Probability Definition and 1000 Threads

I am fairly new to statistics and probability, I'm currently studying for a career change. I was asked this question but I don't know how to navigate through and get to any kind of an answer. Previously, the government conducted a lottery to award visas to 20,000 advanced-degree holders first...

Hi all, I was wondering if there exist any theorems that allow one to relate any joint distribution to its marginals in the form of an inequality, whether or not ##X,Y## are independent. For example, is it possible to make a general statement like this? $$f_{XY}(x,y) \geq f_X (x) f_Y(y)$$...

Homework Statement A computer program is tested by 5 independent tests. If there is an error, these tests will discover it with probabilities 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. Suppose that the program contains an error. What is the probability that it will be found by at least two...

Homework Statement A man owns two old cars, A and B, and has trouble starting them on cold mornings. The probability both will start is 0.1; the probability B starts and A does not is 0.1; the probability that neither starts is 0.4 a) Find the probability that car A will start. b) Find the...

Homework Statement I was reading this example in a book... " Two competing software companies are after an important contract. Company A is twice as likely to win this competition as company B. Hence, the probability to win the contract equals 2/3 for A and 1/3 for B." Homework EquationsThe...

I have only recently started studying information theory and I am stuck even trying to understand the basic problem and concept of information theory. My question is why does information need a probability? I have read some text that starts by introducing the idea that the amount of information...

If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation. Hint:The joint density function can be obtained by differentiating the...

My attempt I used negative binomial to solve the problem, however I'm left with a polynomial that is difficult to solve? Is there any other way to approach this problem? I used the inequality because I'm trying to find the range of p. Since the probability of winning the series for team...

##E = \frac{1}{2}(kx^2+m \dot{x}^2)## ## \frac{2E - kx^2}{m}=\dot{x}^2## ##\frac{dx}{dt} = \sqrt{\frac{2E - kx^2}{m}}## or ## dt = \sqrt{\frac{m}{2E - kx^2}}dx ## ⇒##= \frac{1}{\sqrt{\frac{2E - kx^2}{m}}}dx## My Question please help me. 1. I know ##T = 2\pi\sqrt{\frac{m}{k}} .## but i don't...

I would like to know the difference between the ##\chi^{2}## distribution and the PTE (Probability-To-Exceed) ? I must compare 2 data sets A and B and in the article I am reading, they talk about this PTE : For the moment, I only know the ##\chi^{2}## distribution with ##k=2## degrees of...

Hey There! I've finished reading The Introduction to Probability and Statistics book. Now I'm looking for another book as awesome as that was. Can anyone suggest me a book? Thanks in advance.

Hi. What is the probability of throwing three dice at the same time while we are getting at least one of the particular number? I mean, for example, the probability of getting at least one number 2 among the three dice thrown Is it 1/6+1/6+1/6 = 3/6 = 1/2 ? How to explain to my daughter that...

Homework Statement (i) Construct the cumulative distribution table for the number of heads when the four coins are tossed. Coins are fair. (ii) Find the Median.Homework EquationsThe Attempt at a Solution (i) x 0 1 2 3...

I Dont understand how light polarized along a certain axis in a prepared state can then pass through a second polarizer that is at an angle to the first. The experiment I’m trying to understand starts with unpolarized light which then passes through a first filter which vertically polarizes the...

If I have ten dice and pick one at random, given that I don't know anything about them the probability of getting a dice of value 4 would be 1/6? Given that I know that there is two dice of value 4, the probability of getting a 4 would be 2/10 = 1/5? Which would be correct in repeated trials?

What is the best procedure to determine Lambda to calculate the Poisson probability? Say I want to calculate P(X ≥1) of an accident occurring next day. For this I would calculate the average of daily accidents and divide it by 10. The question is, should I take the previous 10 days? Or calculate...

Hi :) Here's my problem along with what I've done. Here is the problem: That is the p.d.f. of a random variable X. I have to find the cdf. I don't know which I should do so I tried it two ways. First: $\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...

Homework Statement [/B] Given a group of 100 married couples, let X1 be the number of sons and X2 the number of daughters the couple has. P(X1 = 0, X2 = 2) = f(0, 2) = 8 /100 = 0.08 2. Homework Equations The Attempt at a Solution I tried to look for a similar example online, I found this...

Hello All, I have the following question in one of my tutorials. I need some help in resolving this. Background: A manufacturing company developed 40000 new drugs and they need to be tested. Question The QA checks on the previous batches of drugs found that — it is 4 times more likely that a...

Let there be a sphere whose inner surface consists of a perfectly reflecting surface. It has a hole on it which allows a ray of light to enter. Give the angle made with the normal of the hole when the ray of light enters such that the probability that the ray comes out is the least?Assuming the...

Homework Statement In a short time dt, the probability that a car passes an observer is dt/τ. In a random time interval of length T, what is the probability P(n) that exactly n cars pass the observer? The Attempt at a Solution I can only see how to find P(0) the probability that no cars pass...

A coin had tossed three times. Let ##X##- number of tails and ##Y##- number of heads. Find the expected value and variance ##Z=XY##. My solution: We know, that ##Y=3-X##, so ##Z=(3-X)X## for ##X=0,1,2,3##. ##Z=2## for ##X=1,2## and ##Z=0## for ##X=3,0## So, ##E(Z)=E((3-X)X))= 2 \cdot ⅜ +2 \cdot...

Homework Statement The following pedigree shows that the trait is autosomal recessive. The probability of carriers in the population is 1/3. What is the probability that the child of III-3 and 4 is affected? 2. Relevant info The answer provided is 1/36. The Attempt at a Solution The...

