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 Coin Tossing Surprise I
  3. A Fair Game of Chance
  4. A Pair of Probability Games for Beginners
  5. Problem 25 from the Spring 2018 Mathcounts
  6. Problem 8 from the Spring 2018 Mathcounts
  7. A Problem of Three Liars
  8. A Problem of Two Liars
  9. A Proof by Game for a Sum of a Convergent Series
  10. A Question about the Median
  11. A Search for heads and Its Consequences - After Miller And Sanjurjo
  12. A Three Group Split
  13. A Triangle out of Three Broken Sticks
  14. Acting As a Team I
  15. Amoeba's Survival
  16. Are Most Triangles Obtuse?
  17. Artificially Unintelligent
  18. Aspiring Tennis Club Candidate
  19. Average Number of Runs
  20. Average Number of Runs in a Sequence of Random Numbers
  21. Average Visibility of Moviegoers
  22. Averaging Raindrops - an exercise in geometric probability
  23. Averages of Terms in Increasing Sequence
  24. Balls of Two Colors
  25. Balls of Two Colors II
  26. Barycentric Coordinates and Geometric Probability
  27. Bear cubs problem
  28. Bear Born on a Tuesday
  29. Benford's Law and Zipf's Law
  30. Bertrand's Paradox [Java]
  31. Birds On a Wire [Java]
  32. Birthday Coincidence
  33. Black Boxes in a Chain
  34. Book Index Range
  35. Bubbling of Sorts
  36. Buffon's Needle Problem
  37. Buffon's Noodle [Java]
  38. Careless Mailing Clerk
  39. Checkmate Puzzle
  40. Chess Players Truel [Java]
  41. Chevalier de Méré's Problem
  42. Chickens in Boxes
  43. Choosing the Largest Random Number
  44. Clubs or no Clubs
  45. Clumps on a One Lane Road
  46. Coin Tossing Contest
  47. Concerning Even Number of Heads
  48. Converting Temperature From C° to $F^{\circ}$
  49. Crossing a River after a Storm
  50. Crossing Bridge in Crowds
  51. Determinants in $\mathbb{Z}_2$
  52. Diminishing Hopes
  53. Diminishing Hopes II
  54. Distributing Balls of Two Colors in Two Bags
  55. Dropping Numbers into a 3x3 Square
  56. Expectation of Interval Length on Circle
  57. Expectation of Pairings
  58. Expectation of the Largest Number
  59. Expected Number of Happy Passengers
  60. Fair Duel
  61. Family Size [JavaScript]
  62. Family Statistics [Java]
  63. Flat Probabilities on a Sphere
  64. Four Letters
  65. Four Random Points on a Sphere
  66. Galton's Paradox
  67. Gambling in a Company
  68. Getting Ahead by Two Points
  69. Getting from A to B via C
  70. Given the Probability, Find the Sample Space
  71. Gladiator Game
  72. Guessing Hat Numbers
  73. Hemisphere Coverage
  74. How Long Will It Last?
  75. How to Ask an Embarrassing Question
  76. In Praise of Odds
  77. Incidence of Breast Cancer
  78. Integer Rectangle [Java]
  79. Integer Sequence with Given Statistical Parameters
  80. Number of Trials to First Success
  81. Lewis Carroll's pillow problem [JavaScript]
  82. Lights on a Christmas Tree
  83. Loaded Dice
  84. Loaded Dice II
  85. Losing Socks Over a Year
  86. Lost Boarding Pass
  87. Lucky Contest Winners
  88. Lucky Times at a Moscow Math Olympiad
  89. Marking And Breaking Sticks [JavaScript]
  90. Matching Socks [JavaScript]
  91. Mathematics and Biology [Java]
  92. Metamorphosis of a Quadratic Function
  93. Matching Socks in Dark Room
  94. Misuse and Misconception of Statistics
  95. Monty Hall Dilemma
  96. Multiple of 3 out of the Box
  97. Number of Wire Loops
  98. Numbered Balls Out Of a Box
  99. Numbers in a Square
  100. Odds and Chances in Horse Race Betting
  101. Overlapping Random Intervals
  102. Parrondo Paradox [Java]
  103. Pauling's joke
  104. Pencil's Logo
  105. Planting Trees in a Row
  106. Playing with Balls of Two Colors
  107. Playing with Integers and Limits
