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 TOPICS This page:   probability    Search   Dr. Math See also the Internet Library:   probability COLLEGE Algorithms Analysis Algebra    linear algebra    modern algebra Calculus Definitions Discrete Math Exponents Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean Imaginary/Complex   Numbers Logic/Set Theory Number Theory Physics Probability Statistics Trigonometry Browse College Probability Stars indicate particularly interesting answers or good places to begin browsing. See also College Statistics. Four of a Kind in a 13-Card Hand [03/24/2001] What is the probability of receiving four-of-a-kind when dealt 13 cards from a regular 52-card deck? The Fox and the Numbat [07/19/1999] What is the probability of each animal winning the game? The rules are... Generating a Random Point within a Circle [04/02/2007] Given a circle C with center (xc, yc) and radius R, what's the best way to have a computer generate a random point within the circle? Getting More Heads [05/29/2000] If Bob flips 850 fair coins and Alice flips 851 fair coins, what is the probability that Alice will get more heads than Bob? Gnomes and Hats [11/13/2001] Ten gnomes in the dungeon of a castle of a tyrannical king are given a chance of survival... Gnomes in Trouble (Again!) [04/26/2002] Ten gnomes have been captured by a Sultan, who forces them to solve a puzzle to escape impalement. Independent and Mutually Exclusive Events [01/03/2007] Can you explain the difference between events that are independent and events that are mutually exclusive and give examples of each? Independent Probablilty Distributions [12/15/2005] I have two independent random variables A and B with known continuous density and distribution functions and their pdf's overlap. How do I calculate the probability that variable A will be lower than variable B? Infinite Monkeys on Infinite Typewriters Producing Shakespeare [01/02/2006] An interesting discussion of the classic monkey problem which focuses on how we know that the works of Shakespeare will eventually be produced. Interesting Counting Probability Problem [11/19/2004] In an election two candidates, Atif and Bryan, have in a ballot box a and b votes repectively with a > b, for example 3 and 2. If ballots are randomly drawn and tallied, what is the chance that at least once after the first tally the candidates have the same number of tallies? Inverse Transformation Method and Random Variables [02/03/2009] I am looking for help on the proof of using the Inverse Transformation Method to simulate a random variable having a continuous distribution. Law of Large Numbers and the Gambler's Fallacy [05/07/2001] Should you get the same total, on average, when you make three throws of three dice each as when you throw nine dice at once? Also, can you explain the Law of Large Numbers and the Gambler's Fallacy? Logistic Distribution [10/08/2002] I'm having trouble with a particular logistic probability distribution function... A Markov Chain Example [07/01/1998] We are given P pots with N balls each and take B balls out of them, where B < N. On average, how many pots do we touch? Mean and Standard Deviation [03/01/1999] Prove that the mean (u) and standard deviation (s) for a binomial distribution are np and the square root of (npq). Measuring the Randomness of a Shuffled Deck [12/19/2003] I'd like to compare how well humans and computer programs can shuffle a deck of cards, but I need some way to measure the randomness of a shuffle. How can I do this? Moment-generating Function of a Binomial Random Variable [10/27/1999] How can I find the moment-generating function (MGF) of a binomial random variable? Moment Generating Functions [06/20/1998] Can you help me with the following proofs on moment generating functions? Moment Generating Functions [03/30/1999] What can be established about the sum of negative binomial random variables? Normal Distribution and the Lottery [6/20/1995] I wrote a Visual Basic program to generate "Pick 6" lottery numbers, producing six random numbers from 1 to 46... the result being a somewhat normal distribution, but I can't understand why. Odds of Winning in Dice Game of Craps [02/20/2007] How would I calculate the odds of winning in the dice game Craps? Paradox Involving Swapping Dollars [08/01/2005] A discussion of a seeming paradox involving two people comparing amounts of money in their wallets with the greater amount being given to the holder of the lesser amount. At first glance it seems that the game favors both participants, but a closer look reveals that is not really the case. PDFs Explained: From Histograms to Calculus [12/21/2014] A student familiar with the