Probability Problems

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
The Orderly Pursuit of Pure Disorder.
Discover, January, 1981

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 1, 2, 3 numbers. This is how the function works.

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 1 ≤ n ≤ 3. The formula is

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

  1. 100 Prisoners and a Light Bulb
  2. A Fair Game of Chance
  3. A Pair of Probability Games for Beginners
  4. A Proof by Game for a Sum of a Convergent Series
  5. Amoeba's Survival
  6. Are Most Triangles Obtuse?
  7. Aspiring Tennis Club Candidate
  8. Average Number of Runs
  9. Averaging Raindrops - an exercise in geometric probability
  10. Balls of Two Colors
  11. Barycentric Coordinates and Geometric Probability
  12. Bear cubs problem
  13. Bear Born on a Tuesday
  14. Benford's Law and Zipf's Law
  15. Bertrand's Paradox [Java]
  16. Birds On a Wire [Java]
  17. Birthday Coincidence
  18. Book Index Range
  19. Buffon's Needle Problem
  20. Buffon's Noodle [Java]
  21. Careless Mailing Clerk
  22. Checkmate Puzzle
  23. Chess Players Truel [Java]
  24. Chevalier de Méré's Problem
  25. Clubs or no Clubs
  26. Diminishing Hopes
  27. Family Size [JavaScript]
  28. Family Statistics [Java]
  29. Four Letters
  30. Getting Ahead by Two Points
  31. Given the Probability, Find the Sample Space
  32. Gladiator Game
  33. How to Ask an Embarrassing Question
  34. Incidence of Breast Cancer
  35. Integer Rectangle [Java]
  36. Lewis Carroll's pillow problem [JavaScript]
  37. Lost Boarding Pass
  38. Lucky Contest Winners
  39. Marking And Breaking Sticks [JavaScript]
  40. Matching Socks [JavaScript]
  41. Mathematics and Biology [Java]
  42. Misuse and Misconception of Statistics
  43. Monty Hall Dilemma
  44. Multiple of 3 out of the Box
  45. Parrondo Paradox [Java]
  46. Pauling's joke
  47. Pencil's Logo
  48. Probabilities in a Painted Cube
  49. Probability and Infinity
  50. Probability of Degenerate Random Matrix in Z(2)
  51. Probability of Increasing Sequence
  52. Probability of Two Integers Being Comprime [JavaScript]
  53. Random Clock Hands [Java]
  54. Rectangle on a Chessboard [Java]
  55. Short Runs from an Urn
  56. Sick Child and Doctor
  57. Simpson's paradox
  58. Probability of Divisibility
  59. Probability of Two Integers Being Coprime
  60. Sample Probability Problems from AMC
  61. Shuffling Probability
  62. The 2016 ARML Competition, Problem 7
  63. Three pancakes problem [JavaScript]
  64. Two Envelopes Paradox
  65. Two Friends Meeting
  66. Two Solutions: One Correct, One Illuminating. An Example
  67. Weighted Dice Problem [JavaScript]
  68. What is the Color of the Remaining Ball? [JavaScript]

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