Probability Definition and 1000 Threads

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. M

    M&Ms Probability Help: Understanding P(B|A) in Random Selection Scenarios"

    Homework Statement [/B] A bowl contains a large number of M&Ms. A single M&M is chosen at random, its colour is observed, and then it is returned to the bowl. A second M&M is chosen at random and its colour is observed. Let A be the event that the first M&M is yellow and B be the event that the...
  2. M

    Double Dice Probability: Event A and B in Sample Space | Solved

    Homework Statement In a probability experiment, a fair die is rolled twice. • If the first roll is odd, the outcomes are recorded as they appear. • If the first roll is even, the recorded outcome for the second die is doubled. For example, if the first die was 2 and the second 4, the...
  3. M

    Help with probability question

    Homework Statement (a) Suppose a fair six-sided die is rolled once. Let A be the event that an even face occurs and B be the event that a face less than 4 occurs. Are the events A and B independent? Show this mathematically. (b) A fair coin is tossed three times. Let A be the event that the...
  4. G

    The probability of dice coming up 6 twice in twelve rolls

    Homework Statement Explain the formulas used to obtain the solution for the question above (What is the probability of two tetrahedral dice landing on a 6 if twelve dice are rolled?) 1 - (3/4)^12 - 12 * ((3/4)^11) * (1/4) = 0.842 Homework Equations If we were looking for one dice landing on 6...
  5. B

    MHB Probability Question - Lottery

    H Everyone, Can someone please tell me what the following chances of winning is:A customer receives 1 ballet for every \$5,000 they spend with our company during a promotion. There are a total of \$3,000,000 in purchases over the course of the promotion and therefore 600 ballets are handed...
  6. E

    B Calculating Probability with Random Variables and Constants

    Hello all, I have a question: Suppose I want to find the following probability: Pr\left[\frac{\alpha_1}{\alpha_2+1}\leq\gamma\right] where ##\alpha_i## for i=1, 2 is a random variable, whatever the distribution is, and ##\gamma## is a constant. Can I write it as...
  7. B

    MHB Probability Activity Word problem for relative frequency

    I am drowning in math right now and would love some help bc I am horrible at math. Here is what I have got. Question: 1. The four major blood groups are designated A, B, AB and O. Within each group there are two types; positive and negative. Find data on the relative frequency of these eight...
  8. F

    I Can the bird return home?(about probability)

    Here's a question makes me confused. A bird leaves its home to find some food. It won't return if it doesn't find any food. Suppose the probability it returns in the first day is 1/2 and returns in the second day is 1/4 and returns in the third day is 1/8 ... returns in the nth day is (1/2)^n...
  9. L

    B How are permutations and probability related?

    This may already be widely taught and I could be stating the obvious here, but I noticed how closely related permutations and probability are, and this gives an intuitive way to think about permutations. For example, take a deck of 52 cards. How many possible permutations are there for the...
  10. L

    I Yet another Bayesian probability question

    But a very simple one. Just to check I'm not getting it wrong. Suppose you have a very large enclosure with 100 animals. 70 of these animals are cats, 30 are dogs. There is enough food for all the animals, but you introduce a new type of food, to see whether either cats or dogs will show a...
  11. T

    I Finding probability of changing states

    I am following the derivation shown in this link on adiabatic passage. I have posted one part below: I am simply wondering how this expression was derived and how it indicates the probability of being in a state that is different from the initial state? How exactly is this represented by...
  12. N

    B Why is probability defined in space rather than at a point in quantum mechanics?

    I have started quantum mechanics on my own using online lectures. So i have very basic doubts: 1) probability of electron is defined in space rather than a point. My question is why don't we comment about probability at a point. I thought two possible explanation that: 1.1)ψ2dx the dx term...
  13. Adgorn

    I Feynman's probability lecture -- a few questions

    I am reading lecture #6 on the first volume of "feynman's lectures on physics", and I understood quite well the first half of the lecture. However after he proved that D(rms)= √D^2= √N, I started to lose him, and so I have quite a few questions: 1) "The variation of Nh from its expected value...
  14. S

    MHB Why is there a probability paradox in this bag of balls?

    A bag contains two balls. Either ball can be black or white. Without drawing any balls, determine the colors of the balls.Solution The bag could contain any of three contents, each with probabiity \tfrac{1}{3}. . . \boxed{B B} \qquad \boxed{BW} \qquad \boxed{WW} Add a black ball to the bag...
  15. Chris Frisella

    I Relationship of Light's Wavelengthvs and Probability Wave?

    http://physics.stackexchange.com/questions/103904/size-of-a-photon I can follow the equitations from John Rennie's answer in the above thread, but considering real 3D space, I don't understand how wavelength would make the location of a photon less precise. Can anyone explain that more?
  16. ecoo

    Rearrangement of the letters "mathematics" (probability)

    Homework Statement What is the probability that a random rearrangement of the letters in the word "mathematics" will begin with the latters "math"? Homework Equations Probability = (# of desired results) / (# of total results) The Attempt at a Solution The solution I got was (2*2*2*1) /...
  17. DiracPool

    B Math: Probability of Accessing CNN News from 2 Sites w/80% Reliability

    How do I express the following scenario mathematically? I have access to CNN news online from two different internet sites, each of which have about an 80% reliability of actually providing the feed when I log onto the site. If I only had access to one of the sites, I'd know that I had an 80%...
  18. P

    MHB What Is the Probability That Any Bucket Receives Exactly n Red Balls?

