Probability Definition and 1000 Threads

Homework Statement Hi all, I have some difficulty understanding the following problem, help is greatly appreciated! Let ##U_1, U_2, U_3## be independent random variables uniform on ##[0,1]##. Find the probability that the roots of the quadratic ##U_1 x^2 + U_2 x + U3## are real. Homework...

X is a random variable that follows the Log-Normal probability density function. n indipendent trials are carried out. We want to know the probability density function of the random variable Y, that is defined as the average value of the “n” outcomes of the trials described above.

Homework Statement I want to know why probability disintegration per second of a radioactive nucleus does not depend on time lived by it. Homework Equations N/N(initial)=e^(-λt) The Attempt at a Solution According to the above equation, the probability should increase with the passage of...

Homework Statement The wavefunction at t = 0 is given by $$\Psi = N*e^{-\frac{r}{a_0}}$$ where ##r = |\mathbf{x}|##. ##a_0## is a constant with units of length. The electron is in 3 dimensions. Find the approximate probability that the electron is found inside a tiny sphere centered at the...

According to a survey in 2017, nearly half of the Americans believe, that aliens visit earth.However, I think that a lot of conditions have to be met, so that aliens would visit earth. First of all, they have to exist. Meaning their home planet has to be a habitable planet (but only a very...

Hi all, I am rather confused about the following concept. Assistance is greatly appreciated! A time-dependent probability amplitude can be written as $$\langle a_k| e^{-\frac{i}{\hbar}\hat{H}t} |\psi\rangle$$ where ##a_k## is an eigenvalue. Suppose I want the x-representation of the ket, I can...

Homework Statement "Toss a coin repeatedly. Denote the event of getting heads or tails on the ##i##-th try by ##H_i## and ##T_i## respectively, where ##P(H_i)=p## and ##P(T_i)=1-p##, for some ##0\leq p \leq 1##. Now denote by ##E## the event of getting ##r## consecutive heads before ##s##...

Homework Statement A password has numbers 0-9 in it. The password is 5 digits, repeats are not allowed, and order doesn't matter (I just have to have the correct 5 digit buttons depressed). A. If I guess at the password, what is the probability that the box will open? B. If I have completely...

Hi everyone, I am currently working through the textbook Statistical Inference by Casella and Berger. My question has to do with transformations. Let ##X## be a random variable with cdf ##F_X(x)##. We want to find the cdf of ##Y=g(X)##. So we define the inverse mapping, ##g^{-1}(\{y\})=\{x\in...

Homework Statement [/B] Hi all, I have an issue understanding the concepts pertaining to the following problem, assistance is greatly appreciated. I understand the "flow" of the problem; first find the probability of obtaining balls of the same colour, then use the geometric distribution...

Homework Statement [/B] A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x) = (1/2) kx2 . If the spring constant of the oscillator is suddenly doubled, then the probability of finding the particle in ground state of new potential will be? (A)...

Hello. I am trying to decipher the formula, making sure I understand what exactly is going on in each part of the expression. I will be grateful for your guidance, corrections and help. Below I show the formula and the example for only 3 possible outcomes (in general it would be k) \(p =...

Homework Statement Team 0 and Team 1 have played 1000 games and Team 0 has won 900 of them.[/B] When the two teams play next, knowing only this information, which team is more likely to win? Homework Equations P(X,Y) = P(YlX) x P(X) = P(XIY) x P(Y) (Not Sure) The Attempt at a Solution Hi, I...

Hello. I am reading an online stats book, and there is the following question, which I solved incorrectly, and I think I understand what is my mistake, but I will be grateful for your explanation, if I have incorrectly detected the logic behind my mistake. I am weak at math (trying to improve it...

Homework Statement Homework Equations Chebyshev's Theorem: The percentage of observations that are within k standard deviations of the mean is at least 100(1 - (1/k2))% Chebyshev's Theorem is applicable to ANY data set, whether skewed or symmetrical. Empirical Rule: For a symmetrical...

