Card, Dice & Coin Problems

Read these first

1. 01Kelly Sizing in a Coin-Flip Game

   Given a biased coin and permission to bet any fraction of your bankroll on repeated flips, how much should you actually bet? Too little leaves growth on the table; too much and you eventually blow up. The Kelly fraction is the exact answer.

2. 02Rerolling a Die: When to Take the Number

   You roll a die and can either keep the number shown or pay a cost to reroll, once or repeatedly. The optimal rule is a threshold: keep any number above a cutoff, reroll anything below it, and the cutoff is just an expected-value comparison.

3. 03The Red-Black Card Stopping Game

   A shuffled deck is turned over one card at a time. You can stop whenever you want and bet the next card is red. Is there a strategy that beats 50/50, and if so, what edge does it give you?

Then the rest

1. 04Counting Poker Hands
2. 05Craps: Working Out the House Edge
3. 06Detecting a Biased Coin From Few Flips
4. 07Expected Cards Until the First Ace
5. 08Expected Longest Streak of Heads
6. 09Expected Maximum of Several Dice
7. 10How Many Riffle Shuffles Randomise a Deck
8. 11Hypergeometric Draws From a Deck
9. 12Non-Transitive Dice
10. 13Optimal Card Guessing When You Count What Is Gone
11. 14Sicherman Dice
12. 15Simulating a Die With a Coin and Back Again
13. 16The Dice Game Pig and Optimal Stopping
14. 17Two Decks Turned Over Together
15. 18Why the Doubling Betting System Fails