What is Probability: Definition and 1000 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. T

    MHB Probability question involving a lift

    Dear friends, I am stuck at the following probability problem. Will appreciate your help. An office block has five floors (ground, 1, 2, 3 and 4), all connected by a lift. When it goes up to any floor (except 4), the probability that after it has stopped it will continue to rise is 3/4. When...
  2. mr.tea

    Prob/Stats Probability with measure theory

    Hi, I am looking for a book for studying probability theory using measure theory. This is the first course I am taking of probability. Notions and theorems from measure theory are part of this course. As it turns out, this is a catastrophic disaster, and the textbook for this course is also not...
  3. T

    MHB Probability question where an object is chosen randomly out of two objects

    Dear friends, I am stuck at the following probability problem. Will appreciate your help. Two opera singers, Mario and Clarissa both perform on the same night, in separate recitals. The independent probabilities that two newspapers X and Y publish reviews of their recitals are given below...
  4. T

    MHB Male Customers Preferring Non-Fiction: Proving Independence

    Given the scenario that a bookstores customers is 40% male and 50% of male customers prefer non-fiction, what is the probability that a customer is male and prefers non-fiction. So, since being male and preferring non-fiction are independent, we can multiply them and get 20%. How can we prove...
  5. T

    MHB Stuck with probability question involving tree diagram?

    Dear friends, I'm unable to solve the following probability question. Please help me solve it. Thanks in advance. The answer given in the book is: 5/9 [for part (b)]. Don't know even if the answer is correct. Suzi has taken up golf, and she buys a golf bag containing five different clubs...
  6. J

    Probability of equal roots in equation

    Homework Statement A quadratic eqn of form ax2 + bx + c = 0 is selected. The values of a, b and c are distinct and selected from 1, 2, 3, 4, 6, 8, 9. What is probability of chosen equation to have equal roots? Homework Equations root = (-b +/- Sqrt(b2-4ac)) / (2a) For equal roots b2 = 4ac[/B]...
  7. M

    MHB Find the probability that E' occurs at least once

    Data Below are the possible observations of a random experiment in a tree diagram.Each time it is observed whether the event E defined in part i occurs or not. Where do I need help Find the probability that E' occurs at least once Many Thanks :)
  8. TheSodesa

    Tsebyshev's probability

    Homework Statement The average lifetime of a product ##T=7.5## (years). The variance of the lifetime ##\sigma^{2} = 0.41##. Using Tsebyshev's inequality, determine the lower bound for the probability, that the product lasts at least 5 years. Homework Equations Tsebyshev's inequality...
  9. e2m2a

    I Sequential vs simultaneous probability

    If I toss 1000 coins, one at a time, the laws of probability predict about half will end up heads and the other half tails. But what does probability say if I toss all 1000 coins at the same time? I assume the probability remains the same: half will end up heads, the other half tails. So, my...
  10. T

    MHB How Do I Calculate the Probability of Two Events in Email Marketing?

    Let say I am a company and I email 1 million customers to try to get them to make a purchase on my website, of which 50000 click the link in the email and of those 1000 makes a purchase. We can say that clicking the link is event A, and making a purchase is event B. What is $P(B | A)$ or the...
  11. falcon555

    Is Fermi-Dirac Probability Part B Solvable?

    Hi dear friends Please reffer to my work , I did part ( a ) Can someone help me to solve part( b ) Please
  12. falcon555

    Calculating Fermi Dirac Probability - Part B Guide

    Hi dear friends Please reffer to my work , I did part ( a ) only Could you please help me to do part ( b ) I don't know how to do it.
  13. G

    B Understanding Probability and Fairness in Coin Tosses

    Hi, i was wondering if some one could explain to me some things about probability. How is probability determined is it based on a average or a small set of experimental data? If you were to flip a coin Randomly 50 times and you get heads 40 times and tails 10 times could you then say you have...
  14. S

    A Probability of photon emission from quantum dot

    Hi guys, I am looking for a formula which I am sure exits but I cannot locate it. The problem is that a quantum dot absorbs a photon of wavelength λ0(dot is semiconductor or could be any other material). Assuming that it reemits a photon, what is the probability that this emitted photon will...
  15. C

    B Proton decay probability (Bayesian or frequentist?)

