Bayes Flip: Heads Decides Coin

This question presents a simple-seeming coin-flipping experiment where one of the possible coins behaves very differently from the others. After making a random choice from a mostly ordinary collection, you observe an extreme run of identical outcomes. The candidate is asked to reason about how this surprising evidence should change their belief about which underlying object was selected. Versions of this setup are popular in quantitative trading, data science, and tech interviews, because they highlight how to think about rare mechanisms that produce very characteristic signals.

Solving it leans heavily on conditional probability and Bayes' theorem, together with careful distinction between prior probabilities and likelihoods. The interviewer is looking for someone who can translate the story into a clean probabilistic model, compute the relevant conditional terms, and then combine them correctly rather than relying on intuition. Clear articulation of hypotheses, correct handling of small probabilities, and the ability to explain why the posterior shifts so dramatically are just as important as getting the final numerical value.

What it tests

This problem class is governed by Bayesian inference, where you update your belief about an underlying cause (which coin you picked) after observing evidence (the sequence of flips). The key structure is that rare causes with certain outcomes can become as plausible as common causes with rare outcomes, once you condition on surprising evidence. The pattern is that the posterior probability balances the prior likelihood of each hypothesis against how well each explains the observed data. When the likelihood of the evidence under a rare hypothesis is much higher than under a common one, the rare hypothesis can become probable, even if it started out unlikely. This is why Bayes' Theorem often turns intuition on its head: it quantifies how evidence should shift your beliefs, not just confirm them.