10 heads then two-headed coin chance

This question presents a simple-looking coin-flipping experiment designed to test how you update beliefs after seeing an unlikely run of outcomes. You are told that there is a large collection of mostly ordinary coins plus a single special coin, and that you pick one at random, flip it many times, and see only heads. The core task is to reason about how that streak should change your view of which type of coin you probably picked. Similar conditional probability and Bayesian updating puzzles appear in interviews for quant research, trading, and some data science roles, where interpreting surprising data correctly is central.

To tackle it, you need to translate the story into competing hypotheses about the underlying coin, attach prior probabilities to each hypothesis, and compute how likely the observed streak would be in each case. The solution leans on Bayes' theorem, likelihood ratios, and comfort with exponentials and very small probabilities. Interviewers watch whether you clearly distinguish prior and posterior, avoid intuitive traps like ignoring base rates, and articulate how accumulating evidence gradually overwhelms the initial odds. They may also probe follow-up variations to see if your reasoning generalizes.

What it tests

Whenever you must update your belief about which of several possible underlying causes produced an observed outcome, the governing structure is Bayesian inference. The key is to weigh the prior probability of each cause by how likely it is to produce the observed evidence, then normalize across all possibilities. This is not just about plugging into Bayes' theorem: the principle is that evidence which is much more likely under one hypothesis than another can rapidly outweigh even a small prior. The pattern holds because, mathematically, the posterior probability is proportional to the product of prior and likelihood, so rare observations (under most hypotheses) can drastically shift your beliefs toward the hypothesis where the observation is common. This is the foundation of rational belief updating in the presence of new data, and it applies no matter how many hypotheses or how unlikely the evidence.