Probability Definition and 1000 Threads

Hello everyone! I'm studying the physics of complex systems and I'm approaching probability theory. I understand that we need a ## \sigma-algebra ## and the Kolmogorov axioms in order to define a probability space but then I bumped into the following relation: $$ p(A_1 \cup A_2 ) = p( A_1 ) + p(...

Homework Statement If we throw with two dice a and b, than we make two spaces call them space A and B. Space A = a+b and space B = a*b. Next the question there are a couple of questions: 1) What is the probability P(A > 7) 2) What is the probability P(B = odd) 3) What is the probability P( A n...

I was reading a book about innumeracy and one of the chapters was on probability. This weather woman said 'there is a 50% chance of rain on Saturday, and a 50% chance of rain on Sunday, so the chance of rain this weekend is 100%' Obviously she was wrong, but it got me thinking how would one...

Hello! I have the following problem I'm trying to solve: Homework Statement An Hydrogen atom in the state |100> is found between the plates of a capacitor, where the electric field (weak and uniform) is: E(t) = \epsilon e^{-\alpha t / \tau}. Calculate the parameters of the potential...

I study control theory and robotics. Recently I figured out that I have a much deeper understanding of probability and statistics compared to my colleagues. Is this 'talent' valuable in my field and if so, where? We used this theory to define white noise, but nothing more...as of now. Also I am...

Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?

Given $$P(B \land C)$$ will it always be true that $$P(B \land C | A) P(A) + P(B \land C | \lnot A) P( \lnot A)$$ (regardless what $P(A)$ would be)? How can I prove this?

Hey guys! I have been working on this project during some time. First off, do you all think that this is possible? Of course, as we all may know, the roulette is a chaotic system, very sensitive to initial conditions, although deterministic. This means that small errors in the...

Hello All I was wondering if someone could help explain what the probability density function tells you. I am trying to learn about surveying and the PDF keeps cropping up and I do not fully understand it. For example I have:- measured a single angle 15 times calculated my Standard Deviation...

Hey! :o I am looking at the following: An airline assumes that $5 \%$ of all passengers that have booked for a flight will not appear for departure. They therefore book a flight with $50$ seats by selling $52$ tickets. It is assumed that one passenger independently of the other cancels his...

I have a statistics problem that is probably not too difficult for someone who knows what they are doing, but I still need help with it. Here’s the scenario. There is a town with a highway built at the end of 2013. The mayor is concerned because of the number of traffic accidents on the...

I have some data that I want to do simple linear regression on. However I don't have a lot of datapoints, and would like to know the uncertainty in the parameters. I.e. the slope and the intercept of the linear regression model. I know it should be possible to get a prob. distribution of the...

Homework Statement Imagine you are playing a game with me, of drawing balls from a box. There are two blue balls and two red balls. They are picked with equal probability, and are drawn without replacement. If you draw a blue ball, I give you $1. If you draw a red ball, you pay me $1.25. What...

Homework Statement In a manufacturing plant, a sample of a 100 items is taken from an assembly line. For each item in the sample, the probability of being defective is .06. What is the probability that there are 3 or more defective units in the sample? Homework Equations z = (x -...

Hey everyone, first, let me say I understand the complement rule. Where I am confused is over the integration. My professor said that suppose you have a continuous cumulative distribution function F(x) = 1-e-x/10, if x > 0 (0, otherwise). And suppose you want to find P(X>12) you can use the...

Is there a precise definition for the statement that two differently worded probability problems are "equivalent"? One technique of (purportedly) solving a controversial probability problem is to propose an "equivalent" problem whose solution is not controversial. (e.g. The Sleeping Beauty...

Hello everyone ! I would like to know if prob. theory is critical to learning Qm !

Homework Statement Let 0≤p≤1. Let there be k distinct numbers (they can be natural numbers) a1, a2, ... , ak, each repeating respectively b1, b2, ... , bk times. Let q < ∑r=1k br Determine the minimal values of b1 ... bk such that the probability of q numbers chosen out of ∑r=1k br numbers...

Hello! I am a bit confused about the interpretation of probability density in QFT. Let's say we have the Klein-Gordon equation. I understand that this is the field equation for a spin-0 charged particle. So if we find a solution ##\phi(x)## of the Klein-Gordon equation, as far as I understand...

Homework Statement A small manufacturing firm sells 1 machine per month with 0.3 probability; it sells 2 machines per month with 0.1 probability; it never sells more than 2 machines per month. If X represents the number of machines sold per month and the monthly profit is 2X2 + 3X + 1 (in...

Hi All This is in relation to the folllowing paper: https://arxiv.org/pdf/1402.6562.pdf See section 3 on examples where standard probability theory is discussed. Is it valid? To me its rather obvious but I had had a retired professor of probability say probability theory doesn't have a state...

Homework Statement I want to generate two random variables, one is normally distributed N ~N(10, 25) and the other one, E, is exponentially distributed with mean 1. I was not given a particular correlation coefficient.Homework Equations normal cdf, exponential cdf, inverse transform method...

Not my homework exactly but it's homework like so.. 1. Homework Statement There are some red and white counters in a bag. At the start there are 7 red and the rest white. Alfie takes two counters at random without putting any back. The probability that the first is white and the second red is...

Hey guys! first off, thank you all for your attention!. Okay I am a bit naïve on this, and also this question may seem a bit strange. But here it goes. Let's say that a mechanic roulette throws the ball at different preprogrammed velocities. Each velocity will lead to a determined final time...

