. CaptainX

    B Conditional Probability

    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...
  47. The unbelievable solution to the 100 prisoner puzzle.

    The unbelievable solution to the 100 prisoner puzzle.

  48. Biochemgirl2002

    Calculating general normal random probability

    a) P(X<18) = (18-20)/sqrt25 =-2/5 =-0.4 then you use the standard normal table and find that; P(X<18)=0.3446 b) P(X>27) = (27-20)/5 = 7/5 = 1.4 P(Z>1.4) =P(Z<-1.4) =0.0808 C) =(13<X<23) =13-20/5 , 23-20/5 =-7/5 , 3/5 =-1.4 , 0.6 P(Z<0.6)-P(Z<-1.4) =0.7257-0.0808 =0.6449
  49. P

    I Are these events independent?

    Hello everyone. Let us consider 3 events A,B,C such that: $$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$ Notice that the second term is a union and not an intersection. Are they independent? And what if the assumption was: $$P(A \cap( B \cup C))=P(A)*P(B)*P(C)$$? I know that the independence condition...
  50. christang_1023

    I How to understand this property of Geometric Distribution

    There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$. I understand it in such a way: ##X## is independent, that's to say after there are ##(n+k-1)## successive failures, ##k## additional trials performed afterward won't be impacted, so these ##k##...