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. U

    Probability generating function

    (a) I find the geometric distribution $$X~G0(3/8)$$ and I find p to be 0.375 since the mean 0.6 = p/q. So p.g.f of X is $$(5/8)/(1-(3s/8))$$. (b) Not sure how to find the p.g.f of Y once we know there are 6 customers?
  2. Killtech

    I Quantum physics vs Probability theory

    Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems...
  3. P

    Biology Genetics Probability (Autosomal Recessive)

    I think both Jane and Bill are carriers. I'm unsure if the answer is 1/4 (25%).
  4. snatchingthepi

    I Step unclear in Peebbles' Probability Book

    If anyone could help me understand how Peebles gets from line one of the autocorrelation to the second line, I'd be most grateful. I don't understand what identity or property is being used to go from a product in the expectation value to a sum in the expectation value.
  5. K

    A Martingale, calculation of probability - Markov chain needed?

    I am trying to estimate probability of loosing (probability of bankrupt ##Pb##) using Martingale system in betting. I will ilustrate my problem on the following example: Let: ##p## = probability of NOT getting a draw (in some match) We will use following system for betting: 1) We will bet only...
  6. S

    Average speed of a probability density wave and wave packets

    Show that ##v_{av}=\frac{\hbar k_2 + \hbar k_1}{2m}## is equal to ##v_{av}=\frac{\omega_2 - \omega_1}{k_2-k_1}##. Which of the identities listed above (if any) would make the sign change between ##k_2## and ##k_1##? One can attain a "wave packet" by superposing two or more sinusoidal waves...
  7. Peter Alexander

    Probability from the tolerance of a capacitor (Gaussian distribution)

    Given the upper data, if the nominal value for capacitance is 33nF and tolerance of 20%, then values can range between 26.4nF and 39.6nF. With the bottom margin being set at 30nF, this means that the interval takes approximately 72% of all values. Is this the correct procedure to solve this...
  8. The unbelievable solution to the 100 prisoner puzzle.

    The unbelievable solution to the 100 prisoner puzzle.

    • Swamp Thing
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    • probability
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    • Category: Pop Science
  9. M

    MHB Need help with probability question. probability of dependent events.

    Would like to know what method, or distribution to use when solving a problem like this:I start from level 0. There is a probability p chance to drop to level -1 and a (1-p) chance to increase to level 1. The levels range from level -n to level n. When it reaches level -n or level n, it resets...
  10. Biochemgirl2002

    Calculating general normal random probability

    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
  11. Monoxdifly

    MHB [ASK] Probability of Getting the Main Doorprize

    There's an event which is joined by 240 members. The Event Organizer prepares 30 doorprize with one of them being the main ones. If Mr. Aziz's family has 15 tickets, the probability that Mr. Aziz gets the main doorprize is ... A. 1/16 B. 1/8 C. 1/4 D. 1/2 I thought the answer was 15/240 (the...
  12. S

    Probability of a fluctuation/entropy decrease

    Suppose we have a gas in the room at some temperature which is room temperature or higher. In some references the probability is given by -ΔS, which is indeed a tiny number and makes sense. However, in other references the probability is given by the Boltzmann factor plus the number of...
  13. S

    Calculating Probability of Choosing Vowels: Common Mistakes and Solutions

    My attempt: case 1: get one vowel (A) from word HOORAY = 1/6 x 2/3 x 4/5 = 4/45 case 2: get one vowel (A) from word MATHS = 3/6 x 2/3 x 1/5 = 1/15 case 3: get two vowels (2A) from word HOORAY and MATHS = 1/6 x 2/3 x 1/5 = 1/45 Total probability = 8/45 Answer key = 1/3 Where is my mistake...
  14. srallaba

    B Probability of spin in the +x direction

    Consider a qubit. Given a state equation, how to calculate the probability of finding spin in +x direction?
  15. L

    MHB Compute probability closeness between points in a 2D surface

    Hi all, Sorry, in my first message, I posted this question in the Basic Probability section, and so I moved it to this section. I have a surface (for example, a blank paper). In this surface, I have some elements of the set "A" randomly distributed. In this surface, I also have some elements...
  16. megf

    I Probability Density Function of the Product of Independent Variables

    How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...
  17. Philip Koeck

    A The probability of finding R out of N bosons in one half of a volume

    For the probability of finding R out of N (indistinguishable) bosons in one half of a volume with a total of 2g states (g in each half) I get the following expression: PR = WR / WT where WT is the number of ways of distributing N particles in the total volume: WT = (N+2g-1)! / (N! (2g-1)!)...
  18. Biochemgirl2002

    What Are the Tests for Independence in Probability Statistics?

