Probability Definition and 1000 Threads

What is the probability that a randomly chosen natural number, which is not smaller than a given natural number x, is greater than x? I think it must be ##1## because there are infinitely numbers that greater than ##x## but how can I proof this? Thank you.

does the sample space of this kind of sure event contain only one event?

Hello All, I would like to find out what the percentage is of what happened to us in a drawing - quite amazing and would like to know what the percentage is in doing this. My wife and I were looking into starting another business but it looked like we were getting roadblocked at every turn. My...

Suppose this expression : ((A ∨ ~B) ∨ (C -> D)) This simplifies to : A + ~B + ~C + D , using boolean algebra. Now suppose i know that this expression is true: p( ((A ∨ ~B) ∨ (C -> D)) ) = 1; and I also know that p(A) = 0.3 , p(B) = 0.99 , p(C) = 0.92 . but the probability of the expression is...

Hello. Is there a name for distribution that describes "broken" Galton box like this? And distribution formula.. Edit: I've also attached how it looks like in simulation. Thanks.

Hi PF, I'm really struggling with an intricate probability problem that I'm investingating just for fun and I don't know how to approach it correctly. So, there's this 1996 arcade car racing PC game called Death Rally that works like this: 20 racers are competing in a car racing championship...

Hello everyone, I’ve been trying to solve this probability problem, but got stuck. Here’s the problem and my attempt: Problem In the Squid Game marble challenge, a bag contains 10 red marbles, 8 blue marbles, and 7 green marbles. A player is allowed to draw two marbles without replacement. If...

I have two solutions, but I have no idea which solution correct. Both of them seem correct to me. Help please. First solution: There is an infinite number of natural numbers. Product of digits is odd only if all digits in number are odd. Counting natural numbers with all odd digits: 1-digit...

Hello! If I have a 2 level system with the energy difference between the levels ##\omega## and a photon source, producing ##n## photons per second, with a frequency ##\omega/3##. What is the probability of having a 3 photon excitation and how can I derive it?

Will the probability density of a wave function be affected by length contraction due to special relativity?

I thought that the expected utility is simply the utility of an outcome multiplied by the probability of that outcome. I thought about the following: set $$p:=\Pr(Z=1 | X=x) = \frac {e^{(x-\mu)e^{-\gamma}}}{1+e^{(x-\mu)e^{-\gamma}}}$$ and then...

I don't understand this accusation. Given a single system prepared in some state ##\psi = \sum c_i|i\rangle##, the probability ##\mathrm{tr} |\psi\rangle\langle\psi|i\rangle\langle i|##, according to Griffiths, is the probability that the system has the property ##i##. This is in contrast to...

What would be the sample space for this? Wouldn't we have to make groups of like (2,3,0,0,0)/(1,1,1,1,1)/etc I will cite the source below but they take the sample space as n(S)=##5^5## Isnt that wrong? We do not have 5 options in marbles for each of the baskets? i As we fill in, the number of...

I read a news article shouting for more regulations on trucks as a study revealed that compared to accidents involving cars, the death rates for accidents involving trucks was higher (among cyclists). I imagined this scenario (it squares with the news article mentioned): P(accident involving...

I don't have good references yet and am hoping you all can help me understand my pop-sci reading. I have read that quantum mechanics implies that there is a very small probability that I can fall right through my chair, or suddenly appear on the moon among other crazy things. I subscribe to the...

Below is a scenario where at every step, you are choosing the more favourable option, but yet you would end up worse off definitely. How could it be? Suppose you start playing a fair game (or a game slightly to your advantage) with $10 and bet $1 each round. You tell yourself that you would...

I made this question up so I have no guarantee that there is a clean answer. It seems like there should be a simple approach though, I’m just not seeing it. First attempt: Find the chance that only the first k boxes are left empty. Then we can multiply by ##{n \choose k}## to get the total...

The probability that a randomly chosen integer is divisible by a given integer p is 1/p, regardless of whether p is prime. The probability that 2 distinct randomly chosen integers are divisible by the same prime p is 1/p2. I am not sure however whether the probability that 2 distinct randomly...