    If an event has probability occurring say 1/100000000000000000000000000000000 times.. if you do the experiment 100000000000000000000000000000000 times.. you are supposed to get the hit at least once? This is the proton decay experiment.. it's more than the above probability figure.. but if you...
  16. O

    MHB Probability: road junctions

    There are four road junctions labelled A,B,C,D and four towns labelled W,X,Y,Z,. Driver is approaching junction A from Town W, when he realizes that he does not know how to get to the factory. He decides that at road junction he will choose a road to take random , but he will not go back to the...
  17. TheSodesa

    Chances of player A winning a chess tournament?

    Homework Statement 4 Players (A, B, C, and D) enter a chess tournament. The probabilities of them winning each other have been calculated based on their previous games. The results are in the below table: \begin{array}{|c|cccc|} \hline\\ \\ & \text{A wins} & \text{B wins} & \text{C wins} &...
  18. M

    A Three lines two neurons and M×M receptors.

    Dear all, I have lately been interested in AI. One of my thoughts has been as follows: What if we have two 'neurons' and the world consists of vertical, horizontal and diagonal lines. How would we go about making sure only one neuron is assigned to either vertical or horizontal lines and...
  19. E

    A Stopping rule for quality control problem

    Problem: Suppose I have a production process that yields output in batches of n items. For each batch, I can test whether they are of good or bad quality. Let q_i ∈ {1,0} be the quality of tested item i. If more than half of the items are ‘bad’, the batch should be discarded. In other words...
  20. TheSodesa

    How Do You Calculate Conditional Probability with Complements?

    Homework Statement [/B] P(A | \overline{B}) = ? Homework Equations Multiplicative rule: \begin{equation} P(A | B) = \frac{P(A \cap B)}{P(B)} \end{equation} Additive rule: \begin{equation} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{equation} Difference: \begin{equation} A \backslash B = A...
  21. TheSodesa

    Probability of choosing stale donuts out of 24

    Homework Statement There are 6 stale donuts in a set of 24. What is the probability of: a) there being no stale donuts in a sample of 10? b) there being 3 stale donuts in a sample of 10? c) What is the chance of a stale doughnut being found? Homework Equations N \, permutations = N! The...
  22. Pallatinus

    Prob/Stats What Probability book suits me better?

    I found those books in my library and I want to know which suits better for me. (Want to learn rigorous Probability AND applies it to statistical mechanics) Previous knowledge: Calculus 1 and 2, Introductory Statistics and Combinatorics. Books: Probability Theory: The Logic of Science - E.T...
  23. B

    B Real Probability: Rational vs Irrational Numbers

    So from what I've been reading rational numbers are a countable infinity, while the irrationals are an uncountable infinity. So the number of irrational numbers > the number of rational numbers. Irrational numbers can "normal irrationals" or transcendental numbers, or at least that is what I've...
  24. K

    B Card-drawing probability problem

    I've been trying how to figure out how to figure out the probability of different situations in card-choosing, and I am having trouble getting my head around how to approach this: Let's say we have two people drawing 26 of the 52 cards in a deck. What is the probability that either team has...
  25. L

    Probability and Bernoulli trials

    Homework Statement A box contains m white and n black balls. Suppose k balls are drawn. Find the probability of drawing at least one white ball. Homework Equations Probability of one success = P({1 successful trial}) = n * p * qn-1 p = probability where q = 1-p Fundamental theorem of...
  26. A

    The probability of a probability?

    TEXT : - We have an event called X with two possible outcomes that are outcome A and outcome B . the probability for the outcome to be A is noted as p , where p is a real number in the range [0.0 ; 1.0]. we also note p' a real number representing the probability that p is in range [0.9 ; 1.0] ...
  27. M

    Probability insurance company problem

    Homework Statement (4) (a) Let X be a random variable defined by the probability mass function P(X = x). The possible values X can take (denoted x) and the probability of those values occurring P(X = x) can be seen below x 0 1000 2000 5000 P(X = x) 0.94 0.03 0.02 0.01 Find E(X). (b) An...
  28. M

    What is the expected profit for a car rental insurance company?

    Homework Statement (4) (a) Let X be a random variable defined by the probability mass function P(X = x). The possible values X can take (denoted x) and the probability of those values occurring P(X = x) can be seen below x 0 1000 2000 5000 P(X = x) 0.94 0.03 0.02 0.01 Find E(X). (b) An...
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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?
  44. 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) /...
  45. 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%...
  46. P

    MHB Binomial Probability - Again.

    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...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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