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. Black Boxes in a Chain
  21. Book Index Range
  22. Buffon's Needle Problem
  23. Buffon's Noodle [Java]
  24. Careless Mailing Clerk
  25. Checkmate Puzzle
  26. Chess Players Truel [Java]
  27. Chevalier de Méré's Problem
  28. Chickens in Boxes
  29. Choosing the Largest Random Number
  30. Clubs or no Clubs
  31. Coin Tossing Contest
  32. Concerning Even Number of Heads
  33. Crossing a River after a Storm
  34. Determinants in $\mathbb{Z}_2$
  35. Diminishing Hopes
  36. Dropping Numbers into a 3x3 Square
  37. Family Size [JavaScript]
  38. Family Statistics [Java]
  39. Flat Probabilities on a Sphere
  40. Four Letters
  41. Getting Ahead by Two Points
  42. Given the Probability, Find the Sample Space
  43. Gladiator Game
  44. Hemisphere Coverage
  45. How to Ask an Embarrassing Question
  46. Incidence of Breast Cancer
  47. Integer Rectangle [Java]
  48. Lewis Carroll's pillow problem [JavaScript]
  49. Loaded Dice
  50. Loaded Dice II
  51. Lost Boarding Pass
  52. Lucky Contest Winners
  53. Marking And Breaking Sticks [JavaScript]
  54. Matching Socks [JavaScript]
  55. Mathematics and Biology [Java]
  56. Metamorphosis of a Quadratic Function
  57. Misuse and Misconception of Statistics
  58. Monty Hall Dilemma
  59. Multiple of 3 out of the Box
  60. Numbered Balls Out Of a Box
  61. Numbers in a Square
  62. Odds and Chances in Horse Race Betting
  63. Overlapping Random Intervals
  64. Parrondo Paradox [Java]
  65. Pauling's joke
  66. Pencil's Logo
  67. Points on a Square Grid
  68. Practical Inevitability of Clustering
  69. Practical Inevitability of Empty Spaces
  70. Probabilities in a Painted Cube
  71. Probability à la Tristram Shandy
  72. Probability and Infinity
  73. Probability of $2^n$ Beginning with Digit $1$
  74. Probability of an Odd Number of Sixes
  75. Probability of Four Random Integers Having a Common Factor
  76. Probability of a Cube Ending with 11
  77. Probability of Degenerate Random Matrix in Z(2)
  78. Probability of Increasing Sequence
  79. Probability of Two Integers Being Comprime [JavaScript]
  80. Quotient Estimates
  81. Quotient Estimates II
  82. Random Clock Hands [Java]
  83. Random Intervals with One Dominant
  84. Recollecting Forgotten Digit
  85. Rectangle on a Chessboard [Java]
  86. Red And Green Balls in Red And Green Boxes
  87. Rolling a Die
  88. Semicircle Coverage
  89. Short Runs from an Urn
  90. Sick Child and Doctor
  91. Simpson's paradox
  92. Probability of Divisibility
  93. Probability of Two Integers Being Coprime
  94. Probability of Visiting Grandparents
  95. Probability with Factorials
  96. Sample Probability Problems from AMC
  97. Shuffling Probability
  98. Simulating Probabilities
  99. Six Numbers, One Inequality
  100. Six Numbers, Two Inequalities
  101. Six Numbers, Three Inequalities
  102. The 2016 ARML Competition, Problem 7
  103. The Coffee Shop Game
  104. The Expected Number of Fixed Points
  105. The Marriage Problem
  106. Three pancakes problem [JavaScript]
  107. Three Random Points on a Circle
  108. To Bet Or Not To Bet
  109. Two Envelopes Paradox
  110. Two Friends Meeting
  111. Two Solutions: One Correct, One Illuminating. An Example
  112. Tying Knots In Brazil
  113. Waiting for an Ace
  114. Waiting for Multiple Heads
  115. Weighted Dice Problem [JavaScript]
  116. What is the Color of the Remaining Ball? [JavaScript]

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