The Biased Signal

What this Bayesian inference interview question tests

This is a medium-difficulty probability question that asks you to update your belief about an unknown parameter in light of observed data. It is a canonical example of Bayesian reasoning, a cornerstone of quantitative finance and the kind of thinking trading firms like Two Sigma use to evaluate signal quality.

The setup gives you two competing hypotheses (fair coin vs. biased coin), a prior belief about each, and then sample data. Your task is to compute a posterior probability using Bayes' theorem—that is, to reverse the direction of conditional probability. You must calculate the likelihood of the observed data under each hypothesis, weight by the prior, and normalize. This tests whether you can reason clearly about model comparison and evidence.

• Prior and posterior probabilities
• Likelihood calculation for a binomial outcome
• Bayes' theorem and model selection
• Normalizing to a valid posterior distribution