Probability Definition and 1000 Threads

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...

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...

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...

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...

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)...

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...

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...

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...

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 $$...

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...

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...

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...

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...

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)...

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...

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...

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...

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

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...

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...

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...

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.

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...

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

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...

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...

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!

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...

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...

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...

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? -...

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.)

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}##...

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...

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...

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%...

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...

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

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...

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

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.

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...

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...

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...

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...

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.

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...

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...

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)\...