Probability Definition and 1000 Threads

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)\...

I try to list all the possible sequences: 1 2 3 1 3 5 1 4 7 2 3 4 2 4 6 2 5 8 3 4 5 3 5 7 4 5 6 4 6 8 5 6 7 6 7 8 I get 12 possible outcomes, so the probability is ##\frac{12 \times 3!}{8^3}=\frac{9}{64}## But the answer key is ##\frac{5}{32}## . Where is my mistake? Thanks

Not really a homework question. I was reading a book on card tricks and it said that it's almost certain that in a shuffled deck of cards, there will be at least two consecutive cards of the same value. I just wanted to know the actual probability of that. So, here's my question: in a standard...

Problem: A system contains two identical spinless particles. The one particle states are spanned by an orthonormal system ##|\phi_k>##. Suppose that particle states are ##|\phi_i>## and ##|\phi_j>## (##i \neq j##). (a) Find the probability of finding the particle in the state ##|\xi>## and...

Hi, Each page of a book contains N symbols, possibly misprints. The book contains n =500 pages and r =50 misprints. Show that (a) the probability that pages number 1, 2, . . . , n contain, respectively , $r_1, r_2 , . . . , r_n $ misprints equals $$\frac{\binom{N}{r_1}\binom{N}{r_2}. ...

ln Will Kurt's Kurt's book "Bayesian Statistics The Fun Way" he gives a problem at the end of a chapter " Raw eggs have a 1/20,000 probability of having salmonella. If you eat two raw eggs what is the probability that you ate a raw egg with salmonella." The online answer he gives: "For this...

For this problem, Is it as simple as using the probability density function, P = Ψ2 and plugging in the radius value given to me? So essentially I am just squaring the wave function and plugging in?

Find the solution here; Ok my interest is on part (b) and (c) only. Let's start with (b), My take is, $$\int_4^5 \dfrac{2}{75}x\,dx=\left.[\frac{x^2}{75}]\right|_4^5$$ $$=(0.33333333-0.21333333)+\frac{2}{15}×5$$ $$=0.12+0.6666666666=0.78666666$$ note that at ##f(x)##=##\dfrac{2}{15}##, the...

if I select two integers at random between 1 and 1,000, what is the probability that their sum will be prime?

Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix P given below: Now, I have already figured out the solutions for parts a,b and c. However, I don't know how to go about solving part d? I mean the question says we can't use higher powers of matrices to justify our...

I just want to be certain, i think the inequality indicated is not correct...ought to be less than. Kindly confirm...This is a textbook literature.

I calculated the probability of box 1/2/3 given white. P(Box 1 | white) = P(Box 1)*P(White | Box 1)/(P(Box 1)*P(White | Box 1) + P(Not Box 1)*P(White | Not Box 1)) = ((1/3)*(1/2)) / (((1/3)*(1/2) + (2/3)*(5/7) = 7/27 P(Box 2 | white) = P(Box 2)*P(White | Box 2)/(P(Box 2)*P(White | Box 2) +...

Find the question and solution here ( sorry its a bit blurred) ... given using the hypergeometric method...i wanted to understand what is the clear distinction between hypergeometric and binomial distribution? Could it be in in reference to no replacement and replacement...? My approach on this...

Problem: In a dresser there are 3 drawers. In one drawer there are two black socks and one white sock, in the second drawer there are two white socks, and in the third drawer there is a black and white sock. Suppose I chose a drawer randomly ( meaning, in a uniform distribution ) and I took a...

I have information that $$\rho_{ab}=\sum_{j}p_{j}\ket{\Psi_{j}^{ab}}\bra{\Psi_{j}^{ab}}$$ and $$Pr(o_{j}^{(a)}|\Psi_{ab})=Tr_{ab}(\ket{\Psi_{ab}}\bra{\Psi_{ab}}(\ket{o_{j}^{(a)}}\bra{o_{j}^{(a)}}\otimes \mathbb{I}_{2})) \text{.}$$ I started by representing the density operator for pure states...

My attempt: $$P(\text{B is positive}|\text{A is positive})=\frac{P(\text{B is positive} \cap \text{A is positive})}{P(\text{A is positive})}$$ $$=\frac{P(\text{B is positive})\times P(\text{A is positive})}{P(\text{A is positive})}$$ $$=P(\text{B is positive})$$ $$=0.01 \times 0.99 + 0.99 \times...

Reif says " ... variable ##u## which can assume any value in the continuous range ##a_{1}<u<a_{2}##. To give a probability description of such a situation, one can focus attention on any infinitesimal range of the variable between ##u## and ##u+d u## and ask for the probability that the variable...

I am trying to determine the likelihood of a driver winning a race based on an associated rating as well as the team he drives for. The probability that Driver A beats Driver B = .8504 The probability that Team A beats Team B = .7576 How do I combine these two probabilities, where the outcome...

Please tell me some good books on Probability for first course.

this is the question Here is a tutorial video but his steps are very confusing to me. I personally know bayes theorem and have already studied probability and got good marks in it(It may not be a metric for being quality in it given that it is nepal we are talking about.)...

