# Probabilities Definition and 58 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.

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1. ### Looking for a particular function

TL;DR Summary: I want to find a function with f'>0, f''<0 and takes the values 2, 2^2, 2^3, 2^4,..., 2^n Hello everyone. A professor explained the St. Petersburgh paradox in class and the concept of utility function U used to explain why someone won't play a betting game with an infinite...
2. ### I Roulette system -- which optimization is better?

Hey, gotta do some explanation first: I assume you know how roulette works. (if you dont: ball is thrown into a pit and it can either land on red, black or zero, each having a certain likeliness to land there. you can bet on where the ball will land) let's assume unrealistically you have the...
3. ### B The law of total probability with extra conditioning

Hello, I am studying probability and came across this theorem, it's the law of total probability with extra conditioning, I tried to work out a proof but couldn't ,does anyone know the proof for this : thanks!
4. ### B About the naive definition of probability

hello, I took an introductory course about statistics, we viewed the naive definition of probability which says "it requires equally likely outcomes and can't handle an infinite sample space ", I understood that it requires finite sample space but I didn't understand "equally likely outcomes "...
5. ### A A game of seemingly mutually independent events

There are 4 players numbered 1 to 4. There is a room with an entrance door to one side and exit door to opposite side. Inside the room, there are 4 boxes numbered 1 to 4. Inside each box, there is a chit containing a number (with equal probability) from 1 to 4 (inclusive). No two chits can have...
6. ### Normal Distribution -- Hypokalemia diagnosis with single versus multiple tests

TL;DR Summary: Finding the probability with one measurement and multiple measurements on separate days. Question: Hypokalemia is diagnosed when blood potassium levels are low, below 3.5 mmol/L. Let’s assume we know a patient whose measured potassium levels vary daily according to N(µ = 3.8...
7. ### I Is there a way to calculate expected value from probabilistic data?

So I ran a python simulation of 1,000 games of toss (50/50 odds) where each game consists of 100,000 consecutive flips. The result was this: 1000 is our starting balance and as expected, there's a nice normal distribution around it. I also calculated the average value after all the games and it...
8. ### Using Poisson random variables to calculate this probability

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...
9. ### I Probability of 10 consecutive tails with 30 coin flips

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...
10. ### Winning and losing distributions

I started by trying to write a normal distribution function for losing and not sure if that is possible.
11. ### What is the ideal number of guesses to maximize your marks in MCQs?

To approach this, I first assumed the case when the students attempts all the remaining questions. Probability that they gain 4 marks for a guess = ##\frac 1 4## Probability that they lose 1 for a guess = ##\frac 3 4## Now let us say the number of correct guesses = ##r## Now we should have at...
12. ### I Probability that two points are on opposite sides of a line

I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $[0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction...
13. ### I Prove that the norm squared of a superposition of two states is +ve

This is what I have so far: $$|\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + \alpha^*\beta\Psi_1^*\Psi_2 + \alpha\beta^*\Psi_1\Psi_2^*$$ $$=> |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + 2Re(\alpha^*\beta\Psi_1^*\Psi_2)$$ I am...
14. ### Markov chain on state {1, 2, 3, 4, 5, 6 , 7}

I need this for a programming project. Could you help?
15. ### A Building a better ranking system (probability)

Suppose we have four games and the probability that a player will win the game are as follows: Game 1: 71% Game 2: 55% Game 3: 58% Game 4: 16% Suppose player b won these games with the following percentages of time: Game 1: 100% Game 2: 96% Game 3: 87% Game 4: 67% In other words, he's a very...
16. ### Prove expression for N! using inclusion-exclusion principle

Well ##\cup_{i} E_i## is just the event that at least one color is not used, so its probability is given by ##1- (1/N)^N##. Now if I is a subset of {1,...,N} where ##\left | I \right | = l## then ##Pr(\cap_{i\in I} E_i) = (1-l/N)^N## (I'm guessing this is where I'm making a mistake?). So then we...
17. ### Probability question: picking colored marbles out of a bag

there are 5 marbles in a bag: 4 are blue, and 1 is red. Solution there are 5 in total 4 are Blue so 4/5 or80% Is it right? thanks
18. ### I David Deutsch (1985) attempt to solve the incoherence problem

