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
- Non-transitive 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 50-50 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?
1 | 2 | 3 |
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
- Average Visibility of Moviegoers
- Averaging Raindrops - an exercise in geometric probability
- Balls of Two Colors
- Balls of Two Colors II
- 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
- Black Boxes in a Chain
- 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
- Chickens in Boxes
- Choosing the Largest Random Number
- Clubs or no Clubs
- Coin Tossing Contest
- Concerning Even Number of Heads
- Crossing a River after a Storm
- Determinants in $\mathbb{Z}_2$
- Diminishing Hopes
- Dropping Numbers into a 3x3 Square
- Family Size [JavaScript]
- Family Statistics [Java]
- Flat Probabilities on a Sphere
- Four Letters
- Getting Ahead by Two Points
- Given the Probability, Find the Sample Space
- Gladiator Game
- Hemisphere Coverage
- How to Ask an Embarrassing Question
- Incidence of Breast Cancer
- Integer Rectangle [Java]
- Lewis Carroll's pillow problem [JavaScript]
- Loaded Dice
- Loaded Dice II
- Lost Boarding Pass
- Lucky Contest Winners
- Marking And Breaking Sticks [JavaScript]
- Matching Socks [JavaScript]
- Mathematics and Biology [Java]
- Metamorphosis of a Quadratic Function
- Misuse and Misconception of Statistics
- Monty Hall Dilemma
- Multiple of 3 out of the Box
- Numbered Balls Out Of a Box
- Numbers in a Square
- Odds and Chances in Horse Race Betting
- Overlapping Random Intervals
- Parrondo Paradox [Java]
- Pauling's joke
- Pencil's Logo
- Points on a Square Grid
- Practical Inevitability of Clustering
- Practical Inevitability of Empty Spaces
- Probabilities in a Painted Cube
- Probability à la Tristram Shandy
- Probability and Infinity
- Probability of $2^n$ Beginning with Digit $1$
- Probability of an Odd Number of Sixes
- Probability of Four Random Integers Having a Common Factor
- Probability of a Cube Ending with 11
- Probability of Degenerate Random Matrix in Z(2)
- Probability of Increasing Sequence
- Probability of Two Integers Being Comprime [JavaScript]
- Quotient Estimates
- Quotient Estimates II
- Random Clock Hands [Java]
- Random Intervals with One Dominant
- Recollecting Forgotten Digit
- Rectangle on a Chessboard [Java]
- Red And Green Balls in Red And Green Boxes
- Rolling a Die
- Semicircle Coverage
- Short Runs from an Urn
- Sick Child and Doctor
- Simpson's paradox
- Probability of Divisibility
- Probability of Two Integers Being Coprime
- Probability of Visiting Grandparents
- Probability with Factorials
- Sample Probability Problems from AMC
- Shuffling Probability
- Simulating Probabilities
- Six Numbers, One Inequality
- Six Numbers, Two Inequalities
- Six Numbers, Three Inequalities
- The 2016 ARML Competition, Problem 7
- The Coffee Shop Game
- The Expected Number of Fixed Points
- The Marriage Problem
- Three pancakes problem [JavaScript]
- Three Random Points on a Circle
- To Bet Or Not To Bet
- Two Envelopes Paradox
- Two Friends Meeting
- Two Solutions: One Correct, One Illuminating. An Example
- Tying Knots In Brazil
- Waiting for an Ace
- Waiting for Multiple Heads
- Weighted Dice Problem [JavaScript]
- What is the Color of the Remaining Ball? [JavaScript]
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