Two Bankers Meet at Noon

This interview question considers two people arriving within the same time window, each staying for a fixed duration, and asks for the probability that their visits overlap. The arrival times are modeled as independent and uniformly distributed over a given interval, and "meeting" is defined in terms of their time difference being small enough that both are present simultaneously. Variants of this setup appear in probability and brainteaser rounds for quant, trading, and risk roles, because it connects an intuitive real-world story (two bankers at a station) with a clean probabilistic model.

Solving it relies on translating the timing condition into geometry on a square representing all joint arrival times. The key technique is to recognize that independence and uniformity make every point in this square equally likely, so the desired probability is just an area ratio. Candidates are expected to identify the region where the overlap condition holds, understand its symmetry, and compute its area cleanly. Interviewers watch for correct conditioning, comfort moving between algebraic inequalities and geometric regions, and an ability to explain the reasoning clearly without getting lost in the story.

What it tests

Whenever two independent events are uniformly distributed over the same interval, and an outcome depends on their difference being within a fixed bound, the problem reduces to finding the area of a band around the diagonal in the square of possible outcomes. The key is that the joint probability is proportional to the area of the region where the condition holds, since uniformity means every point is equally likely. The width of the band is determined by the bound on the difference, and the remaining area outside the band often forms symmetric shapes (like triangles) that are easier to calculate. This geometric approach is powerful because it translates an abstract probability into a concrete area calculation. The principle holds because the independence and uniformity ensure that the probability is just the fraction of the total area where the event occurs.