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 Coin Tossing Surprise I
- A Fair Game of Chance
- A Pair of Probability Games for Beginners
- Problem 25 from the Spring 2018 Mathcounts
- Problem 8 from the Spring 2018 Mathcounts
- A Problem of Three Liars
- A Problem of Two Liars
- A Proof by Game for a Sum of a Convergent Series
- A Question about the Median
- A Search for heads and Its Consequences - After Miller And Sanjurjo
- A Three Group Split
- A Triangle out of Three Broken Sticks
- Acting As a Team I
- Amoeba's Survival
- Are Most Triangles Obtuse?
- Artificially Unintelligent
- Aspiring Tennis Club Candidate
- Average Number of Runs
- Average Number of Runs in a Sequence of Random Numbers
- Average Visibility of Moviegoers
- Averaging Raindrops - an exercise in geometric probability
- Averages of Terms in Increasing Sequence
- 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
- Bubbling of Sorts
- 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
- Clumps on a One Lane Road
- Coin Tossing Contest
- Concerning Even Number of Heads
- Converting Temperature From C° to $F^{\circ}$
- Crossing a River after a Storm
- Crossing Bridge in Crowds
- Determinants in $\mathbb{Z}_2$
- Diminishing Hopes
- Diminishing Hopes II
- Distributing Balls of Two Colors in Two Bags
- Dropping Numbers into a 3x3 Square
- Expectation of Interval Length on Circle
- Expectation of Pairings
- Expectation of the Largest Number
- Expected Number of Happy Passengers
- Fair Duel
- Family Size [JavaScript]
- Family Statistics [Java]
- Flat Probabilities on a Sphere
- Four Letters
- Four Random Points on a Sphere
- Galton's Paradox
- Gambling in a Company
- Getting Ahead by Two Points
- Getting from A to B via C
- Given the Probability, Find the Sample Space
- Gladiator Game
- Guessing Hat Numbers
- Hemisphere Coverage
- How Long Will It Last?
- How to Ask an Embarrassing Question
- In Praise of Odds
- Incidence of Breast Cancer
- Integer Rectangle [Java]
- Integer Sequence with Given Statistical Parameters
- Number of Trials to First Success
- Lewis Carroll's pillow problem [JavaScript]
- Lights on a Christmas Tree
- Loaded Dice
- Loaded Dice II
- Losing Socks Over a Year
- Lost Boarding Pass
- Lucky Contest Winners
- Lucky Times at a Moscow Math Olympiad
- Marking And Breaking Sticks [JavaScript]
- Matching Socks [JavaScript]
- Mathematics and Biology [Java]
- Metamorphosis of a Quadratic Function
- Matching Socks in Dark Room
- Misuse and Misconception of Statistics
- Monty Hall Dilemma
- Multiple of 3 out of the Box
- Number of Wire Loops
- 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
- Planting Trees in a Row
- Playing with Balls of Two Colors
- Playing with Integers and Limits
- Points in a Semicircle
- Points on a Square Grid
- Practical Inevitability of Clustering
- Practical Inevitability of Empty Spaces
- Probability à la Tristram Shandy
- Probability of an Odd Number of Sixes
- Probability and Infinity
- Probabilities in a Painted Cube
- Probability in Dart Throwing
- Probability in Scoring
- Probability in the World Series
- Probability in Triangle
- Probability of $2^n$ Beginning with Digit $1$
- Probability of Equal Areas on a Chessboard
- Probability of First Digits in a Sequence of Powers
- Probability of Four Random Integers Having a Common Factor
- Probability of a Cube Ending with 11
- Probability of a Meet in an Elimination Tournament
- Probability of a Random Inequality
- Probability of Average
- Probability of Degenerate Random Matrix in Z(2)
- Probability of Divisibility
- Probability of Doubles
- Probability of Equilateral Triangle
- Probability of First Digit in Product
- Probability of Having 5 in the Numerator
- Probability of Majorization II
- Probability of Two Integers Being Coprime
- Probability of Visiting Grandparents
- Probability with Factorials
- Probability of Increasing Sequence
- Probability of No Distinct Positive Roots
- Probability of No Two-Tail Runs
- Probability of Random Lines Crossing
- Probability of Successive Integers
- Probability of the Second Marble
- Probability of Two Integers Being Comprime [JavaScript]
- Quotient Estimates
- Quotient Estimates II
- Random Arithmetic Progressions
- Random Clock Hands [Java]
- Random Intervals with One Dominant
- Random Numbers And Obtuse Triangle
- Random Points on a Segment
- Random Sum
- Randomly Placed Letters in Envelopes
- Recollecting Forgotten Digit
- Rectangle on a Chessboard [Java]
- Red And Green Balls in Red And Green Boxes
- Red Faces of a Cube
- Right Strategy for a Weaker Player
- Rolling a Die
- Rolling Defective Die
- Semicircle Coverage
- Short Runs from an Urn
- Sick Child and Doctor
- Simpson's paradox
- Snake Permutations And Their Number
- Sample Probability Problems from AMC
- Shelving an Encyclopaedia
- Shuffling Probability
- Simulating Probabilities
- Six Numbers, One Inequality
- Six Numbers, Two Inequalities
- Six Numbers, Three Inequalities
- Taking Chances with Your Medicine
- Taking Turns to Toss a Die
- The 2016 ARML Competition, Problem 7
- The Coffee Shop Game
- The Expected Number of Fixed Points
- The Marriage Problem
- The Most Likely Position
- Three pancakes problem [JavaScript]
- Three Random Points on a Circle
- To Bet Or Not To Bet
- Training Bicyclists on a Mountain Road
- Two 6s in a Row
- Two Balls of the Same Color
- Two Coins: One Fair, one Biased
- Two Balls Out Of Four
- Two Dice Repetition
- Two Envelopes Paradox
- Two Friends Meeting
- Two in a Row
- Two Solutions: One Correct, One Illuminating. An Example
- Two Varsity Divisions
- Tying Knots In Brazil
- Waiting for a Larger Number
- Waiting for a Train
- Waiting for All Six Outcomes
- Waiting for an Ace
- Waiting for Multiple Heads
- Waiting to Exceed 1
- Walking Randomly - How Far?
- Weighted Dice Problem [JavaScript]
- What is the Color of the Remaining Ball? [JavaScript]
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