A passcode having 2 fives implies that order does not matter (5 7 5 is the same passcode as it would be if we switched the 2 fives). There are 8 available digits in total, but 5 is being used twice, so we only have 7 options for the third number in the code. There are 3 possible cases for the...

So, I have these two seemingly wildly improbable events that I observed. I once saw my parents next door neighbor waiting for a subway in DC. We both live 100's of miles away, and we see each other waiting for this train while randomly sightseeing. To me it seems to be a tiny probability...

Okay, i was able to solve it by trial and error, i am seeking for a more concrete approach. Can combination work here? or a more solid approach using sequences? or probability itself? My trial and error, ##P_{green} = \dfrac{9}{12}×\dfrac{8}{11} ×\dfrac{7}{10}×\dfrac{6}{9}×\dfrac{3}{8} =...

When there is a probability involved with an energy state, e.g. with partition function, why is total energy the same as average energy (if it is). Just want to make sure - is this just a definition? Thanks

Say an archer has a probability p of hitting the target. Given n shots at the target, the number of hits = np The standard deviation of hits = ##\sqrt{np(1-p)}## Say p = 0.7 Given 100 shots, my expected/average number of hits = ##100 \times 0.7 = 70## The standard deviation for the number of...

For this, The solution is I have doubt where they got the reflection probability formula from. Someone may know how to find it. I think that ##R + T = 1##. But I'm not sure where the transmission probability formula comes from either. Kind wishes

X and Y are discrete random variables with the following joint distribution: a) Calculate the probability distribution, mean, and variance of Y. My attempt: I have calculated the probability for different values of Y and X using the following equation: ##\text{Pr(Y = y)}## = ##\sum_{i=1}^l##...

The gain in odds that aliens are visiting Earth (A) due to ##n## independent reports of close encounters (C) is given by: $$\frac{\rm Odds(A|C)}{\rm Odds(A)}=\left[\frac{\rm Prob(C|A)}{\rm Prob(C|\bar A)}\right]^n.$$ Let us assume that we have good cases such that an alien explanation (##a##) is...

∣r⟩,∣l⟩,∣i⟩, and ∣o⟩ can all be expressed as expressions for ∣u⟩ and ∣d⟩. So, given the state vector ∣ψ⟩ = α∣u⟩ + β∣d⟩, is it possible to know not only the probability of ∣u⟩ but also the probability of ∣r⟩ and ∣i⟩? ∣ψ⟩ can be expressed as an expression for ∣r⟩, ∣l⟩ or ∣i⟩, ∣o⟩.

I am currently studying Economics at undergraduate level and want to enhance my knowledge about probability and statistics in order to better understand econometrics.

Let ##X## be the event that the first draw is a picture card Let ##Y## by the event that the second is a picture card Then the probability that at least one of the cards is a picture card is the probability of ##X## union ##Y## and has the formula Equation 1 ## P\left(X \cup Y \right) = P...

I am looking for some academical concept to work on 3 parts : 1) Real and imaginary analysis of two functions describing 2 events 2) If the first event's function is imaginary and the second is real , how can we analyse the intersection that show how the imaginary function turned out...

A bag contains ##4## red balls, ##3## green balls and ##2## blue balls. A random ball is selected from this bag. P(ball is red) = ##P(R) = \frac{4}{9}## P(ball is green) = ##P(G) = \frac{3}{9}## P(ball is blue) = ##P(B) = \frac{2}{9}## P(ball is not red) = ##P(\neg R) = \frac{3}{9} +...

Confetior ... I'm a layman with minimal physics background, but the most happening place in physics, going by media articles, seems to be Einstein's relativity and Max Planck et al's quantum physics. I did a little reading on Schrodinger's cat and what I could gather is unless an observation is...

I don't even understand what question is being posed here. The answers given by the author are as follows: These are numbers, potentially very large ones.

