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. Average Visibility of Moviegoers
  10. Averaging Raindrops - an exercise in geometric probability
  11. Balls of Two Colors
  12. Balls of Two Colors II
  13. Barycentric Coordinates and Geometric Probability
  14. Bear cubs problem
  15. Bear Born on a Tuesday
  16. Benford's Law and Zipf's Law
  17. Bertrand's Paradox [Java]
  18. Birds On a Wire [Java]
  19. Birthday Coincidence
  20. Book Index Range
  21. Buffon's Needle Problem
  22. Buffon's Noodle [Java]
  23. Careless Mailing Clerk
  24. Checkmate Puzzle
  25. Chess Players Truel [Java]
  26. Chevalier de Méré's Problem
  27. Clubs or no Clubs
  28. Coin Tossing Contest
  29. Crossing a River after a Storm
  30. Determinants in $\mathbb{Z}_2$
  31. Diminishing Hopes
  32. Family Size [JavaScript]
  33. Family Statistics [Java]
  34. Four Letters
  35. Getting Ahead by Two Points
  36. Given the Probability, Find the Sample Space
  37. Gladiator Game
  38. Hemisphere Coverage
  39. How to Ask an Embarrassing Question
  40. Incidence of Breast Cancer
  41. Integer Rectangle [Java]
  42. Lewis Carroll's pillow problem [JavaScript]
  43. Loaded Dice
  44. Lost Boarding Pass
  45. Lucky Contest Winners
  46. Marking And Breaking Sticks [JavaScript]
  47. Matching Socks [JavaScript]
  48. Mathematics and Biology [Java]
  49. Misuse and Misconception of Statistics
  50. Monty Hall Dilemma
  51. Multiple of 3 out of the Box
  52. Numbered Balls Out Of a Box
  53. Numbers in a Square
  54. Odds and Chances in Horse Race Betting
  55. Overlapping Random Intervals
  56. Parrondo Paradox [Java]
  57. Pauling's joke
  58. Pencil's Logo
  59. Practical Inevitability of Clustering
  60. Practical Inevitability of Empty Spaces
  61. Probabilities in a Painted Cube
  62. Probability à la Tristram Shandy
  63. Probability and Infinity
  64. Probability of $2^n$ Beginning with Digit $1$
  65. Probability of Four Random Integers Having a Common Factor
  66. Probability of a Cube Ending with 11
  67. Probability of Degenerate Random Matrix in Z(2)
  68. Probability of Increasing Sequence
  69. Probability of Two Integers Being Comprime [JavaScript]
  70. Random Clock Hands [Java]
  71. Recollecting Forgotten Digit
  72. Rectangle on a Chessboard [Java]
  73. Semicircle Coverage
  74. Short Runs from an Urn
  75. Sick Child and Doctor
  76. Simpson's paradox
  77. Probability of Divisibility
  78. Probability of Two Integers Being Coprime
  79. Probability of Visiting Grandparents
  80. Sample Probability Problems from AMC
  81. Shuffling Probability
  82. Simulating Probabilities
  83. Six Numbers, One Inequality
  84. Six Numbers, Two Inequalities
  85. Six Numbers, Three Inequalities
  86. The 2016 ARML Competition, Problem 7
  87. Three pancakes problem [JavaScript]
  88. Three Random Points on a Circle
  89. Two Envelopes Paradox
  90. Two Friends Meeting
  91. Two Solutions: One Correct, One Illuminating. An Example
  92. Tying Knots In Brazil
  93. Waiting for an Ace
  94. Weighted Dice Problem [JavaScript]
  95. What is the Color of the Remaining Ball? [JavaScript]

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