Homework Statement Assume a job has 12 applicants, and 4 job openings. I want me and my 3 friends to all get the job. What is this probability. Homework Equations ! Factorial and Permuation & combinatoins The Attempt at a Solution Number of possible solutions: 12C4 = 495 Possible ways Number...

Homework Statement Discrete random variables ##X,Y,Z## are mutually independent if for all ##x_i, y_j, z_k##, $$P(X=x_i \wedge Y=y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_k )$$ I am trying to show (or trying to understand how someone has shown) that ##X,Y## are also independent as a result...

Homework Statement Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance. Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?Homework Equations The Attempt...

Hi all, I have a few questions regarding the issue of independence. Many thanks in advance. ##\textbf{1}## If I find that some events ##A, B, C## obey the following formula $$P(A \cap B \cap C ) = P(A)P(B)P(C)$$ it does not necessarily mean that a) they are mutually independent and b) ##A##...

<Moderator's note: Moved from homework.> Hi all, I have an issue understanding a statement I read in my text. It first states the following Proposition (Let's call it Proposition A): The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n...

Okay, so I just found out about the Rayleigh distribution being the radial distribution of a point composed of normal distributed cartesian components. And this is because of the area element, right? But how then can the joint density of the cartesian component's distributions equal that of the...

Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...

Homework Statement We are investigating hydrogen in a plasma with the temperature 4500 ºC. Calculate the probability per atom and second for stimulated emission from 2p to 1s if the lifetime of 2p is 1.6 ns Homework Equations ##A=\frac{1}{\Sigma \tau}## $$A_{2,1} = \frac{8*\pi *h *...

In the first volume of his lectures (cap. 6-5) Feynman asserts that these 2 can be the PDF of velocity and position of a particle. Under which conditions it's possible to model velocity and position of a particle using these particular PDFs ? ps: Is the "Heisenberg uncertainty principle"...

Hi Imagine we have a lottery, with chance of winning 1 in 1000 (1/1000). I have made computer simulations in order to find confidence levels for winning. At 1000 bought lottery tickets, the confidence of winning is 64.1% and 2000 bought lottery tickets the confidence of winning is 87.1% By...

There are five hexagons. The edges of each hexagon have been colored with one of three colors randomly. If you pick two hexagons randomly without replacement, what is the probability that they are the same? (Rotation is okay). The total space or denominator is 3^(2×6), therefore we have...

The probability, I was taught, for red to appear at roulette (on a European table, with a single green zero) is 18/37; and the probability for non-red to appear (black or green zero) is 19/37. If we live in a deterministic universe (I appreciate that that’s a big if for some, not so much for...

My question is whether there is a lowest possible probability for something to possibly (physically) occur on a cosmic basis? That is, is there a threshold 'lowest' probability below which something cannot occur? I'm not referring to 'zero' as the lowest probability. That's obvious. Rather, a...

I would like to know the solution to Liouville equation ∂ρ/∂t=-{ρ,H} given the initial condition ρ(t=0)=δ(q,p) where δ(q,p) is a dirac delta centered in some point (q,p) in phase space. I have the feeling, but I'm not sure, that the solution is of the form ρ(t)=δ(q(t),p(t)) where q(t) and...

<Moderator's note: Moved from a technical forum and thus no template.> Hi all, I'm looking at an exercise in probability and I have a little doubt. So the exercise goes like this: "Two players are playing against each other. They have the same probability of winning a single game. In order to...

It seems to me that so far in quantum mechanics, because we have yet to establish probability patterns for, say, what the spin of a photon is, we have made it some mystical thing that we can only know by measuring it. I don't really buy that; is the result of a lottery impossible to know until...

Imagine a gambler playing a casino game with fixed bet, fixed odds, no skill, and a starting bankroll, ##M_0##. She plays until she can no longer afford to bet and records only how many bets she was able to make, ##N_0##, until she could not afford to bet. Each day she goes back to the casino...

##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...