    A) (0.5)-(0.3) =0.2 b) c) i have no clue how to determine from just a set of numbers if they are independant or mutually exclusive
  19. S

    MHB Probability calculation of dependent events with limitations

    Dear forum, First time posting and as English is not my native language, I'd like to apologize in advance for any linguistic errors I make. Yesterday, I received a case which sounded really easy to calculate but for some reason I can't get my head around it. This is the case: In a shipping...
  20. Haynes Kwon

    I Probability flux inside the finite potential barrier

    In Bransden textbook, it is stated that the probability current density is constant since we are dealing with 1-d stationary states. It gives probability flux outside the finite potential barrier which I verified to be constant with respect to x, but it doesn't provide the probability current...
  21. S

    MHB Maximizing Probability of Finding Explosive Device in Bounded Domain

    I am trying to solve the following problem. Let us take a bounded domain $S$ in which an explosive device is located. A team is deployed to locate and disable the device before a certain time T (when the device explodes). There are several criteria to be satisfied: 1. The domain $S$ is...
  22. warhammer

    Question on Quantum Physics- Probability of finding a particle

    I calculated the complex conjugate of both the given wavefunctions. For ψ1: ∫re^((-2)mod(r)x)dx=1 with upper limit ∞ & lower limit -∞. I replaced the upper and lower limit after breaking down the function inside integration as follows- r*∫e^(2rx)dx from -1/r to 0 and r*e∫e^(-2rx)dx from 0 to...
  23. M

    MHB Solving 2 Problems with Exponential RVs: CLT & Probability

    I have the following two problems that I need to solve: 1. Suppose that the service time for a student enlisting during enrollment is modeled as an exponential RV with a mean time of 1 minute. If the school expects 500 students during enrollment period, what is the probability that the...
  24. P

    I Do These Probability Conditions Imply Independence of Events A, B, and C?

    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...
  25. Wi_N

    I I have a real-life Probability problem

    I'll try to keep this short. Kazakhstan just hosted the world championship wrestling tournament and I noticed that they did exceptionally well this time around, hometown psychological advantage aside they finished in 2nd place. Last year they finished 13th in Budapest. So immediately I knew...
  26. jaumzaum

    Is symmetry necessary for solving triple integrals in probability calculations?

    I just started learning triple integrals. I don't know if this is right (I'm only concerned about the limits of the Integral) Consider the case ## a>=b>=c ## $$ P = \lim_{M \rightarrow +\infty} \frac { \int_{a=0}^M \int_{b=\frac a {1.1}}^a \int_{c= \frac a {1.1}}^b \, da \, db \, dc} {...
  27. ckirmser

    Question on Probability - been way too long since college

    Summary: Probability of an event based on a data table Good morning, all - I'm working on a question involving obesity based on alcohol and tobacco consumption. The question is based on a table with five variables; • (age) An age group (10-25, 26-50, 51-75, 76+) • (alc) An alcohol...
  28. Z

    Comp Sci Slotted Aloha: Another probability question

    Hi, probability that one node succeeds = p probability that one node does not succeed = (1-p) probability that 3 nodes do not succeed = (1-p) ^3 Total probability = probability that one node succeeds * probability that 3 nodes do not succeed Total probability = p(1-p)^3 Is it correct? Zulfi.
  29. Z

    Comp Sci Probability Question: Slotted Aloha

    Can some body please explain me this solution? Zulfi.
  30. christang_1023

    I How to understand this property of Geometric Distribution

    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##...
  31. M

    MHB Find the probability all 5 dice rolls are the same

    Given 5 dice rolls that are independent from each other, what is the probability for the following results? (order of roll does not matter) 1. all 5 dice rolls are the same 2. 4 dice rolls are the same 3. the dice rolls are in sequence (1-5 or 2-6) -order does not matter 4. two pairs of dice...
  32. L

    MHB Probability of Having a Disease

    According to the CDC 1 out of 102 persons who are males of a particular demographic have HIV. That mean's there is 1/102 = .98 * 100 = 1% chance of having HIV. There is a swab HIV test available that is 91.7% accurate at being able to identify HIV positive people. This means that 12 out of 13...
  33. U

    I Box System Probability Problem

    Good morning, I have a system that consists of a huge number of boxes, it is not important to know how many, each containing a binary number: In every cell, the probability that there is a '1' is p, so the probability that there is a '0' is equal to (1-p). What I do is take them in groups of N...
  34. entropy1

    B Is an Ensemble of Measurements a True Reflection of Probability?

    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?
  35. J

    I Does the postulate of equal a priori probability apply only to equilibrium?