In experiment A: I observe an event 2 times in 2 trials. In experiment B: I observe an event 100 times in 100 trials. In both cases, I calculate a frequency of 100% In both cases, I calculate a 95% confidence interval of (1, 1). But intuitively the result of experiment B is "stronger" than...

There is a race meeting with a number of races. Each race has horses entered and each horse has a (saddlecloth) number, 1….x, with x being the number of horses entered in each race. Each horse has a known probability of winning a race (reflected in its price). I’d like to set a market where...

Hello. So I got a question about heredity . Let's say the probability of inheriting schizophrenia is 6 % if one parent is affected. So i know that for 6 % probability, there is 1.2 kid out of 5 who will inherit that illness . So is it better not to have kids in this case ?

Hello everybody. I have used probability in troubleshooting electronic pcbs , but the complexity of the designs forced me to apply it in other branches of electrical engineering. What branch in electrical other than maintenance uses probability theory ?

The claim is Proof. If it was ##P(X = x ) > 0## for uncountably many ##x##, then there would exist an ##N\in\mathbb N## such that ##P(X = x) > \frac{1}{N}## for uncountably many ##x## (1). This in turn would imply that there exist countably infinite many ##x## such that ##P(X=x)>\frac{1}{N}##...

Questions: P (JohnCalls|Burglary) ? Why? Source of the image: Artificial Intelligence: A Modern Approach - Third Edition, by Stuart Russell and Peter Norvig. My attempt at solving: using Bayes' Theorem = P (A|B) = ( P(B|A) * P(A) ) / P(B) P(JohnCalls|Burglary) = P(J|B) = ( P(B|J) * P(J) /...

I am doing a study of the possibility of transition between 12 different events. I have a dataframe with these key events (listed from 1 to 12) over a period of time. I constructed a transition probability matrix between these events (photo of the matrix is attached below). As I don't have a...

I know it is a simple problem, but I am confused by the fact that we can look at the problem from two points of view. In general, we differentiate between the balls of the same colour, I mean we could mark them B1, B2, ... and B6, the same way R1, R2, R3 and R4. Then we could say that the...

https://anxieties.com/self-help-resources/fear-of-flying/how-safe-is-flying/ "In fact, based on this incredible safety record, if you did fly every day of your life, probability indicates that it would take you nineteen thousand years before you would succumb to a fatal accident." The...

How many times do I need to have sex each month to ensure it will happen one of those months? Does it statistically have to happen sooner or later? Say I live for 250 years and copulate absurd amount of times.

I have a probability distribution over the interval ##[0, \infty)## given by $$f(x) = \frac{x^2}{2\sqrt{\pi} a^3} \exp\left(- \frac{x^2}{4a^2} \right)$$From this I want to derive a formula for the inverse cumulative density function, ##F^{-1}##. The cumulative density function is a slightly...

Assume there are two tall building with same height, and the materials are the same, having same distance away from the storm cloud. Will the one building with extreme high consumption of electricity cause higher chance to get struck by lightning than the one building without ? Thanks!

I am not a fan of random and statistics. I know it is extremely useful and probably the mathematical branch more applicable to real life to understand the world around us but I am a Calculus and Vectors boy. This problem though I find interesting. I would like to find a generalized solution for...

I've a small molecule ABCD made of building blocks A,B,C,D. The molecule can get cleaved at any 'bond' between the building block one bond at a time. Researchers have compiled probabilities from dissociation studies of large number of molecules( made up of many other building blocks) and the...

Hi everyone, This is an example of binary variable called as logical NOT https://www.fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/chap2s1.html?scroll=sseclognot ...and this is the complement rule of probability...

Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is...

TL;DR Summary: Looking for help on a Intro to QM Problem Hi All, THIS IS A GRADED PIECE OF WORK AT MY UNIVERSITY PLEASE DO NOT JUST GIVE ME THE ANSWER , I have made this post to see if what i've calculated seems reasonable, it sounds unlikely as 0.4 - 0.5L is in the middle of the well. The...