Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
Hi,
Given a spin in the state ##|z + \rangle##, i.e., pointing up along the z-axis what are the probabilities of measuring ##\pm \hbar/2## along ##\hat{n}##?
My problem is that I'm not sure to understand the statement. It seems like I have to find the probabilities of measuring an eigenvalue...
Let W be the event that a prize is received. Then ##p(W) + p(not ~ W) = 1##. We need to find ##p(not W)## and so let's try to find ##p(W)## and then we can subtract it from 1 to get ##p(not ~ W)##.
The experiment is buying 2 tickets. So, $$p(W) = \frac { {}^{10}C_2} { {}^{10000}C_2}$$
Thus...
I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up.
$$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$
$$ S = \{ (i,j) : i...
Hello everyone, I found this problem online about probability, for me, I think that to have a 2 letter palindrome is less likely because we need to have the same letter in the 2 places which gives us 26 possibilities (aa , bb, cc ....) however for words with 3 letters we have 26 possibilities...
We have three Random variable or vector A,B,C. Condition is A & B are independent as well as B & C are independent RVs . But A & C are the same random variable with same distribution . So How can determine E{ABC}. Can I write this E{ABC}= E{AE{B}C}?
Theorem: Let ## X ## be a random variable. Then ## \lim_{s \to \infty} P( |X| \geq s ) =0 ##
Proof from teacher assistant's notes: We'll show first that ## \lim_{s \to \infty} P( X \geq s ) =0 ## and ## \lim_{s \to \infty} P( X \leq -s ) =0 ##:
Let ## (s_n)_{n=1}^\infty ## be a...
When the expectation value of spin in the z direction for one particle is zero and I make measurements for an even number of particles in the same state, do I get exactly half to be spin up and half to be spin down along the z direction? More generally, what does spin expectation value for one...
If I want to get the spin angular momentum of a particle using the Stem-Gerlach experiment, I think I will find the spin 1/2 particle either spin up or spin down, but not both. I however want to ask this : Is there a non-zero probability that a particle which is spin-up in the z direction to be...
My attempt to answer this question: With the radii in the ratio ## 1: \frac12: \frac13 ##, the area of the corresponding circles will be in the ratio of ##1: \frac14: \frac19 ##. The areas of the three rings will be in the ratio of ## \frac34 : \frac{5}{36}: \frac19 ##
So, if three shots are...
Hello all, I would like to check my understanding and get some assistance with last part of the following question, please.
For part (d), would I use f(x | y) = f(x, y) / f(y) ?
Problem statement:
My attempt at a solution, not too confident in my set-up for part (d). I drew a sketch of the...
Hello all, I am wondering if my approach is coreect for the following probability question? I believe the joint PDF would be 1 given that the point is chosen from the unit square. To me, this question can be reduced down to finding the area of 1/4 of a circle with radius 1. Any help is appreciated!
My attempt for part (a) is as given below. I will attempt part (b) after getting part (a) correct.
(a) Based on what is asked, we can identify 3 independent events as follows: (i) select any 2 bags followed by (ii) select a ball from one bag followed by (iii) select a ball from the other bag...
I tried to solve this problem using the chart given below. But I get a different answer of ##\frac {2}{3}## rather than ##\frac {3}{4}##. Maybe the answer given is incorrect?
I determine from the chart the number of ways in which A could win given that A has already won 2 of first 3 points...
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...
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 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...
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...
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...
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...
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...
Hello!
I am looking for textbooks to relearn Combinatorics, Permutations Combinations and Probability and also Matrix algebra( decomposition, etc). I had done these many years ago and the course/books provided to me at that time weren't that great. So I want to relearn this with a more...
Suppose there are two persons A and B such that both have a personal communication system which can transmit and receive bits. B has a biased coin whose bias is not known. A asks B to toss the coin 2000 times, send a 0 when a tail comes up and a 1 when a head comes up. It is known that whatever...
In Oz Lotto, balls are numbered 1 to 45. Nine are selected, seven of which are winning numbers and two being supplementary numbers. Players select seven numbers.
The odds of winning can be found here: https://www.lottoland.com.au/magazine/oz-lotto-everything-there-is-to-know.html
I tried...
Hi,
I was looking at this problem and just having a go at it.
Question:
Let us randomly generate points ##(x,y)## on the circumference of a circle (two dimensions).
(a) What is ##\text{Var}(x)##?
(b) What if you randomly generate points on the surface of a sphere instead?
Attempt:
In terms of...
