Three Straight Heads in Five Flips

This classic combinatorics question is about patterns inside short sequences, a core theme in many quant interviews. Instead of focusing on individual flips, it focuses on the structure of the entire sequence and how a specific run can appear within it. Candidates must keep track of global constraints while reasoning locally about positions, which is central to many probability puzzles used in quant prep.

It trains pattern containment, exclusion, and careful case differentiation under tight constraints. You must manage overlapping configurations in a small state space and be precise about "exactly" versus "at least" when describing events. This hones discrete probability intuition, combinatorial counting discipline, and attention to edge cases, all of which are core skills in quant interviews.

For top trading firms, this kind of question matters because it exposes whether a candidate can think rigorously about randomness without brute force. Interviewers see how you translate an informal description into a mathematically clean event, control overcounting, and keep a clear mental model of all valid configurations. These are the same skills used later in modeling execution paths, rare events, and discrete-time trading strategies in real quant roles.

What it tests

Problems about runs of consecutive outcomes in sequences (like coin tosses) are governed by the principle of pattern containment and exclusion. The key is to count sequences that contain the desired run (here, exactly three consecutive heads) while carefully excluding those that extend the run beyond the specified length. This is because the combinatorial structure of sequences allows overlapping runs, so naive counting can easily overcount cases where the run is longer than required. The principle holds because the presence or absence of specific outcomes before and after the run determines whether the run is 'exactly' of the desired length, not just 'at least.' Understanding the boundaries of the run—what comes immediately before and after—is essential to avoid overcounting or undercounting.