Waiting for Patterns: HT vs. HH

Expected waiting time for coin-flip patterns: HT vs. HH

This is an easy-to-medium probability problem that tests your intuition about waiting times for patterns in a sequence of independent trials. It appears frequently in quant interviews because the answer is counterintuitive and reveals whether you can reason carefully about state-based processes rather than rely on symmetry arguments.

To solve it, you need to set up equations for the expected number of flips required to see each pattern, accounting for partial progress. The key insight is that some failed attempts at one pattern leave you closer to success than failed attempts at another. This requires thinking through state transitions—when you flip a coin, how much of your target pattern are you carrying forward toward the next flip?

• Markov chains and state-based recurrence relations
• Partial overlaps and reset states
• Why symmetry can be misleading in pattern-waiting problems