  108. Points in a Semicircle
  109. Points on a Square Grid
  110. Practical Inevitability of Clustering
  111. Practical Inevitability of Empty Spaces
  112. Probability à la Tristram Shandy
  113. Probability of an Odd Number of Sixes
  114. Probability and Infinity
  115. Probabilities in a Painted Cube
  116. Probability in Dart Throwing
  117. Probability in Scoring
  118. Probability in the World Series
  119. Probability in Triangle
  120. Probability of $2^n$ Beginning with Digit $1$
  121. Probability of Equal Areas on a Chessboard
  122. Probability of First Digits in a Sequence of Powers
  123. Probability of Four Random Integers Having a Common Factor
  124. Probability of a Cube Ending with 11
  125. Probability of a Meet in an Elimination Tournament
  126. Probability of a Random Inequality
  127. Probability of Average
  128. Probability of Degenerate Random Matrix in Z(2)
  129. Probability of Divisibility
  130. Probability of Doubles
  131. Probability of Equilateral Triangle
  132. Probability of First Digit in Product
  133. Probability of Having 5 in the Numerator
  134. Probability of Majorization II
  135. Probability of Two Integers Being Coprime
  136. Probability of Visiting Grandparents
  137. Probability with Factorials
  138. Probability of Increasing Sequence
  139. Probability of No Distinct Positive Roots
  140. Probability of No Two-Tail Runs
  141. Probability of Random Lines Crossing
  142. Probability of Successive Integers
  143. Probability of the Second Marble
  144. Probability of Two Integers Being Comprime [JavaScript]
  145. Quotient Estimates
  146. Quotient Estimates II
  147. Random Arithmetic Progressions
  148. Random Clock Hands [Java]
  149. Random Intervals with One Dominant
  150. Random Numbers And Obtuse Triangle
  151. Random Points on a Segment
  152. Random Sum
  153. Randomly Placed Letters in Envelopes
  154. Recollecting Forgotten Digit
  155. Rectangle on a Chessboard [Java]
  156. Red And Green Balls in Red And Green Boxes
  157. Red Faces of a Cube
  158. Right Strategy for a Weaker Player
  159. Rolling a Die
  160. Rolling Defective Die
  161. Semicircle Coverage
  162. Short Runs from an Urn
  163. Sick Child and Doctor
  164. Simpson's paradox
  165. Snake Permutations And Their Number
  166. Sample Probability Problems from AMC
  167. Shelving an Encyclopaedia
  168. Shuffling Probability
  169. Simulating Probabilities
  170. Six Numbers, One Inequality
  171. Six Numbers, Two Inequalities
  172. Six Numbers, Three Inequalities
  173. Taking Chances with Your Medicine
  174. Taking Turns to Toss a Die
  175. The 2016 ARML Competition, Problem 7
  176. The Coffee Shop Game
  177. The Expected Number of Fixed Points
  178. The Marriage Problem
  179. The Most Likely Position
  180. Three pancakes problem [JavaScript]
  181. Three Random Points on a Circle
  182. To Bet Or Not To Bet
  183. Training Bicyclists on a Mountain Road
  184. Two 6s in a Row
  185. Two Balls of the Same Color
  186. Two Coins: One Fair, one Biased
  187. Two Balls Out Of Four
  188. Two Dice Repetition
  189. Two Envelopes Paradox
  190. Two Friends Meeting
  191. Two in a Row
  192. Two Solutions: One Correct, One Illuminating. An Example
  193. Two Varsity Divisions
  194. Tying Knots In Brazil
  195. Waiting for a Larger Number
  196. Waiting for a Train
  197. Waiting for All Six Outcomes
  198. Waiting for an Ace
  199. Waiting for Multiple Heads
  200. Waiting to Exceed 1
  201. Walking Randomly - How Far?
  202. Weighted Dice Problem [JavaScript]
  203. What is the Color of the Remaining Ball? [JavaScript]

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