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

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