Can anyone elaborate on Deutsch's attempt to solve the incoherence problem? He postulates a continuously infinite set of universes, together with a preferred measure on that set. And so when a measurement occurs, the proportion of universes in the original branch that end up on a given branch...
19. ### I Probability that a Random String is a Word

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...
20. ### Geometric Law of Probability with Dice

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...
21. ### I Conditional probability choosing from the objects

Hello. I am reading an online stats book, and there is the following question, which I solved incorrectly, and I think I understand what is my mistake, but I will be grateful for your explanation, if I have incorrectly detected the logic behind my mistake. I am weak at math (trying to improve it...
22. ### I Probability of a random walk reaching the point X; maximal c

https://ibb.co/guBuPd As the title indicates, I want to calculate the Probability of a stock price reaching a determined point, by considering the system as a random walk model, and after that, to compute the so called "maximal curves". I found the whole explanation in this article...
23. ### Pedestrian at a road crossing

Homework Statement Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A...
24. ### Probability and pedestrian wait time density function

Homework Statement Pedestrians approach to a signal at the crossing in a Poisson manner with arrival rate ##\lambda## arrivals per minute. The first pedestrian arriving the signal starts a timer ##T## then waits for time ##T##. A light is flashed after time T, and all waiting pedestrians who...
25. ### B Given a success rate of 1% and 100 tries....

What is the probability of no success? 1 success? 2 successes? No success: 0.99^100 = 36.6% chance? 1 success: (0.99^99) + 0.01 = 37.97% chance? or (0.99^99) * (0.01)^1 = 0.369% chance ? You would think it would be fairly likely 2 successes: ? -Depends on formula for 1 success Can...
26. ### B What is the sum of multiple probabilities

If I have an asset that has a 10% chance to fail and I have ten of these assets in a basket, then what is the chance that one will fail in this basket? 10%?:partytime: What is the chance of 10 failing? 0,01%? Please also explain in some laymans terms. I am a total noob when it comes to...
27. ### A Probability of obtaining general quantum measurement outcome

The Fundamental Theorem of Quantum Measurement is stated as follows: Every set of operators ##\{ A_n \}## ##n =1,...,N## that satisfies ##\sum_n A_n^{\dagger}A_n = I## describes a possible measurement on a quantum system, where the measurement has ##n## possible outcomes labeled by ##n##. If...
28. ### Find the cdf given a pdf with absolute value

Homework Statement Consider a continuous random variable X with the probability density function fX(x) = |x|/5 , – 1 ≤ x ≤ 3, zero elsewhere. I need to find the cumulative distribution function of X, FX (x). 2. Homework Equations The equation to find the cdf. The Attempt at a Solution FX(x)...
29. ### Logarithm and statistical mechanics

Hello, I'll try to get right to the point. Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.
30. ### I Average profits and losses on a roulette table

Recently, youtube suggested a video to me on roulette betting theory. The essence of the idea: Start with a small bet. If you win, bet the same, if you lose, double up. If you lose a second time, double again (or bet the sum of your previous bets) until you win. Start over at the small bet. The...
31. ### I Conditional Expectation Value of Poisson Arrival in Fixed T

Assume a Poisson process with rate ##\lambda##. Let ##T_{1}##,##T_{2}##,##T_{3}##,... be the time until the ##1^{st}, 2^{nd}, 3^{rd}##,...(so on) arrivals following exponential distribution. If I consider the fixed time interval ##[0-T]##, what is the expectation value of the arrival time...
32. ### Average magnetic moment of the system

I was reading the statistical physics textbook and was really confused with the notation: I don't understand the last part of the section. Why is that \sum_{\sigma = \pm1} \sigma P(\sigma) equals to \left< \sigma \right>? And what does \left< \sigma \right> actually mean? Is it the average...
33. ### Confusion between z and t values

Homework Statement Hi, Alright so I have some confusion on when to use specific tests and the z vs t test. Given this example (not my homework) could someone please clarify. Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation...
34. ### Help on determining independent events

Homework Statement Hi, I have this question that I've been pondering for a while, I keep flipflopping on what I think is right. I only need help on the last part on whether the events are independent or not, the rest of the text is backstory to the question. I know for events to be independent...
35. M

### I Bootstrap percolation and percolation

I read about bootstrap percolation and I would like to find links and similarities between bootstrap percolation and percolation (the initial model). I wonder if there is any result in percolation that is still valid in the bootstrap model.
36. ### A Queueing Problem

First post and I'm wondering if I could get some help. I'm new to queueing theory so I'm not sure how to solve this problem. Population-------------> Queue--------------> Server I have a calling population that is infinite or a vast amount. The queue capacity is limited at 13. There is 1...
37. ### I Beat the roulette wheel with this?

