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126 questions · full solutions

Expected Value

Pricing games is the job, and these questions teach it: waiting times, optimal stopping, Wald's identity, and the reroll games that secretly rehearse option pricing.

  1. 01Price a Die RollEasy
  2. 02Flips to First HeadEasy
  3. 03Rolls to a SixEasy
  4. 04One Reroll AllowedMedium · Private
  5. 05Two Rerolls AllowedMedium · Private
  6. 06Collect All Six FacesMedium · Private
  7. 07First HHMedium · Private
  8. 08Expected MaximumMedium · Private
  9. 09The St. Petersburg GameMedium · Private
  10. 10The Two-Envelope ParadoxHard · Private
  11. 11Three Draws, Stop When You LikeHard · Private
  12. 12Draws Until the Sum Exceeds 1Hard · Private
  13. 13Two Distinct FacesEasy
  14. 14Cards Between the AcesEasy
  15. 15Lowest of Three DiceEasy
  16. 16Same-Color NeighborsEasy
  17. 17How Long Does the Rain Last?Easy
  18. 18Price the Carnival Dice GameEasy
  19. 19Price the ProductEasy
  20. 20First Repeated FaceMedium · Private
  21. 21Rolls to Reach TenMedium · Private
  22. 22Position of the First AceMedium · Private
  23. 23A Dollar a RollMedium · Private
  24. 24Paid the Whole SumMedium · Private
  25. 25Paid the Sum, Minus the SixMedium · Private
  26. 26Best of Four RollsMedium · Private
  27. 27Three Boxes, Keep or PassMedium · Private
  28. 28Stop on a Head CountMedium · Private
  29. 29Counting RunsMedium · Private
  30. 30Local Peaks in a Random OrderingMedium · Private
  31. 31How Many Distinct Values?Medium · Private
  32. 32Record HighsMedium · Private
  33. 33Walk Around the HexagonMedium · Private
  34. 34Equal Waits, Unequal RaceHard · Private
  35. 35Selling the HouseHard · Private
  36. 36Stop on a FractionHard · Private
  37. 37How Long Until Ruin or Riches?Hard · Private
  38. 38The Casino That Prices PatternsHard · Private
  39. 39Drift to a TargetHard · Private
  40. 40Profit Target or Stop-LossHard · Private
  41. 41Coming Home in Two DimensionsHard · Private
  42. 42The Longest StreakHard · Private
  43. 43Price a Raffle TicketEasy
  44. 44Balance the BetEasy
  45. 45First the Coin, Then the DiceEasy
  46. 46Reroll, For a FeeMedium · Private
  47. 47Bid Your NumberMedium · Private
  48. 48St. Petersburg, TamedMedium · Private
  49. 49Keep the Best TwoMedium · Private
  50. 50One Flip of InformationMedium · Private
  51. 51A Call Option on DiceMedium · Private
  52. 52The Offer Might VanishMedium · Private
  53. 53How Fast Does Kelly Double?Medium · Private
  54. 54Bidding on a LemonMedium · Private
  55. 55Call It, Then I ChooseMedium · Private
  56. 56Buying DrawsMedium · Private
  57. 57Price the Cubed RollMedium · Private
  58. 58Four Interviews, One OfferHard · Private
  59. 59The Offers Stay on the TableHard · Private
  60. 60Park or Drive OnHard · Private
  61. 61Tomorrow It's Worth LessHard · Private
  62. 62Quit After the First LossHard · Private
  63. 63Cash Out the DeckHard · Private
  64. 64You Against the ProphetHard · Private
  65. 65Shade the BidHard · Private
  66. 66The Winner OverpaysHard · Private
  67. 67Half Kelly by the NumbersHard · Private
  68. 68Turn One Into FiveHard · Private
  69. 69Bet Big or Bleed OutHard · Private
  70. 70Bank It or Roll AgainHard · Private
  71. 71Call Every ColorHard · Private
  72. 72Hide and SnipeHard · Private
  73. 73Log Utility Meets St. PetersburgHard · Private
  74. 74Is the Second Peek Worth It?Hard · Private
  75. 75Two Prices, One BuyerHard · Private
  76. 76Two Keeps, Five DrawsHard · Private
  77. 77Price the BluffHard · Private
  78. 78The Lowball BankerHard · Private
  79. 79Try the Unknown CoinHard · Private
  80. 80Reroll With Open CardsHard · Private
  81. 81Five Presses of the ButtonHard · Private
  82. 82The Doubling System, AuditedHard · Private
  83. 83Price the Series LengthHard · Private
  84. 84What a Stop-Loss Buys YouHard · Private
  85. 85Overbook the FlightHard · Private
  86. 86Pay Per LookHard · Private
  87. 87Pick Your PenaltyHard · Private
  88. 88Beat the Coin With a ThresholdHard · Private
  89. 89Chasing the Full SetHard · Private
  90. 90Odds and Evens, PricedHard · Private
  91. 91The Last True SignalHard · Private
  92. 92Quit or ClimbHard · Private
  93. 93The Press That BreaksEasy
  94. 94The Best Mark of a Short RunMedium · Private
  95. 95Two Regimes, Two PlaybooksMedium · Private
  96. 96Price the Deductible PolicyMedium · Private
  97. 97Pay to Jump the QueueMedium · Private
  98. 98Paid on the Log ScoreMedium · Private
  99. 99Split the JackpotMedium · Private
  100. 100Replace It While It Still EarnsHard · Private
  101. 101Minor Fix, Major OutageHard · Private
  102. 102The Shuttle Gap You Land InHard · Private
  103. 103Trade Through It or Stand Down?Hard · Private
  104. 104Join the Line or Walk AwayHard · Private
  105. 105Two Auctions, One BudgetHard · Private
  106. 106The Bonus Is a Call OptionHard · Private
  107. 107Fees Above the High-Water MarkHard · Private
  108. 108Price the Tower, Layer by LayerHard · Private
  109. 109Self-Insure What You Can CarryHard · Private
  110. 110Pick Your Variance by the Prize TableHard · Private
  111. 111Gamble Early, Bank LateHard · Private
  112. 112Two Quote Streams, One SearchHard · Private
  113. 113Going Back Costs TenHard · Private
  114. 114Two Buys Against Your GutHard · Private
  115. 115Drawdown of a Five-Step WalkHard · Private
  116. 116What the Stop Costs Your EdgeHard · Private
  117. 117Unwind It Today or Risk TomorrowHard · Private
  118. 118Two Bets Into One Risk BudgetHard · Private
  119. 119Worth a Retry, But Only Up HighHard · Private
  120. 120The Scoring Rule That Pays for BravadoHard · Private
  121. 121Price the Band on a Hundred FlipsHard · Private
  122. 122Your Bet Moves Your Own OddsHard · Private
  123. 123Pay to PeekMedium · Private
  124. 124Reroll One of TwoMedium · Private
  125. 125Hire the Best of a HundredHard · Private
  126. 126Ant Across the CubeMedium · Private