The Biased Coin Factory

What this Bayesian inference interview question tests

This is an easy probability question that appears frequently in quant interviews to assess whether a candidate can reason about posterior inference — that is, how to update beliefs about an unknown parameter (the coin's bias) in light of observed data.

The question requires combining two key ideas: recognizing that the coin's true bias is unknown and must be inferred from the flip outcomes, and then computing a predictive probability for the next flip given what you've learned. This is a canonical setup in Bayesian statistics, and the clean answer rewards clear thinking about conditional probability and the law of total expectation.

• Bayesian updating and posterior distributions
• Prior and likelihood specification
• Predictive probability and exchangeability