Probability · Maths EE idea · Solid
Penney's game: beating any coin sequence
A research question to start from
Why can the second player in Penney's game always choose a three-coin sequence that is more likely to appear first, and with what probability do they win?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A non-transitive surprise you can calculate exactly with states.
Mathematics you would need
- Conditional probability and states
- Expected waiting times
- Non-transitivity
- Conway's leading numbers (optional)
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Compute waiting times for each sequence.
- Compute head-to-head win probabilities.
- Prove the second player's strategy works in every case.
Scope and difficulty
Solid. Solid.
Pitfalls
- Simulation only.
- Confusing waiting time with winning probability.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Penney's game probabilities; Conway algorithm leading numbers. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).