Four Heads in a Row Chance

This classic probability question is about understanding sequences of events under independence. It forces you to think precisely about repeated coin flips and what it really means for outcomes to "line up" across time. In quant prep, such problems build intuition for how paths of a stochastic process behave, which is directly relevant to pricing and risk scenarios involving repeated random shocks.

It trains conditional probability, independence, and careful event definition. You practice translating an informal requirement about streaks into a well-defined event in a probability space. It also builds comfort with simple path-based reasoning that later scales to Markov chains, random walks, and hitting times, core tools in quantitative finance interviews.

This matters in quant interviews because firms expect you to reason cleanly about randomness, not just plug formulas. This style of question reveals how you think about events, paths, and constraints, which underlies trading strategy design, backtesting, and risk modeling.

What it tests

When calculating the probability of a sequence of identical outcomes in repeated independent trials, the key is to recognize that the first outcome can be anything, and the condition is that all subsequent outcomes must match it. This reduces the problem to finding the probability that each of the remaining trials matches a fixed reference, rather than considering all possible sequences. The independence of the trials allows us to multiply the probabilities for each required match. This approach generalizes to any number of trials: after the first, each subsequent trial must independently match the first, so the probability is $(1/2)^{n-1}$ for $n$ coin flips. The principle holds because the first event is unconstrained, and only the alignment of the rest matters.