Probability Problems
 What Is Probability?
 Intuitive Probability
 Probability Problems
 Sample Spaces and Random Variables
 Probabilities
 Conditional Probability
 Dependent and Independent Events
 Algebra of Random Variables
 Expectation
 Probability Generating Functions
 Probability of Two Integers Being Coprime
 Random Walks
 Probabilistic Method
 Probability Paradoxes
 Symmetry Principle in Probability
 Nontransitive Dice
In a world as crazy as this one, it ought to be easy to find something that happens solely by chance. It isn't. 
Kevin McKeen 
American Heritage Dictionary defines Probability Theory as the branch of Mathematics that studies the likelihood of occurrence of random events in order to predict the behavior of defined systems. (Of course What Is Random? is a question that is not all that simple to answer.)
Starting with this definition, it would (probably :) be right to conclude that the Probability Theory, being a branch of Mathematics, is an exact, deductive science that studies uncertain quantities related to random events. This might seem to be a strange marriage of mathematical certainty and uncertainty of randomness. On a second thought, though, most people will agree that a newly conceived baby has a 5050 chance (exact but, likely, inaccurate estimate) to be, for example, a girl or a boy, for that matter.
Interestingly, a recent book by Marilyn vos Savant dealing with people's perception of probability and statistics is titled The Power of Logical Thinking. My first problems will be drawn from this book.
As with other mathematical problems, it's often helpful to experiment with a problem in order to gain an insight as to what the correct answer might be. By necessity, probabilistic experiments require computer simulation of random events. It must sound as an oxymoron  a computer (i.e., deterministic device) producing random events  numbers, in our case, to be exact. See, if you can convince yourself that your computer can credibly handle this task also. A knowledgeable reader would, probably, note that this is a program (albeit deterministic) and not the computer that does the random number simulation. That's right. It's me and not your computer to blame if the simulation below does not exactly produce random numbers.
When you press the "Start" button below, the program will start random selection. Every second it will pick up one of the three numbers  1, 2, or 3. You can terminate the process anytime by pressing the "Stop" button. Frequencies of selections appear in the corresponding input boxes. Do they look random?

Remark
Actually, the process of selection includes no selection at all. As a mathematician Robert Coveyou from the Oak Ridge National Laboratory has said,
The generation of random numbers is too important to be left to chance. Instead, I have a function that is invoked every second. Each time it's invoked, it produces one of the three
I start with an integer seed = 0. When a new random number is needed, the seed is replaced with the result of the following operation
seed = (7621 × seed + 1) mod 9999
In other words, in order to get a new value of seed, multiply the old value by 7621, add 1, and, finally, take the result modulo 9999. Now, assume, as in the example above, we need a random selection from the triple 1, 2, 3. That is, we seek a random integer n satisfying
n = [3 × seed/9999] + 1.
Taking it step by step, dividing seed by 9999 produces a nonnegative real number between 0 and 1. This times 3 gives a real number between 0 and 3. Brackets reduce the latter to the nearest integer which is not greater than the number itself. The result is a nonnegative integer that is less than 3. Adding 1 makes it one of the three 1, 2, or 3.
See Seminumerical Algorithms by Donald Knuth for more details.
Problems
 100 Prisoners and a Light Bulb
 A Fair Game of Chance
 A Pair of Probability Games for Beginners
 A Proof by Game for a Sum of a Convergent Series
 Amoeba's Survival
 Are Most Triangles Obtuse?
 Aspiring Tennis Club Candidate
 Average Number of Runs
 Averaging Raindrops  an exercise in geometric probability
 Balls of Two Colors
 Barycentric Coordinates and Geometric Probability
 Bear cubs problem
 Bear Born on a Tuesday
 Benford's Law and Zipf's Law
 Bertrand's Paradox [Java]
 Birds On a Wire [Java]
 Birthday Coincidence
 Book Index Range
 Buffon's Needle Problem
 Buffon's Noodle [Java]
 Careless Mailing Clerk
 Checkmate Puzzle
 Chess Players Truel [Java]
 Chevalier de Méré's Problem
 Clubs or no Clubs
 Diminishing Hopes
 Family Size [JavaScript]
 Family Statistics [Java]
 Four Letters
 Getting Ahead by Two Points
 Given the Probability, Find the Sample Space
 Gladiator Game
 How to Ask an Embarrassing Question
 Incidence of Breast Cancer
 Integer Rectangle [Java]
 Lewis Carroll's pillow problem [JavaScript]
 Lost Boarding Pass
 Lucky Contest Winners
 Marking And Breaking Sticks [JavaScript]
 Matching Socks [JavaScript]
 Mathematics and Biology [Java]
 Misuse and Misconception of Statistics
 Monty Hall Dilemma
 Multiple of 3 out of the Box
 Parrondo Paradox [Java]
 Pauling's joke
 Pencil's Logo
 Probabilities in a Painted Cube
 Probability and Infinity
 Probability of Degenerate Random Matrix in Z(2)
 Probability of Increasing Sequence
 Probability of Two Integers Being Comprime [JavaScript]
 Random Clock Hands [Java]
 Rectangle on a Chessboard [Java]
 Short Runs from an Urn
 Sick Child and Doctor
 Simpson's paradox
 Probability of Divisibility
 Probability of Two Integers Being Coprime
 Sample Probability Problems from AMC
 Shuffling Probability
 The 2016 ARML Competition, Problem 7
 Three pancakes problem [JavaScript]
 Two Envelopes Paradox
 Two Friends Meeting
 Two Solutions: One Correct, One Illuminating. An Example
 Weighted Dice Problem [JavaScript]
 What is the Color of the Remaining Ball? [JavaScript]
Contact Front page Contents Probability Store
Copyright © 19962017 Alexander Bogomolny
62040698 