Biased Coin, Fair Game?

Updating coin-bias probability after observing flips

This is a medium-difficulty probability question that tests your ability to combine prior uncertainty about a parameter with observed evidence to produce a posterior prediction. It is a canonical application of Bayesian inference and appears frequently in quant interviews because it rewards clean reasoning over computation.

The core challenge is recognizing that the coin's bias is not fixed in advance—it is itself random, drawn from a known prior distribution. After you observe the outcomes of the first two flips, you must update your beliefs about what that bias likely is, then use the updated distribution to predict the third flip. This requires you to combine the law of total probability with Bayes' theorem in a way that is both mathematically sound and intuitively transparent.

• Prior and posterior distributions over parameters
• Likelihood of observations under different parameter values
• Law of total probability and marginalization
• Prediction under parameter uncertainty