    In deducing the zeroth law of thermodynamics in micro-canonical ensemble, there is a frequently-mentioned example. Suppose we put two isolated system, system 1 and 2, in contact and allowing them to exchange heat. The total energy of the combined system is $$E = {E_1} + {E_2}$$ The total...
  36. christang_1023

    I What is the addition law for probability with multiple elements?

    The additional law with two elements can be expressed $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$, while the law with three elements can be $$P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C)$$ Now I wonder if there is the more general form of addition law, which applies...
  37. P

    I Stuck on the probability of rolling 'p' with 'n' s-sided dice

    Hi, I've been following the derivation of wolfram mathworld for this problem and I'm running into some trouble regarding the summation indices. Currently I am at the step where we have found that (it's pretty much just binomial expansion and taylor series to get to this point) $$ f(x) =...
  38. A

    MHB Probability of cancer because of microwaves

    Please help understand how to solve this question. Sorry for the terrible English. If there’s 10% chance of getting cancer from microwaves, and 3% of the population gets cancer, what is the probability that a certain person who have cancer and was exposed to microwaves got it as a result from...
  39. TytoAlba95

    Biology Genetics: Calculating probability

    My Attempt: For the child to be affected, both the parents must be carriers. Hence P(child)= Probability of the father of passing down an affected allele x Probability of the mother of passing down an affected allele = 1/2 x 1/2 =1/4 The...
  40. amjad-sh

    I Transition probability density between two transmitted states

    suppose we are working on a step potential problem, and two transmitted wave functions,corresponding to one particle, are obtained. Let's name them ##|1>## and ##|2>##. How can we interpret physically the case where ##<1|2>##=##-<2|1>##? or in position...
  41. Moara

    Probability of hitting a rotating disc

    There are two points of the smaller disck such that a line is tangent to the two discks and passes by that point. If the particle leaves of any point belongin to the arch that connects these two points, then it will hit the other disck. So I managed to calculate the value of this arch and divide...
  42. entropy1

    I Getting the probabilities right

    If we have a jar with 3 blue balls and 7 white balls, we say that the probability of blindly getting a blue ball out of that jar is 30%. If we have a jar with 2 blue balls and 8 white balls, we say that the probability of blindly getting a blue ball out of it is 20%. Now if we carry out 10...
  43. lfdahl

    MHB Solve Probability Challenge: Prove 2/3 Chance of Only 1 Winner

    Start with some pennies. Flip each penny until a head comes up on that penny. The winner(s) are the penny(s) which were flipped the most times. Prove that the probability there is only one winner is at least $\frac{2}{3}$.
  44. Vicol

    Probability density function in classical mechanics

    Probability density function plays fundamental role in qunatum mechanics. I wanted to ask if there is any analogous density function in classical mechanics. Obviously if we solve Hamilton equations we get fully deterministic trajectory. But it should be possible to find function which shows...
  45. DarMM

    A Representation of Super-Quantum probability

    Classical probability theory can be represented with measure spaces and functions over them. Quantum Probability is given as the theory of Hilbert spaces and operators over them. Both more abstractly are handled by the theory of C*-algebras and their duals. However I know of no structure for...
  46. Manasan3010

    Is an answer possible - Conditional Probability

    I am a noob to this topic so correct me If I made any silly mistake. By plugging in the values I managed to get p(abc)=0.75*0.9*p(c|ab) Here How can I find p(c|ab)? Is this question unsolvable or can I derive it? I also want to know what is meant by p(abc) in literary terms. I also created a...
  47. vvaibhav08

    A Probability distribution: exponential of a quartic

    $$\int_{-\infty}^{+\infty} exp (-[ax + bx^2]^2) dx$$ $$a\&b\in R$$
  48. R

    MHB Calculating Probability of Choosing 11 Reds in a Row

    I realize this is probably quite easy and basic but I just can't get comfortable with calculators to work it our with any certainty. There are 100 socks in a drawer. 84 are Red. 16 are White. If you are blindfolded & have to choose 11 socks randomly, what is the odds or what it more likely...
  49. Z

    MHB Need to find the probability of receiving 8 pairs of identical boxes choosing from 8 items 16 times?

    A friend stated they bought 16 crates all which could contain a random C1-C8 item. He then opened the crates and received exactly 2 each of every C1-C8 items. So, (C1,C1) (C2,C2) (C3,C3) (C4,C4) (C5,C5) (C6,C6) (C7,C7) (C8,C8) is what he ended up with. He stated this was good because...
  50. Anonymous1212144

    B Probability of finding a particle

    Let's say that I observed a free particle at a certain location. Is there any way I can calculate the probability of finding that same particle at another location when I look for it again?
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