50% chance after 100 shots

This question considers a sequence of dependent Bernoulli-style trials in which the success probability at each step is tied to the empirical success rate so far. After fixing the outcomes of the first few shots, the candidate must reason about how this self-referential rule shapes the distribution of total successes after many attempts. The setup belongs to the family of self-reinforcing or feedback-driven processes, and feels similar to Polya urn models often seen in quant trading and research interviews that probe intuition for non-i.i.d. randomness.

To answer it, a candidate needs comfort with conditional probability, Markov chains, and working systematically with state-dependent transition probabilities. The key move is to recognize that the process can be described entirely by the current counts of successes and attempts, and to see how the recursive structure of transition probabilities aggregates into a simple distributional property. Interviewers look for the ability to strip away distracting narrative detail, express the evolution in a clean probabilistic framework, and then leverage symmetry or invariance rather than brute-force computation.

What it tests

This problem class is governed by self-reinforcing stochastic processes, where the probability of a future event depends on the empirical frequency of past outcomes. The key structure is that the process is Markovian: the probability of the next outcome depends only on the current state (number of successes and trials), not the full history. When the success probability at each step is the current empirical mean, the process exhibits a remarkable uniformity: all possible success counts (except the endpoints) become equally likely. This uniformity arises because the process is 'self-balancing': the more successes you have, the harder it is to get more, and vice versa, leading to a flattening of the distribution. The underlying pattern is that feedback mechanisms like this often produce surprising symmetries or uniformities, even when the process appears history-dependent.