probability Definition and Topics - 299 Discussions
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.
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 werent that great. So I want to relearn this with a more intiutive...
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##...
Does an ensemble of measurements yielding outcome A or B yield an approximation to the probability of A and B, or is such an ensemble of measurements something totally different from probability?