Probability Definition and 1000 Threads

Homework Statement Homework EquationsThe Attempt at a Solution [/B] The % of course passing students who fail the prelim test is 38%. So, the required probability is 38 %. Isn't the probability of a student failing the prelim when he his passing the course is known already = probability...

Hey! :o Let $P$ be a probability measure on a $\sigma$-Algebra $\mathcal{A}$. I want to prove or disprove the following statements: $P(A\mid B)=1-P(\overline{A}\mid B)$, for $A, B\in \mathcal{A}$ $P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$ I have done the following...

Homework Statement How to calculate the probability of finding an 1s electron within 1 picometer cubic region located 50pm from the nucleus. Homework Equations The probability of an 1s electron within a spherical volume of radius 'a' from nucleus can be find using the expression...

I need help with this Consider an irreducible Markov chain with $\left|S\right|<\infty $ and transition function $p$. Suppose that $p\left(x,x\right)=0,\ x\in S$ and that the chain has a stationary distribution $\pi .$ Let $p_x,x\ \in S,$ such that $0<\ p_x<1$ and $Q\left(x,y\right),\ x\in...

Hey! :o I want to prove or disprove that for a probability measure $Q$ over a $\sigma$-algebra $\mathcal{B}$ with $A,B\in \mathcal{B}$ the following hold: $Q(A\cup B)=1-Q(\overline{A}\cup \overline{B})$ $1-Q(\overline{A}\cap \overline{B})=Q(\overline{A})+Q(\overline{B})+Q(\overline{A}\cup...

Hey! :o I am looking the following exercise: At a tango course $12$ married couples participate. Find an appropriate measurable space and calculate the probability that exactly nine couples dance together. $5$ women sit at a round table. $3$ men come later. With how many ways can the men...

Homework Statement I am supposed to find probability of staying in x < 0 for a superposition of two Gaussians. The wavefunction is something along the lines of: Homework EquationsThe Attempt at a Solution Usually, the step involved in finding probabilities for 1 particle is just to perform...

Hello, I'm taking a course in Probability for which our only resource is our professor's notes but I find them somewhat hard to follow. He told us he wrote these notes because he could not find a book which contained all the topics we'll be doing, but I still hope someone here can help me find...

Homework Statement X is a Poisson Random Variable with rate of 1 per hour, following the Poisson arrival process a. Find the probability of no arrivals during a 10 hour interval b. Find the probability of X > 10 arrivals in 2 hours c. Find the average interarrival time. d. For an interval of 2...

Homework Statement Imagine a number line -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 A pendulum begins at -1 and then swings towards the LEFT or RIGHT. There is no knowing how fast the pendulum is traveling or how much energy it has but will only swing LEFT or RIGHT. (I) Need to work out the...

Homework Statement X is a geometric random variable with p = 0.1. Find: ##a. F_X(5)## ##b. Pr(5 < X \leq 11)## ##c. Pr(X=7|5<X\leq11)## ##d. E(X|3<X\leq11)## ##e. E(X^2|3<X\leq11)## ##f. Var(X|3<X\leq11)##Homework EquationsThe Attempt at a Solution Can someone check my work and help me? a...

Hey! :o Let $M$ be a measure space and $(a_i)_{i\in \mathbb{N}}\subset M$. I want to show that for positive $p_1, \ldots , p_n$ with $\displaystyle{\sum_{i=1}^np_i=1}$ by $\displaystyle{Q=\sum_{i=1}^np_i\delta_{a_i}}$ a probability distribution is defined. Do we have to show that...

Homework Statement For a 1.0 × 10-26 g particle in a box whose ends are at x = 0 and x = 2.000 Å, calculate the probability that the particle's x coordinate is between 1.6000 and 1.6001 Å if n=1 Homework Equations The Attempt at a Solution I know that since the interval between 1.6000 and...

Hi everyone; A very stupid confusion here. When we want to talk about the most probable radius to find the electron in $1s$ orbital, why do we talk about the radial density and not the probability itself? For instance, the probability of finding the the electron at a radial distance $r$...

I thought of this problem: The roads AB and CD have block E in common. There are 6 blocks in road AB and 13 blocks in road CD. Someone has planted a mine in some of the blocks. He gives us this information: 1. There's only one mine in road AB in one of its blocks. 2. There's only one mine...

