Who’s in the Lead?

The arcsine law and the ballot theorem

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• In fair games, one player usually leads most of the time (the arcsine law).
• Reflection principle: paths touching zero match paths from the mirror-image start.
• Ballot theorem: the winner leads throughout with probability $\frac{a-b}{a+b}$.

Steps

1. A long coin-flip game
2. The arcsine law
3. Lopsided or even?
4. Reflecting paths
5. Election night
6. Never level again