Car Sightings in 5-Minute Windows

This interview question presents a simple-sounding observation process on a highway and asks the candidate to relate probabilities across different time scales. The setup specifies the chance of seeing at least one car over a longer time window and then asks for the corresponding chance over a shorter, nested window, under an assumption of stationarity. It is a typical conditional probability and time-scaling question that appears in quant and trading interviews, often to probe whether candidates can translate an intuitive story about random arrivals into a clean probabilistic formulation without overcomplicating it.

To answer it well, candidates must recognize the complement structure of "at least one" versus "none," and how independence across equal subintervals makes the longer-period probability factorizable. The key ideas include decomposing an interval into smaller independent blocks, working systematically with complements, and manipulating exponents when linking whole-interval and subinterval probabilities. Interviewers watch for clarity in setting up the equations, correct handling of independence and complements, and an ability to reverse-engineer local probabilities from aggregate information.

What it tests

When dealing with the probability of an event occurring at least once over a longer interval, and the process is stationary and independent across subintervals, the total probability can be decomposed into the product of the probabilities for each subinterval. The key is that the probability of 'no event' in the whole interval is the product of the probabilities of 'no event' in each subinterval, due to independence. This structure arises in Poisson processes and similar settings where events are memoryless and identically distributed across time. The general rule is: the probability of at least one event in the whole is one minus the probability of no events in all parts, and the latter is the product of the 'no event' probabilities for each part. This pattern holds because the independence assumption allows multiplication of probabilities, and the 'at least one' phrasing always complements to 'none at all.'