Homework Statement An assembly line is observed until items of both types—good (G) items and items not meeting specification (N)—are observed. Show the sample space. Homework Equations Let G be Good Let N be Not Good The Attempt at a Solution S = {GN, GGN, GG...N, GG..., NG, NNG, NN...G...

the spin, S, for an electron is $$\frac{\hbar}{2}=5.27 \cdot 10^{-35} $$ $$\frac{2MR^2 \omega}{5}=\frac{2MRv}{5}$$ It is said that the speed of an electron is 2200 km per second and can be calculated in classical manners from electrostatic and accelerating forces on the electron from (1.11)...

Ok, so in a logistic regression context, I need to test if the probability of ##Y_{i}=1 ## is the same for two different groups at different ages where age is a continuous variable. This is actually complicated because of nonlinearity. Can I default to testing if the odds of ##Y_{i}=1 ## is...

Andrei has a bag of x sweets. He removes two sweets from the bag simultaneously (without replacement). He now removes a third sweet. The probability that the third sweet is red is (x/2) - 1. How many red sweets were in Andrei's bag to begin with? Could somebody please tell me if (and how) it is...

Q1. Why is the probability current ##j(x,t)=0## at ##x=\pm\infty##? (See first line of last paragraph below.) My attempt at explaining is as follows: For square-integrable functions, at ##x=\pm\infty##, ##\psi=0## and hence ##\psi^*=0##, while ##\frac{\partial\psi}{\partial x}## and hence...

(To any passing moderator: Feel free to move this to "statistics" forum if you feel that would be more appropriate.) Although my "to read" list is already too long, I have lately been getting increasingly interested in learning the basics of conditional probability, including Bayesian analysis...

Homework Statement Given correlation matrix $$M = \begin{bmatrix} 1 & .3 & .5 \\ .3 & 1 & .2 \\ .5 & .2 & 1 \\ \end{bmatrix}$$ And 3 independent standard normals $$N_1, N_2, N_3$$ using cholesky decomposition A) get the correlated standard normals B) and if...

I recently read a question. It was: If 100 birds are sitting in a circle and all of them peck a bird either on their left or on their right randomly, what's the expected number of birds that will be unpecked? The answer to this is 25 birds as probability of not being pecked is 25%. The logic...

Can someone please explain the probability of replacement and without replacement with one simple example for each?

Probability of an event happening = (Number of ways the event can happen)/(Total number of outcomes) Please, explain the above definition How is the above definition applied to the following question. A coin is tossed 100 times. How many heads will pop up? Solution: Let P = probability...

Can someone explain in simple terms the difference between the probability of AND and the probability of OR. Can you provide an example for each? Can you please explain the AND/OR formulas for each probability found in most textbooks?

In the New Testament, the Apostle Paul introduced an event that has come to be known as the rapture of the church. The rapture is an imminent event, which means it could happen at any moment in time. What is the probability that the rapture of the church will take place in our lifetime? Set up...

Substitute teacher, Baraba Rose, convinced a class of kindergarden kids that she would turn into a bird before the school day ends. What is the probability that this event will take place? Solution: Let P = probability P(turning into bird) = 0 The answer is 0 because human beings cannot turn...

There is a 30 percent chance of rain tomorrow. What is the probability of no rain tomorrow. Solution: Let P = probability P(no rain tomorrow) = 1 - P(rain tomorrow) P(no rain tomorrow) = 1 - 0.30 P(no rain tomorrow) = 0.70 or 70 percent Right?

Most students struggle with probability because questions usually involve FUZZY word problems. I am reviewing precalculus right now. In the later chapters of precalculus, a touch of probability is introduced. Can someone explain the basics of probability? I know, for example, that all...

Hello everybody. I have a Markowian homogeneous random walk. Given the transition matrix of the chain, I know that ##P[ X(t) = i | X(t-1) = j ] ≡ P_{j→i}=T_{ij}## where ##T## is the transition matrix and ##X(t)## is the position of the walker...

Homework Statement I have a variable ##s_i## with probability distribution ##w(s_i)## ##(\Delta(s_i))^2## denotes the variance ##=<(s-<s>)^2>=<s^2>-<s>^2## I want to show ## \sum\limits_{i\neq j} <\Delta s_i> < \Delta s_j> =0 ## where ## < > ## denote expectation My book has: ## <\Delta...

Prove the following I literally have no idea where to start or what to do.

1. Given a Markov state density function: ## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ## ##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...

I have heard that the probability of an unstable nucleus decaying is always constant. Is there any way to change this probability?

https://www.physicsforums.com/attachments/197770The path of a photon is a perfect straight line not a sine wave, right? (if the probability amplitude is zero)https://www.physicsforums.com/attachments/197771

Suppose that cars pass a certain intersection at a rate of 30 miles per hour. What is the probability that during a three-minute interval, no cars will pass the intersection? I am really just wondering which distribution to use. I thought is should be Poisson because it is asking for events...

Homework Statement Homework Equations [/B] Definition: A sequence X_1,X_2,\dots of real-valued random variables is said to converge in distribution to a random variable X if \lim_{n\rightarrow \infty}F_{n}(x)=F(x) for all x\in\mathbb{R} at which F is continuous. Here F_n, F are the...

I'm trying to replicate a machine learning experiment in a paper.The experiment used several signal generators and "trains" a system to recognize the output from each one. They way this is do is by sampling the output of each generator, and then building histograms from each trial. Later you...

Homework Statement An electron with a total energy of Eo = 4.4 eV is in the potential well shown above. 1) Find the ratio of the wavelength in Region III to the wavelength in Region I. λ III / λI = 1.772) Given that the wave function of the electron vanishes at the left boundary of Region...

Homework Statement Consider an electron in a state described by angular wavefunction $$\psi(\theta,\phi)=\sqrt{\frac{3}{4 \pi}}\sin \theta \cos \phi$$ Here θ and φ are the polar and azimuthal angles, respectively, in the spherical coordinate system. i. Calculate the probability that a...