References (1) https://www.physicsforums.com/threads/what-is-the-probability-that-the-universe-is-absolutely-flat.971984/ (2) https://www.physicsforums.com/threads/calculating-the-probability-that-the-universe-is-finite.1011826/ Suppose the Friedmann equation is used to analyze two models. (1)...

Hello, I have a question about the following sentence and would appreciate if someone could explain how to read out the conditional probability here. "Each microwave produced at factory A is defective with probability 0.05". I understand the sentence as the intersection ##P(Defect \cap...

I calculated the mean which is 78.4 And the Standard deviation is 5.6 I thought the answer would be (90^(-78.4)/78.4!)*e^-90 But looking back having a decimal factorial doesn't make sense I have the numerical answers for c)= 0.019226 and d)=0.022750 but I my solution was wrong. Any help on...

Reference https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L06.pdf I note that the use of Gaussian probabilities is mentioned many times in the reference. However in many discussions via posts in many threads, there seems to be a consensus that the distribution is...

The 35 member History Club is meeting to choose a student government representative. \item The members decide that the representative, who will be chosen at random, CANNOT be any of the 3 officers of the club. What is the probability that Hiroko, who is a member of the club but NOT an officer...

In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows: "For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...

I really cannot ask this question well. I can only hope its not simply a waste of the readers time. I won't finish every sentance with "maybe I'm wrong", just assume its in my mind every time I hit the period key. An electron on a screen leaves a pixel spot, this pixel spot is a measurement...

Hello, I am trying to figure out the right way to approach this. First of all, other than the different Bohr radius value, does the change to a negative pion make any other difference to calculating the probability? Also what would be the correct way to apply the "small volume"? What I'm...

I am desperate. I've scoured the web for the formula for the probability density function for the interference pattern obtained in the double slit experiment with both slits open. So I want to know the probability density function and not the intensity function. I prefer not to have references...

Hi, i was doing a programming exercise that asked me to simulate te flip of coins until it finds 10 consecutive tails. The program usually needs to flips like 6000/8000 coins before finding 10 tails consecutively, but suddenly i found 10 tails with only 30 coin flips, i think that what happened...

Summary:: Bag A contains 1 white straw, 2 red straws and 2 green straws. Bag B contains 2 white straws, 2 red straws and 1 green straw. One straw is drawn at random from each bag. Find the probabilities that (a) the two straws drawn are of the same colour; (b) one straw is red and the other...

Suppose we've an isolated box having ##N## classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical. Its said that the probability of having the configuration of ##n## particles in the left side is given as...

Would the two masks together be 126% effective?

Hey! :giggle: $N\in \mathbb{N}$ christmas presents will be distributed. In every gift there is additional either a chocolate snowman or a chocolate reindeer. It will be independent of each other and with the same probability of giving out gifts with a chocolate snowman or chocolate reindeer...

we haven't learned howw to do parts d-f yest. could you please give me a hand? (a) $$E(X)=E(Y)=\int_0^3\int_0^3\frac{1}{9}dxdy=\frac{3}{2}$$ (b)its typoe suppoesed to be W=Y-2 $$E(Z)=E(X-2)=E(X)-E(2)=-\frac{1}{2}$$ $$E(W)=E(Y-2)=E(Y)-E(2)=-\frac{1}{2}$$ (c) i guess joint pdf from ##E(Z)## and...

Two numbers are bring chosen by random from the set {1,2,3,4,5}. If the sum of the two numbers is even, you win 100 dollars, otherwise you win nothing. In order to participate in the game, you pay $80. What is the expected value and variance of the profit after 17 games ? I solved this one...

There is a box with 2324784 bullets. Balls are divided into 5 groop and there is a definition for each groop and how many balls and what is the probability of getting each groop. The game with the return of the balls List the probability and quantity groop 1 906192 0.389796213 groop 2 1006880...

The probability that two randomly chosen integers to be coprimes is known to be equal to ## \prod_{2}^{\infty}(1-\frac{1}{p^2})=6/\pi^2## I tried to conceptualize the problem in the following way but got different results. Suppose that we pick up an integer at random which could be either prime...

(a) The probability mass function of X and Y: $$p_{X,Y}(1,1)= \frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}$$ $$p_{X,Y}(1,0)=\frac{1}{2}\cdot \frac{1}{4}=\frac{1}{8}$$ $$p_{X,Y}(0,1)= \frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}$$ $$p_{X,Y}(0,0)= \frac{1}{2}\cdot \frac{1}{4}=\frac{1}{8}$$ or $$...

I'm a freelance web developer and right now I'm stuck into a mathematical(probability) issue where I'm developing an answer filtering tool. This tool works as follows: 1) There are six questions that the user will be asked 2) Based on the answer of first question, my server will decide which...

Hi guys, I have a maths question about Probability, hope somebody can help me out, thank you very much. There are 5 red, 4 blue and 3 green balls in a bag. Two balls are taken from the bag, one after the other. If the first ball is not replaced, and it was not green. What is the probability...