For a somwehat simple version of the problem, imagine a group of ten people (n=10), for which we observe two binary attributes (d= 2). One of the attributes tells us if any given person wears glasses (d_1) the other if any given person wears a hat (d_2). We know that in total 5 people wear hats...

a class has 9 students who are to be assigned seating by lot. What is the probability that students will be arranged in order from shortest to tallest? (assume that no two stusents are the same height.)

Hi, Say L is a human language (e.g. German, Chinese, etc.) and w is a string in L of length n>1. Is it known for different languages what the probability is that w is a word in L? And if S is an ordered set of strings, the probability that S is grammatically correct in L? I mean, I know or have...

Homework Statement Let X be a random variable. It is not specified if it is continuous or discrete. Let g(x) alway positive and strictly increasing. Deduce this inequality: $$P(X\geqslant x) \leqslant \frac{Eg(X)}{g(x)} \: $$ where x is real. Homework Equations I know that if X is discrete...

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a small system that has 2 normal modes of vibration, with natural frequencies $$\omega_1$$ and...

Studying probabilistic density, I know that a function that is integrated between two limits presents a probability. But how should I think to solve a problem where I need to determine the probability of a particle being seen being that its moment liner is a constant value

Calculation of probability with arithmetic mean of random variables There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the probability of the event that...

Homework Statement Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and $$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$ Prove also the ##10## in the inequality can't be replaced...

Hi All, I would like to know who was the first scientist to use probability amplitudes in solving either math or physics problems. Best wishes, DaTario

Hello All, Firstly, what a wonderful place for people to help one and other. I am glad that I have stumbled across : ) I have posted this query in advanced because to me there it is advanced but if it is basic to others then please excuse me. I can easily work out the probability of...

The probability P of k consecutive tails occurring in n coin tosses is 1 - (1 / F) where F is element n+2 in the k-step Fibonacci series divided by 2n. As n approaches infinity, P approaches 1 for any value of k. Is the above statement true? EDIT: sorry, mistyped heading question, can't edit...

A tank contains 50 balls. 10 are frosted (F) and labeled (L), 23 are mild (M) and unlabeled, 7 are frosted and not labelled (N), 10 are mild and labeled. 1. A ball is randomly selected from the tank and it was frosted. What is the probability that the ball was labeled? 2. The 1st ball was...

Hi, I'm sorry if this question has already been answered somewhere and I'm just too incompetent to find it, buuut: As the title already says, I really do not get that part of quantum physics (if you can even say I'm getting ANY part at all...). As I searched all Google for an answer I just...

In a homicide case, 7 different witnesses picked the same man from a lineup. The line up contained 5 men. If the identification were made by random guesses, find the probability that all 7 witnesses would pick the same person.

A study conducted at a certain college shows that 55% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that among 7 randomly selected graduates, at least one finds a job in his or her chosen field within a year of graduating

A IRS auditor randomly selects 3 tax returns from 45 returns of which 15 contain errors. What is the probability that she selects none of those containing errors? Round to four decimal places.

If we have two sequences s1 and s2, both of N coin tosses, is the entropy of getting two sequences that are exactly the same then lower than sequences of which can be said that they differ by x incident tosses? Is the entropy of getting sequences s1 and s2 that differ by N/2 tosses the highest...

Homework Statement (b) A nuclear research reactor produces radiation for neutron scattering measurements. A safety procedure shuts the reactor down if a radiation level monitoring detector measures more than 3 counts per minute. In a test, 156 counts are recorded during a random 24 hour...

Homework Statement Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##. a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##. b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...

Homework Statement We have a normal 6 sided dice marked from 1 to 6. There is an equal chance to get each number at every roll. Let's put 1&2 as A type, 3&4 as B type and 5&6 as C type. We roll the dice over and over until we get a number of every type. Let X be the number of rolls. We are...

Homework Statement "Mr. A and Mr. B fire at each other until one of them is hit. Each duelist has the same probability of hitting each other each time a shot is fired. Mr. A has probability ##a## of hitting, and Mr. B has probability ##b## of hitting. Given Mr. A shoots first, calculate the...

I chose the thread level for what I imagine to be the difficulty of this problem, not my own math ability. Obviously (?) there are 256^32 possible series. So knowing how (i.e., how to calculate) how many of these are in ascending order would give the answer. Is there some polynomial time algorithm?

Homework Statement If two identical dice are thrown, what is the probability that the total of the numbers is 10 or higher? [Hint: list the combinations that can give a total of 10 or higher.] [2] Two dice have been thrown, giving a total of at least 10. What is the probability that the throw...

In order to check if a quantum particle crosses a barrier or not, isn't calculating ##|\psi(x)|^2## enough in that particular region ? Why do we need to calculate probability current density for that matter?

Homework Statement An ice-cream store makes 150 ice-cream balls every day. The cost of making each ice-cream ball is $3. The price of an ice-cream ball is $8. The demand distribution is as follows: 100 ice-cream balls with probability 25%, 150 ice-cream balls with probability 50%, and 200...

Homework Statement f(x) = (3/4)(-x^2 + 6x - 8) for 2 < x < 4 (0 elsewhere) A) Find F(x) integral 2 to 4 ((3/4)(-x^2 + 6x - 8))dx B) Use F(x) to find P(3 < X < 3.5) integral 3 to 3.5 ((3/4)(-x^2 + 6x - 8))dx 11/32 C) Use F(x) to find P(X > 3.5) 1-( P(3 < X < 3.5)) = 21/31 D) Find E(X)...

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...