Ten Heads: Fair or Double-Headed?

This classic quant interview question is about how evidence reshapes beliefs when there are competing explanations. You start with two different types of coins in the population and then see an extreme outcome. The core issue is how strongly that outcome should make you favor the "special" coin over an ordinary one. It captures the heart of conditional probability and how prior information and observed data interact.

It trains Bayesian thinking, posterior reasoning, and comfort with updating probabilities after seeing unlikely strings of events. It also sharpens intuition about how quickly odds can move when an event is far more consistent with one hypothesis than another. As part of quant prep, it reinforces clear separation between prior, likelihood, and updated belief.

This matters in quant interviews because real trading and risk problems rarely come with obvious models. You must infer hidden states and parameters from noisy, sometimes extreme data. Interviewers use this style of question to test whether you can turn intuitive notions of "surprising" observations into rigorous conditional probabilities. Strong performance here signals that you can handle model calibration, regime detection, and statistical inference under uncertainty in actual quant roles.

What it tests

This problem class is governed by Bayesian inference, where we update our beliefs about an underlying cause (which type of coin was chosen) in light of observed evidence (the sequence of heads). The core structure is that each hypothesis (which coin you picked) predicts the observed data with a certain likelihood, and the posterior probability for each hypothesis is proportional to its prior probability times its likelihood. The reason this pattern holds is that observing rare events under one hypothesis (like 10 heads from a fair coin) makes alternative explanations (like a biased coin) much more plausible, even if they were initially unlikely. The key is that the evidence is much more probable under some hypotheses than others, so the posterior can shift dramatically from the prior.