Mystery Coin

What this probability question tests

This is a medium-difficulty probability question that appears frequently on quant trading and finance interviews. It tests your ability to set up and solve equations involving infinite geometric series and conditional probability in a sequential game.

The core skill is recognizing that when two players alternate turns with different success probabilities, you must account for all possible paths to victory—including scenarios where many rounds occur before either player wins. You'll need to balance Joe's first-mover advantage against his unknown coin bias, then use the symmetry condition (equal winning probabilities) to solve for the unknown parameter.

• Geometric series and infinite sums in probability
• Turn-based game analysis and recursive probability
• Setting up equations from symmetry or fairness constraints