Homework Statement Bayes' risk is ##L^*=0## for a classification problem. ##g_n(x)## is a classification rule (plug-in) such that ##g_n=0## is ##\eta_n(x)\leq 1/2## and ##g_n=1##$ otherwise. The function ##\eta##is given by ##\eta(x)=\mathbb{E}(Y|X=x)##. Then ##\mathbb{P}(g_n(X)\neq Y)\leq...

Homework Statement Assume that in a traffic junction, the cycle of the traffic signal lights is 2 minutes of green (vehicle does not stop) and 3 minutes of red (vehicle stops). Consider that the arrival time of vehicles at the junction is uniformly distributed over 5 minute cycle. The expected...

I scored in the 88th percentile in a certain personality trait and am trying to figure out the probability of that given that I'm male. I'm trying the likelihood that I would land in the 88th percentile given that I'm male. Definitions: T = trait, M = males, F = female. Given: P(T|M) = 0.3...

I'll start off by saying this is technically a coursework question, just not mine, nor of the present. I work in a tutoring center and it's been a while since I have taken stats and don't quite understand this person's professor's reasoning. The general idea of the question was that there was a...

https://www.quora.com/How-can-I-find-the-probability-density-function-of-a-continuous-random-variable-in-a-given-problem/answer/Maxime-Denis-2 How can I find the probability density function of a continuous random variable in a given problem? A mass m swings at the end of a rope (of length L)...

In classical mechanics and EM, the energy carried by a wave is the amplitude squared. In QM the (complex) amplitude squared of the position-space wave function is the position probability density. Do physicists regard this as anything more than just an interesting coincidence? Has anybody...

Homework Statement I am in a lab course studying Brownian motion. I have gathered data for for movement in two dimensions. I currently have a fairly large data set for ∆x and ∆y and have taken the mean and standard deviation of each. The lab asks what is the probability that these two sets of...

Homework Statement Given X_1,\dots,X_n a simple random sample with normal variables (\mu, \sigma^2). We assume \mu is known but \sigma^2 is unknown. The hypothesis is \begin{cases} H_0: & \mu=\mu_0 \\ H_1: & \mu=\mu_1 > \mu_0 \end{cases} Determine the rejection region R...

Hi all This is not a homework question but something work related which I am having difficulty understanding which I was hoping someone from the community could help me with. I am trying to understand how to interpret & create the probability density function plot from a set of data. For...

Homework Statement Let X, Y, and Z have the joint probability density function f(x,y,z) = kxy2z for 0 < x, y < 1, and 0 < z < 2 (it is defined to be 0 elsewhere). Find k. Homework Equations Not sure how to type this in bbcode but: Integrate f(x,y,z) = kxy2z over the ranges of x (zero to...

Homework Statement If event A equals event B, then the probability of their intersection is 1. True or False? Apparently the correct answer is False. The Attempt at a Solution If A=B then they should overlap entirely and their intersection should be 1? The only way I see this working is if...

If I consider a system of only two coins, then there are four microstates, each equally probable. But the probability that the system will have one head and one tail is the most. Describing the system by its all possible configurations is describing it in terms of its microstates. Here, the...

Homework Statement There are 22 students in a class. The professor will divide the class into 4 groups. Group 1 and 2 have 5 members each whilst Group 3 and 4 have 6. Given that the teacher forms the group at random, find the probabilities of : A = event where Paula, Trina, Gia all belong in...

Homework Statement An electron is trapped in a finite potential well that is deep enough to allow the electron to exist in a state with n=4. How many points of (a) zero probability and (b) maximum probability does its matter wave have within the well? Homework Equations For infinite potential...

IF the probability for an event to happen in one try is m/n , what is the probability for the event to happen at least one time in n successive tries. In each try the probability of event is m/n. I care mainly for the formula that I suppose it involves m and n but if you kind enough to provide...

Homework Statement For the hydrogen atom in the first excited state, find the probability of finding the electron in a classically forbidden region. Homework Equations E1 = -Z2μe'4/2n2ħ2 ψ200 = ∫ψ(r)2dr from the limit to ∞ The Attempt at a Solution https://imgur.com/a/fWIMq I feel like this...