Can u help me with this question pls Assume that a gambler plays a fair game where he can win or lose 1 dollar in each round . His initial stock is 200 dollar. He decides a priory to stop gambling at the moment when he either has 500 dollars or 0 dollars in his stock. Time is counted by the...

https://ibb.co/guBuPd As the title indicates, I want to calculate the Probability of a stock price reaching a determined point, by considering the system as a random walk model, and after that, to compute the so called "maximal curves". I found the whole explanation in this article...

Hi there, I am currently trying to understand the theoretical frame work of diffusive shock acceleration. I am having trouble understanding a step in the derivation given by drury 1983 (http://www.oa.uj.edu.pl/user/mio/Ast-Wys-En/Literatura/drury.pdf). In the derivation of eq. 2.47 it is stated...

If we have a series of, say, twenty coin tosses, then each discernable specific series of outcomes has equal probability to occur. However, there is only one discernable specific series consisting of twenty 1's, while there are many more discernable series consisting of ten 1's and ten 0's. So...

I am reading Griffiths' Introduction to Quantum Mechanics, specifically the chapter on scattering. He is discussing the scenario where an incoming beam of particles scatter off an azimuthally symmetric target. At large separation ##r## from the scattering centre, the wavefunction for incoming...

Hey, so I've got this problem that I'm trying to figure out. I've worked out something that I think is probably right through simulation, but I'm not really sure how to tackle it from a purely mathematical probability perspective. So, would anyone know how I should approach this? I've tried a...

Homework Statement A particle with mass m is moving on the x-axis and is described by ## \psi_b = \sqrt{b} \cdot e^{-b |x|}## Find the probability distribution for the particles momentum Homework Equations ## \Phi (p)= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \Psi(x,0) \cdot e^{-ipx} dx##...

In the first Volume of his lectures (cap. 6 first Paragraph), Feynman cites Maxwell : "The true logic of this world is in the calculus of probabilities". Considering the formal and rigorous definition of probability, very often misunderstood by not-scientists, what do you think is the deep...

Homework Statement The frequency of a tsunami large enough to threaten the safety of a nuclear power plant has been estimated to be 1 in 200 years. If the plant’s life is 65 years what is the probability it will be damaged by such a tsunami? What is this probability more commonly termed? [2...

Homework Statement Hello! I'm trying to understand how to solve the following type of problems. 1) Random variables x and y are independent and uniformly distributed on the interval [0; a]. Find probability density function of a random variable z=x-y. 2) Exponentially distributed (p=exp(-x)...

Hi all, I am completely new to this forum. So allow me to introduce myself. I am currently paving my career as a mathematician, particularly in the field of probability theory and financial mathematics. I am currently pursuing a PhD in this subject and could not help but notice how closely...

A point is chosen at random inside a circle.Find the probability 'p' that the point chosen is closer to the center of the circle than to its radius. This comes from the noncountable uniform spaces sections.

Hello! (Wave) Suppose that $X$ has the uniform distribution on the interval $[0,2]$ and $Y$ has the uniform distribution on the interval $[2,4]$. If $X,Y$ are independent, I want to find the probability that the difference $Y-X$ is $\leq 1$. I have thought the following.The density function of...

Homework Statement Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A...

Homework Statement Why is my method not getting the correct answer? What am I missing? Homework EquationsThe Attempt at a Solution I am trying to find the probabilities of different hands in a 5-card poker (Texas hold' em). While the document below shows the answer, I like to use a method...

"Why 2 equally-likely events has each a probability of 0.5 ?" If i explain this saying that is due to the frequences as N goes to infinity, I'm saying a tautology cause it's implied in the definition of probability. So, going to a deeper level, why the probabilities of each of 2...

Homework Statement Pedestrian are arriving to a signal for crossing road with an arrival rate of ##\lambda## arrivals per minute. Whenever the first Pedestrian arrives at signal, he exactly waits for time ##T##, thus we say the first Pedestrian arrives at time ##0##. When time reaches ##T##...