binomial distribution wonders about the probability density function: how and why does the area under a PDF curve represent probability? Starting with elementary histograms, Doctor Peterson builds up an explanation. Phone Number Combinations [05/07/2000] Linda knows the first 3 digits of Jack's phone number, but she doesn't know the other 4... Poisson and Binomial Distributions [02/22/1999] Some assumptions made in Poisson and Binary Distributions. Poisson and Binomial Questions [06/13/1998] Can you explain why the waiting time in a Poisson process is exponential? Also, find the expected value of a binomial distribution? Poisson Distribution Applied to Web Site Page Demand [07/14/2004] A web site delivers 150,000 pages per hour, but demand varies over the hour. How do I predict the peak per second demand during the hour? Poker Probabilities with Seven Card Deal [03/19/2004] What are the probabilities of all poker hands when seven cards are dealt, as in Seven Card Stud or Texas Hold-Em? Poker, Probability, Combinatorics [11/04/1997] If we deal n hands consisting of 2 cards each, what is the probability that there will be no pairs amoung the hands? Predict the Population of Puffins [09/28/2002] Scientists were studying the population of puffins in Alaska. They spotted and tagged 52 puffins in December. Two months later, they spotted 50 puffins. Of those 50 puffins, 3/10 of them had been tagged in December. Predict the population of puffins in Alaska. Probabilities in a Dice Game Using Markov Analysis [03/22/2004] Anne, Bob, and Carmel are going to take turns rolling a fair die, in the order Anne, Bob, Carmel, Anne, Bob Carmel, etc. The first person to roll a 6 will win \$100. a) Find the probability that Anne will win \$100 if the first four numbers rolled are 3, 2, 5, 5. b) Find the probability that Anne will win \$100 if a draw is to be declared with nobody winning \$100 in the event of a 6 not being rolled within the first 8 rolls in total. c) Suppose that a draw is to be declared if ever two 1's are rolled in a row (by two different players). Also suppose that the game is in fact over and did not end in a draw. Find the probability that Anne is the player who won \$100. Probability and Card Shuffling [11/3/1994] Problem: If I have n shuffled cards in k colours numbered from 0 to (n/k)-1, what is the probability that no card will have the same number on it as its position in the deck modulo n/k? Standard example: n = 52, k = 4, cards numbered from 0 to 12. A card with number 3 mustn't be on positions 3, 16, 29, or 42. Probability and Finding the Duration of an Event [04/18/2009] Event A happens constantly every 3 seconds, and each time it happens there is a 15% chance it triggers Event B. Event B lasts 12 seconds. Given that Event B would restart if it's re-triggered while already in effect, what is the average time Event B will run when it is started? Probability and Random Numbers [02/28/2004] Can you use a coin (which has 2 events of equal probability) to devise three events with 1/3rd probability each? I find devising 4 events easy; toss the coin twice and interpret the results 00, 01, 10, 11 (i.e. HH, HT, TH, TT) as the 4 events. But, is it okay to disregard any one result (say 00) and claim that the other three events are of equal probability? I find it hard to believe that a coin can be used to have three events of equal probability. Probability and the Prisoner Problem [07/04/2001] Each of 16 prisoners receives a hat that is either red or blue (the colour is selected randomly; each has a 1/2 probability). All the prisoners must simultaneously either try to guess the colour of their hats, or pass. Probability and Tossing a Coin [10/25/2004] How would I calculate the probability of a tossed coin coming up heads two times in a row before it comes up tails a total of ten times? Probability Distributions [11/04/1997] Supposing a million people, without reference to each other's choices, each choose a 'random' number between 1 and 1,000,000... Probability for a Given Distribution of Objects [7/23/1996] 80 percent of light bulbs last 2400 hours, 20 percent last 2400 hours... Given a collection of screws with a Gaussian distribution of size.... The frequency of a mistake for wires is once in 25 meters... Probability of a Sum [08/16/2002] If the integers m and n are chosen at random between 1 and 100, what is the probability that a number of the form 7^m + 7^n is divisible by 5? Probability of a Sum on Multiple Dice [03/26/2001] How can I write a program for calculating the probability of getting a sum s on n dice with x sides? Do I need to use binomial coefficients? Page: []

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