    Greetings. I would need help as follow up to the answers to my http://mathhelpboards.com/basic-probability-statistics-23/prob-red-ball-buckets-binomial-17282-post79735.html#post79735. After having thrown R red balls over M buckets, the probability that exactly n red balls fall in one, specific...
  19. Fady

    I Probability of getting 3 heads or more in 20 coin flips

    hey folks, My question is how to find probability of getting 3 CONSECUTIVE heads or more in 20 coin flips, what are the odds I have an approach which I need to verify and please clarify in case of missing points. No. of possible outcomes = 2 ^ 20 first I assume that first 3 outcomes are...
  20. MidgetDwarf

    Courses What should I review for an introduction Probability Course

    What should I review for an introductory probability course? Here is the course description. Math 224 I believe is Calculus 3 at this school( multivariable). 380. Probability and Statistics (3) Prerequisite: MATH 224. Frequency interpretation of probability. Axioms of probability theory...
  21. P

    B Probability of finding an electron

    Let us assume that we have an electron belonging to the px orbital. In that case what would be the probability of finding it on the z axis? Would it be zero? My teacher says so, but I think that because we can't predict the boundary where there is 100% possibility of finding an electron, we...
  22. G

    A funny probability problem for the younger ones

    Homework Statement This is a problem that I really liked and that I want to share with you. Firstly because of the story around it, secondly because of the unexpected solution, and finally because it can be investigated with a computer for those who are the least comfortable with maths...
  23. M

    Other How Can I Maintain Interest and Improve My Understanding of Probability?

    I want to have a deep understanding of probability. I've tried William Feller's first book on Probability, and E.T Jaynes' Probability theory - the logic of science (which is very different from most probability books.) But, neither books could hold my interest for long. The first was too...
  24. Y

    MHB Calculating the Probability of Winning $25 in a Bag Game

    Hello all, In a bag there are 18 paper notes. On five of them there is the digit 2, on seven the digit 3, and on six the digit 5. A man takes 3 notes by random. If the multiplicity of the notes is even, he wins 25 dollars. If for each game he pays 6 dollars, what is the average of profit he has...
  25. A

    Find probability of certain event, total probability theorem

    Homework Statement Suppose you're at a college campus. 3/4 of the people on the campus are students or professors from that college, and the rest 1/4 aren't. When asked a question, students and professors from that college will give you a correct answer every time, and those that aren't from...
  26. L

    Probability math confusion sample space (I think)

    Homework Statement high school has 417 students total 186 of total are athletes (play sports) 136 of total are musicians (play music) 74 of total are musicians and athletes. (play music and play sports) a) at which probability does randomly chosen athlete also play music (i.e. be a musician)...
  27. S

    MHB Calculation of odds: probability and deviation

    *WARNING* My understanding of maths is pretty limited and i don't know if i am being really stupid or not. Ive been looking into data analysis of football matches, working out proabilities of certain situations happening and how steady the averages are. i come across a website which as really...
  28. H

    I How to find the inverse of an integral transform?

    I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral...
  29. M

    MHB Probability of Combinations and Permutations

    Hi, I'm a mom trying to help my son understand why he got answers wrong on his online math program. He is taking Geometry, but the last unit in the class is an introduction to Probability and Statistics. After re-reviewing the lesson and re-working the problems he got wrong, we were able to...
  30. W

    B Simple probability with high frequency

    I'm trying to remember simple probability form high school. I'd like to know how the probability of anyone event changes with frequency. So for example the probability of getting one ace of spades is 1/52; what is the probability of getting at least one ace of spades if the card is put back and...
  31. micromass

    Challenge Micromass' big probability challenge

    Probability theory is very nice. It contains many questions which are very easy to state, but not so easily solved. Let's see if you can solve these questions. For an answer to count, not only the answer must be given but also a detailed explanation. Any use of outside sources is allowed, but...
  32. Danielm

    Calculating Winning Probability for a Chess Master in a Competition

    Homework Statement A chess master plays n games at a chess competition, each against a different opponent. Based on their past performances, we can estimate the probabilities of each opponent beating the master: p1, . . . , pn ∈ [0, 1]. Describe an algorithm, which given integer k returns the...
  33. NatFex

    I Sum of Probability Density Function > 1?

