Double-Headed Coin After One Head

This classic quant interview question is about updating beliefs once you see partial information. You start with several possible hidden states and then observe an outcome that is compatible with many of them, but not equally so. It is a clean, concrete setting that makes it natural to discuss uncertainty, evidence, and how information changes odds. Because the setup is simple, it is ideal for fast quant prep and live discussion during interviews at top trading firms.

It trains conditional probability, Bayesian reasoning, and comfort with prior and posterior probabilities. It forces you to distinguish between the chance of seeing the evidence and the chance of a cause being true given that evidence. It also trains candidates to translate a story into events and to reason carefully about likelihoods.

This matters for quant interviews because real trading and risk problems constantly involve inferring hidden states from noisy signals. Interviewers use this kind of question to see if you can think probabilistically under uncertainty, avoid intuitive traps, and articulate a clear, logically consistent argument. Strong performance here signals readiness for more advanced quant prep, including stochastic modeling, signal detection, and Bayesian forecasting used in quantitative finance roles.

What it tests

This problem class is governed by conditional probability and, more specifically, Bayesian inference. The core structure is that you observe an outcome that is more or less likely under different underlying causes (here, types of coins), and you want to update your belief about which cause is responsible. The key is that the probability of observing the evidence (the head) is not the same for each possible cause, so you must weight the prior probability of each cause by the likelihood of the evidence under that cause. This is why Bayes' Theorem is the natural tool: it formalizes how to combine prior beliefs with new evidence to get a posterior probability. The pattern holds because, whenever outcomes are not equally likely across hypotheses, the observed evidence shifts the odds in favor of the hypotheses that make the evidence more probable.