Probability Puzzles

Read these first

1. 01First-Step Analysis: Conditioning on What Happens Next

   Most "expected number of flips/steps until X" puzzles look intractable until you ask one question: what happens after the very first move, and can I write the answer in terms of itself?

2. 02Linearity of Expectation, Even Without Independence

   The expected value of a sum always equals the sum of the expected values, even when the pieces are wildly dependent on each other. That one fact quietly solves half the hard-looking counting puzzles in an interview.

3. 03The Ballot Problem and the Reflection Principle

   Candidate A beats candidate B in the final count, what's the chance A was strictly ahead for the entire tally, from the very first vote counted? A clever mirror-image trick turns a hard counting problem into simple subtraction.

4. 04Waiting Time Until a Pattern Appears

   HTH and HHT are both three-flip patterns with 1/8 probability each, so why does one take longer, on average, to see than the other? The pattern's own internal structure is the hidden variable.

5. 05Solving Waiting-Time Puzzles With Optional Stopping

   Instead of building a state machine, imagine an army of gamblers betting on every flip, the optional stopping theorem turns their fair-game payouts into the waiting time itself, with almost no algebra.

Then the rest

1. 06Collision Bounds Beyond the Birthday Problem
2. 07Expected Maximum of n Uniform Draws
3. 08First-Return Probabilities of a Random Walk
4. 09How Long the Gambler's Ruin Game Lasts
5. 10Penney's Game and Non-Transitive Patterns
6. 11Polya's Urn and Self-Reinforcing Draws
7. 12Simpson's Paradox as an Interview Trap
8. 13Symmetry Shortcuts for Draws Without Replacement
9. 14The Hat-Check Problem and Expected Fixed Points
10. 15The Indicator-Variable Trick
11. 16The Inspection Paradox and the Waiting-for-a-Bus Trap
12. 17The Problem of Points: Splitting an Interrupted Bet
13. 18The Rare-Disease Test Puzzle
14. 19Why HHH Takes Longer to Appear Than HHT
15. 20Monty Hall When the Host Behaves Differently
16. 21The Prosecutor's Fallacy
17. 22The Two-Children Problem