Homework Statement https://imgur.com/a/8deZc Homework Equations P(E) = ∫φ*(r)ψ(r)dr from -∞ to ∞ The Attempt at a Solution To find the probability, I know I have to use this equation: P(E) = ∫φ*(r)ψ(r)dr from -∞ to ∞ My question is, what is the energy eigenstate, φ*(r)? Is it the measured...

Homework Statement According to a survey, of those employees living more than 2 miles from work , 90% travel to work by car . Of the remaining employees, only 50% travel to work by car . It's known that 75% of employees live more than 2 miles from work . Find the probability of that an...

I know that a high frequency light beam is more likely to generate a virtual electron-positron pair than a low frequency one. Can this probability depend on the reference frame? It seems there is a paradox. How do we explain it?

Hey! :o Two usual six-sided dice are thrown. We are given the probability space $\Omega:=\{(1,1),(1,2),..., (1,6), (2,1),..., (6,1)\}$ and the probability function $ P:Pot (\Omega)\rightarrow [0,1], \ P (\{(a,b)\})=\frac{1}{36}$. Calculate for the following random variable $X_i:\Omega->R$ the...

Is it correct to assume that all known particles may be created as virtual particles in the vacuum? If so, is there a higher probability of a particular particle being produced than say some other particle type. For example, is an electron more likely to be created as a virtual particle than a...

I apologize for the simplicity of the question (NOT homework). This is a statistical question (not necessarily a quantum mechanical one). If I have an initial probability function with an associated expected value and then a second probability function is superimposed on the initial...

Homework Statement consider 8 packs of bulb and let x be the number of bulbs in a pack that 'fail' the first time they are used . If 0.02 of all bulbs of this type fail on their first use and each 8-pack is consider random sample , what is the probability that anyone 8-pack has no bulb fail on...

Homework Statement Car colour preferences changes over year . In this year , suppose that 10% of the car are randomly selected , let the sample of cars are 20 . Find the probabilities between thre and five cars ( inclusive ) are black ... I am aksed to do this question using binomial ...

Since it's a PDF, that means the entire area under the curve must be 1, so $\displaystyle \begin{align*} \int_0^1{ a \left( x^2 + b \right) \,\mathrm{d}x } &= 1 \\ a \left[ \frac{x^3}{3} + b\,x \right] _0^1 &= 1 \\ a \left[ \left( \frac{1^3}{3} + b\cdot 1 \right) - \left( \frac{0^3}{3} + b...

Homework Statement Hi, I have this question that I've been pondering for a while, I keep flipflopping on what I think is right. I only need help on the last part on whether the events are independent or not, the rest of the text is backstory to the question. I know for events to be independent...

Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...

If you know the velocity of an object as a function of position Can you use a uniform distribution over one period and the object velocity to perform a change of variables for the positional probability. Example. X(t)=Asin(wt) V(t)=Awcos(wt) V(X)=+-Aw(1-(X/A)^2)^(1/2) P(t)=1/T T=Period Change...

There are 20 people and we are forming 2 teams of 10 people. 3 of the people (3/20) are women, 17 (17/20) are men. Names are drawn from random, in alternating order for teams. What is the probability that all 3 women end up on the same team? Initially I thought it was 25%. 100% probability for...

Homework Statement A jar has 5 marbles, 1 of each of the colors red, white, blue, green and yellow. If 4 marbles are removed from the jar, what is the probability (i) that the yellow one was removed? (i) that the yellow one was not removed? Homework Equations probability = favorable...

I understand that the concept of a atom resembling a solar system is disregarded as being untrue because this is based on the Bohr model which doesn't represent how an electron would actually appear in its probability cloud. However, would it be possible that if there were an observer on the...

Jesus had 12 disciples. What's the probability of selecting John and then James? What is the set up?

Gilligan has 16 green marbles, 50 blue marbles and 60 red marbles. What is the probability of randomly selecting 2 green marbles? What is the set up?

The probability density of the dirac spinor is known to be ∑(Ψ)2 and I know how it is derived. However, I'm just wondering why it should be positive definite. Since the lower two components represent antiparticles, so shouldn't the probability density contribution of those two components be...

Suppose set A is defined as the even integers and set B is defined as for every even integer there are two odd integers, like so: {2,3,3,4,5,5,6,7,7 ... } Can you calculate that the probability of choosing an odd number is 66%?