I am perplexed because it seems that I have come up with a system for beating the roulette which has a positive mathematical expectancy/hope... but only in the calculations, which I think that I've made wrong. So I want to know in what I have failed doing the calculus. Well basically the method...
38. ### I Interpreting "momentum" in WKB approximation

According to WKB approximation, the wave function \psi (x) \propto \frac{1}{\sqrt{p(x)}} This implies that the probability of finding a particle in between x and x+dx is inversely proportional to the momentum of the particle in the given potential. According to the book, R. Shankar, this is...
39. ### Marginal pdf, what am I doing wrong?

Homework Statement f(xy)=49/8*e^(−3.5*y) 0 < y < inf and −y < x < y 0 otherwise a. Find the marginal probability density function of X, fX(x). Enter a formula in the first box, and a number for the second and the third box corresponding to the range of x. Use * for multiplication, / for...
40. ### Statistics probability questions

Homework Statement Each week, Stéphane needs to prepare 4 exercises for the following week's homework assignment. The number of problems he creates in a week follows a Poisson distribution with mean 6.9. a. What is the probability that Stéphane manages to create enough exercises for the...
41. ### How many basic states are there?

Homework Statement Four distinguishable particles move freely in a room divided into octants (there are no actual partitions). Let the basic states be given by specifying the octant in which each particle is located. 1. How many basic states are there? 2. The door to this room is opened...
42. ### I Jaynes on probability in quantum theory

This was off-topic in the thread on vacuum fluctuations where the quote appeared, so I opened a new one. I didn't know the paper before, so first need to read it...
43. ### I About stochastic differential equation and probability density

I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics. a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second...
44. ### Probability of 2 people meeting at given times

Homework Statement i'm having trouble coming up with an equation for this problem [/B] Mulder and Scully agree to meet at the FBI lobby sometime within a period of N minutes (e.g. 40 minutes). Each of them may show up anytime during that period (with a uniform distribution). If Scully arrives...
45. ### Long-run proportion in state ′A′

The question asks: A physical device can be in three states: A,B,C. The device operates as follows (all time units are in hours): The device spends an exponentially distributed amount of time in stateAA (with mean of 12minutes) and then with probability 0.6 goes to state B, and with...
46. ### Computer Server Down Probability

So the problem asks: A computer server runs smoothly for Exp(0.2) days and then takes Exp(0.5)days to fix. The server is running fine on Monday morning, t=0. Find the probability that the server was fixed at least once (i.e. at least one complete repair was done) in the next 7 days and the...
47. ### Roll a fair die 5 times

Homework Statement Roll a fair die 5 times, find the probability that the first two rolls have the same outcomes. Homework Equations The Attempt at a Solution The total outcomes is 6^5, I think we have 6^2 * 6 choose 2 /6^5 since the first two numbers are fixed and we can choose 2 numbers...
48. ### Expected value and variance of max{Y_1,Y_2}

Homework Statement Let Y_1,Y_2 be independent random variable with uniform distribution on the interval [1,2]. Define X=max{Y_1,Y_2}. Find p.d.f., expected value and variance. Homework Equations The Attempt at a Solution Since $X=\max\{Y_1,Y_2\}$, this tells $Y_1$ and $Y_2$ must at most $x$...
49. ### I Proof to the Expression of Poisson Distribution

Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...
50. ### I Two-player card game, combined probability

There is a card game where one player gets three cards and only uses two (the third one is discarded without showing to an opponenet), and the second player gets also three cards and uses only two, discarding the third card in the similar way - without showing. I am trying to enumerate all...