Bayesian Coin Flip Bias Update

This question introduces a simple Bayesian updating problem using a random coin bias as the unknown parameter. A machine produces coins whose probabilities of landing heads vary from coin to coin, and one such coin is observed under different amounts of data. The candidate is asked to describe, in closed form, how the conditional distribution of the coin's bias changes after seeing a single favorable outcome, and how it looks after many repeated trials with a particular empirical frequency. It is a clean, one-parameter setting that forces you to think concretely about how evidence reshapes beliefs, rather than just computing a single point estimate.

To answer well, you need comfort with conditional probability, Bayes' theorem, and working with distributions over parameters rather than outcomes. The problem leans on recognizing conjugacy, manipulating likelihoods, and normalizing densities to obtain a valid cumulative distribution function. Interviewers look for an understanding of how the prior and data interact, how increasing sample size affects concentration of the posterior, and the ability to translate intuitive reasoning about "updating beliefs" into precise mathematical expressions. Candidates who manage the algebra cleanly while keeping the probabilistic interpretation straight tend to stand out.

What it tests

Whenever we have a parameter (like `P`, the probability of heads) drawn from a prior distribution and we observe data generated according to that parameter (like coin flips), the updated belief about the parameter is governed by Bayes' Theorem. The posterior distribution is proportional to the product of the prior and the likelihood of the observed data given the parameter. This mechanism is universal: the more data we observe, the more the likelihood dominates the prior, concentrating the posterior around values of the parameter that make the data most probable. The prior sets the initial plausibility for each value, but the likelihood function 'pulls' the posterior toward parameter values that explain the data well. This is why, as data accumulates, the posterior becomes sharply peaked at the empirical frequency, regardless of the initial prior (unless the prior is extremely strong or excludes the true value).