I want to learn some probability & statistics on my own. I am well versed in Calc 1-3 , elementary ODEs and very little linear algebra. I want a comprehensive , introductory textbook which is NOT COOKBOOK STYLE. I might be self studying AP statistics next term so if the book covers everything I...
Summary:: Year 11 Extension 1 Math problem (Australia)
How many combinations can be made from a 4 digit pin code if we can only use two numbers to form our pin code, and we MUST use 2 distinct numbers. E.g. 1112, 4334, 9944, 3232. But NOT 1111, 2113, 0992 etc. We're using the numbers 0-9 and...
I really don't know what to do for this problem. I looked at similar threads but couldn't seem to grasp the idea of it. I would like help on how to start.
Hey everybody, :smile:
I have a joint density of the random variables ##X## and ##Y## given and want to find out ##P(X+Y>1/2)##.
The joint density is as follows:
$$f_{XY}(x,y) = \begin{cases}\frac{1}{y}, &0<x<y,0<y<1 \\ 0, &else \end{cases}$$
To get a view of this I created a plot:
As...
Summary:: Hello there, I'm a mechanical engineer pursuing my graduate degree and I'm taking a class on machine learning. Coding is a skill of mine, but statistics is not... anyway, I have a homework problem on Bernoulli and Bayesian probabilities. I believe I've done the first few parts...
The summary says it all: why is the probability of an event not calculated by the probability that it is the event AND that it is not any other? Sounds silly, and I am certain the explanation is simple, but I don't see it.
In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...
Can someone help me on this question? I'm finding a very strange probability distribution.
Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
Can MWI account for the probabilities of outcomes? If MWI says all outcomes are realized, is the probability that an outcome occurs then not 100%? How is this explained with the entanglement of the measured object and the measurement apparatus?
Hi folks - I need some help with a tricky probability. Here's the situation:
Let's say there are 4M internet users in Age Group A. (The total set)
Of those 4M, there are 1,000 users who play a specific sport.
Those 1,000 are spread evenly over 125 teams, so 8 players each.
1. What's the...
I decided to take cases.For example-:A gets one 1 duck and B gets 2,3,4,...,51.So i can write this as
50C1(1/2)5051C2(1/2)51+50C1(1/2)5051C3(1/2)51+...
But i was unable to solve it further.
please help.
Here's the problem:
A chef made 500 cookies randomly mixed with 1000 nuts including 600 almonds and 400 hazelnuts in which each nut is the same size. Suppose the number of pieces of nuts in a piece of cookie follows a Poisson distribution.
(a) Suppose cookies are randomly selected one-by-one...
Here's what I think I understand:
First off, the GHZ state ##|GHZ \rangle = \frac {|000\rangle+|111\rangle} {\sqrt 2}##, and ##\sigma_x## and ##\sigma_y## are the usual Pauli matrices, so the four operators are easy to calculate in Matlab.
I'm thinking the expectation values of each operator...
I set hN(1)hN(1) equal to cNcN, but I'm confused on how I'd be able to solve it and because of that I was not able to conclude that 0 is recurrent when qx/px = infinity
In these lecture notes about statistical mechanics, page ##10##, we can see the graph below.
It represents the distribution (probability density function) of the kinetic energy ##E## (a random variable) of all the gas particles (i.e., ##E=\sum_{i}^{N} E_{i}##, where ##E_{i}## (also a random...
1. Definition
If E and F are two events associated with the same sample space of a random experment, the conditional probability of the event E given that F has occurred, i.e. P(E|F) is given by
P(E|F) = (E∩F)/P(F) (P≠0)
2. Properties of conditional probability
Let E and F be events of...
a) P(X<18) = (18-20)/sqrt25
=-2/5
=-0.4
then you use the standard normal table and find that;
P(X<18)=0.3446
b) P(X>27)
= (27-20)/5
= 7/5
= 1.4
P(Z>1.4)
=P(Z<-1.4)
=0.0808
C) =(13<X<23)
=13-20/5 , 23-20/5
=-7/5 , 3/5
=-1.4 , 0.6
P(Z<0.6)-P(Z<-1.4)
=0.7257-0.0808
=0.6449
Hello everyone.
Let us consider 3 events A,B,C such that: $$P((A \cap B )\cup C)=P(A)*P(B)*P(C)$$ Notice that the second term is a union and not an intersection. Are they independent? And what if the assumption was: $$P(A \cap( B \cup C))=P(A)*P(B)*P(C)$$? I know that the independence condition...
There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$.
I understand it in such a way: ##X## is independent, that's to say after there are ##(n+k-1)## successive failures, ##k## additional trials performed afterward won't be impacted, so these ##k##...