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- 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
- 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
- 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
- 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
- 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?