Games, betting and optimal stopping questions

40 problems

Card games, auctions, betting strategies and when to stop. Work backwards from the last move, find the indifference point, and check what the optimal strategy is worth. Each problem has hints and a worked solution, free, no account.

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28 One free restart hard
33 Chuck-a-luck easy
39 Calling the colour of each card medium
52 Guess my number for money medium
53 Hiring the best of five medium
61 Paying to open boxes easy
74 Three cards in a row medium
77 A book you cannot lose easy
89 The option to buy after inspection easy
98 Double six for a hundred easy
99 St Petersburg with a finite bank medium
100 Accept or pass the die medium
102 A d30 against a d20 with a reroll easy
113 Poker with three cards hard
128 Rock-paper-scissors without rock medium
139 Seven pays, even costs easy
141 Down two with one shot left easy
144 Two-finger Morra medium
147 Up to three rolls medium
155 Red pays, black costs hard
163 Split and discard the smaller pile medium
171 How much to bet easy
174 Bidding on what the seller knows easy
177 Sealed bids, three bidders medium
189 A wand that turns two coins over easy
192 The option to double the stake medium
200 A hundred-sided die, one dollar per reroll medium
202 Betting on the series one game at a time medium
217 Keep rolling while it is high easy
222 Four doors, two chances to switch medium
234 Redraw or keep hard
243 Nim with 3, 4 and 5 easy
245 Guess the total, win the total easy
246 Two tetrahedral dice until a tie easy
249 Four strange dice medium
251 The two best of three dice medium
253 Closest to a loaded die medium
257 A three-way duel hard
265 Red balls pay, the yellow one ruins medium
273 Doubles wipe you out medium

Other topics: probability · random processes · order statistics · inference · regression · linear algebra · puzzles · all problems