# What are the odds of winning this Thue-Morse coin flip game?

Ann and Bernard are playing a coin flip game. They take turns according to the Thue-Morse sequence (so they start with Ann, Bernard, Bernard, Ann, Bernard, Ann, Ann, Bernard, then the negation of that order, etc.) and the first person to flip heads wins.

What are the odds of Ann winning the game?

This is $$\frac{1}{2} + \frac{1}{16} + \frac{1}{64} + \frac{1}{128} + \frac{1}{1024} + \frac{1}{2048} + \ ... \ \approx 0.5875459663598924$$. But is there a way to determine an exact value (closed formula)?

The answer to your question is $$1-\tau$$, where $$\tau$$ is the Prouhet-Thue-Morse constant.