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


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.


  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. A Three Group Split
  6. Acting As a Team I
  7. Amoeba's Survival
  8. Are Most Triangles Obtuse?
  9. Aspiring Tennis Club Candidate
  10. Average Number of Runs
  11. Average Visibility of Moviegoers
  12. Averaging Raindrops - an exercise in geometric probability
  13. Balls of Two Colors
  14. Balls of Two Colors II
  15. Barycentric Coordinates and Geometric Probability
  16. Bear cubs problem
  17. Bear Born on a Tuesday
  18. Benford's Law and Zipf's Law
  19. Bertrand's Paradox [Java]
  20. Birds On a Wire [Java]
  21. Birthday Coincidence
  22. Black Boxes in a Chain
  23. Book Index Range
  24. Buffon's Needle Problem
  25. Buffon's Noodle [Java]
  26. Careless Mailing Clerk
  27. Checkmate Puzzle
  28. Chess Players Truel [Java]
  29. Chevalier de Méré's Problem
  30. Chickens in Boxes
  31. Choosing the Largest Random Number
  32. Clubs or no Clubs
  33. Clumps on a One Lane Road
  34. Coin Tossing Contest
  35. Concerning Even Number of Heads
  36. Converting Temperature From C° to $F^{\circ}$
  37. Crossing a River after a Storm
  38. Crossing Bridge in Crowds
  39. Determinants in $\mathbb{Z}_2$
  40. Diminishing Hopes
  41. Dropping Numbers into a 3x3 Square
  42. Expectation of Interval Length on Circle
  43. Expectation of the Largest Number
  44. Expected Number of Happy Passengers
  45. Family Size [JavaScript]
  46. Family Statistics [Java]
  47. Flat Probabilities on a Sphere
  48. Four Letters
  49. Four Random Points on a Sphere
  50. Gambling in a Company
  51. Getting Ahead by Two Points
  52. Given the Probability, Find the Sample Space
  53. Gladiator Game
  54. Guessing Hat Numbers
  55. Hemisphere Coverage
  56. How to Ask an Embarrassing Question
  57. In Praise of Odds
  58. Incidence of Breast Cancer
  59. Integer Rectangle [Java]
  60. Lewis Carroll's pillow problem [JavaScript]
  61. Loaded Dice
  62. Loaded Dice II
  63. Losing Socks Over a Year
  64. Lost Boarding Pass
  65. Lucky Contest Winners
  66. Marking And Breaking Sticks [JavaScript]
  67. Matching Socks [JavaScript]
  68. Mathematics and Biology [Java]
  69. Metamorphosis of a Quadratic Function
  70. Matching Socks in Dark Room
  71. Misuse and Misconception of Statistics
  72. Monty Hall Dilemma
  73. Multiple of 3 out of the Box
  74. Number of Wire Loops
  75. Numbered Balls Out Of a Box
  76. Numbers in a Square
  77. Odds and Chances in Horse Race Betting
  78. Overlapping Random Intervals
  79. Parrondo Paradox [Java]
  80. Pauling's joke
  81. Pencil's Logo
  82. Playing with Integers and Limits
  83. Points on a Square Grid
  84. Practical Inevitability of Clustering
  85. Practical Inevitability of Empty Spaces
  86. Probability à la Tristram Shandy
  87. Probability of an Odd Number of Sixes
  88. Probability and Infinity
  89. Probabilities in a Painted Cube
  90. Probability in the World Series
  91. Probability of $2^n$ Beginning with Digit $1$
  92. Probability of First Digits in a Sequence of Powers
  93. Probability of Four Random Integers Having a Common Factor
  94. Probability of a Cube Ending with 11
  95. Probability of a Meet in an Elimination Tournament
  96. Probability of a Random Inequality
  97. Probability in Dart Throwing
  98. Probability of Degenerate Random Matrix in Z(2)
  99. Probability of Divisibility
  100. Probability of First Digit in Product
  101. Probability of Two Integers Being Coprime
  102. Probability of Visiting Grandparents
  103. Probability with Factorials
  104. Probability of Increasing Sequence
  105. Probability of No Two-Tail Runs
  106. Probability of Random Lines Crossing
  107. Probability of the Second Marble
  108. Probability of Two Integers Being Comprime [JavaScript]
  109. Quotient Estimates
  110. Quotient Estimates II
  111. Random Clock Hands [Java]
  112. Random Intervals with One Dominant
  113. Random Numbers And Obtuse Triangle
  114. Random Sum
  115. Recollecting Forgotten Digit
  116. Rectangle on a Chessboard [Java]
  117. Red And Green Balls in Red And Green Boxes
  118. Red Faces of a Cube
  119. Right Strategy for a Weaker Player
  120. Rolling a Die
  121. Semicircle Coverage
  122. Short Runs from an Urn
  123. Sick Child and Doctor
  124. Simpson's paradox
  125. Snake Permutations And Their Number
  126. Sample Probability Problems from AMC
  127. Shelving an Encyclopaedia
  128. Shuffling Probability
  129. Simulating Probabilities
  130. Six Numbers, One Inequality
  131. Six Numbers, Two Inequalities
  132. Six Numbers, Three Inequalities
  133. Taking Chances with Your Medicine
  134. The 2016 ARML Competition, Problem 7
  135. The Coffee Shop Game
  136. The Expected Number of Fixed Points
  137. The Marriage Problem
  138. The Most Likely Position
  139. Three pancakes problem [JavaScript]
  140. Three Random Points on a Circle
  141. Two Coins: One Fair, one Biased
  142. To Bet Or Not To Bet
  143. Two Envelopes Paradox
  144. Two Friends Meeting
  145. Two in a Row
  146. Two Solutions: One Correct, One Illuminating. An Example
  147. Tying Knots In Brazil
  148. Waiting for a Larger Number
  149. Waiting for All Six Outcomes
  150. Waiting for an Ace
  151. Waiting for Multiple Heads
  152. Waiting to Exceed 1
  153. Walking Randomly - How Far?
  154. Weighted Dice Problem [JavaScript]
  155. What is the Color of the Remaining Ball? [JavaScript]

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