    I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
  34. bapowell

    Insights How Does Bayesian Reasoning Affect Mental Disorders?

    bapowell submitted a new PF Insights post Scientific Inference P3: Balancing predictive success with falsifiability Continue reading the Original PF Insights Post.
  35. EhrEnFeest

    What is the physical dimension/unit of Probability current?

    Homework Statement Question: What is the physical dimension of Probability Current for a particle in 1 dimension? (Quantum Mechanics) Homework Equations Quantum mechanical Probability Current: The Attempt at a Solution I know the physical dimension of mass, that is kg. If I know every...
  36. CynicusRex

    I Poisson process approximation error

    X = # of cars that pass in one hour E(X) = λ = n * p λ cars/1hour = 60min/hour * (λ/60) cars/min In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?" "We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1...
  37. S

    I Why is the standard deviation the error on the singular meas

    I'm a beginner with the study in data analysis in Physics. I'm trying to understand the meaning, in the field of experimental Physics, of the standard deviation ##\sigma## of a series of data. There is one fundamental thing about ##\sigma## that I read but I could not understand. >In a series...
  38. S

    A Normalized Differential Scattering Probability

    The normalized differential quantum-field-theoretic probability ##dP## of scattering is given by ##dP=\frac{|\langle f |S|i\rangle|^{2}}{\langle f|f\rangle\langle i|i\rangle}d\Pi,## where ##|i\rangle## is the initial state, ##|f\rangle## is the final state, ##\langle f|S|i\rangle## are the...
  39. mikebrisingr

    Probability coin and die Question

    I started by looking at the probability of any given situation and maybe making a tree but my professor hasn't explained things very well and most the notes I see online are overwhelming A Game of Coin and Die. This game is played with a fair coin and a die. First player flips a coin. If it...
  40. T

    I Relationship between E-field and Probability Amplitude of Waves

    Electromagnetic waves can be classically described by Maxwell's equations. Photons can be described by probability waves.In this case, what is the relationship between the electric field and the probability amplitude? Are they directly proportional to each other? What about the fact that one...
  41. G

    I What Is the Probability of Rolling Pairs That Sum to Seven with n Dice?

    I'm studying probability and am currently stuck on this question: Let's say we have n distinct dice, each of which is fair and 6-sided. If all of these dice are rolled, what is the probability that there is at least one pair that sums up to 7? I interpreted the above as being equivalent to the...
  42. Smalde

    QM: Time development of the probability of an Eigenvalue

    The problem is actually of an introductory leven in Quantum Mechanics. I am doing a course on atomic and molecular physics and they wanted us to practice again some of the basics. I want to know where I went conceptually wrong because my answer doesn't give a total probability of one, which of...
  43. R

    I Can Conditional Probability Be Solved Generally with PDFs of Variables?

    Is it possible to solve something like this generally or does it depend on the pdf's of the variables? P(x < f(y) | x > -f(y))
  44. lucytranxx

    MHB Bernoulli Trials and Probability

    Let X be a random variable defined as the sum of 5 independent Bernoulli trials in which the probability of each Bernoulli taking the value 1 is given by r. Suppose that prior to the 5 Bernoulli trials, r is chosen to take one of three possible values with the following probabilities: R=r...
  45. S

    A Using probability to predict my sales amount

    Suppose I'm trying to answer the following question: How much more insurance premium will I get, by raising rates by 10%. for this exercise, I"m assuming price is the only factor, just to simplify things. Lets say I give 20,000 quotes per month. Obviously, not all will purchase my policy...
  46. Amine_prince

    Probability question on buckets and balls

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > i am doubting my solution to this problem , therefore i hope someone assists me a bit . it's simple , there is a bucket that contains 5 red balls and 3 white balls , a player picks up 4 balls out...
  47. thegreengineer

    What Is the Probability That Exactly 3 Out of 8 Children Are Girls?

    Right now I'm having a problem with a statistics problem. More specifically with a binomial distribution problem. The problem says: There is a family composed by 8 children. Calculate the probability that 3 of them are girls As far as I know, binomial distribution formula says...
  48. T

    MHB What is the Probability of Getting Three Heads in 30 Coin Tosses?

    Conduct 30 trials and calculate emperical probability of the outcomes. What is the probability of tossing 1 coin three times and getting three heads? Need a flowchart of some sort to show all possible outcomes. This math is new to me and has little understanding of this math lol so any help...
  49. Rampart

    Conditional Probability exercise with dice

    Hey there community, I have a question on an exercise. Actually it is a general question based on it. Here is the exercise: We throw 3 dice. If we know that the sum of these 3 is 10, then what is the probability of at least one of them being 3? Well now, this exercise is very simple. I mean I...
  50. S

    A Discrete Multivariate Probability Distribution

    Homework Statement A fair coin has a ##1## painted upon one side and a ##2## painted upon the other side. The coin is tossed ##3## times. Write down a sample space for this experiment. Let ##X_1## be the sum of the numbers obtained on the first ##2## tosses and ##X_2## be the sum of the numbers...
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