20 分钟 90% 至少 1 辆车：5 分钟概率

The probability of a car passing a certain marker in a 20- minute window is $90\%$ . What is the probability of a car passing the marker in a 5- minute window? Assume that the cars are moving independently of each other.

Remark. This is an example of a problem that is favored by quite a few interviewers. The problem is not very suitable for an exam in a rigorous math course since its formulation is not very precise. In an active interview situation, a candidate has to first make a problem more rigorous by asking relevant questions to the interviewer. This discussion between the interviewer and the candidate is considered to be a part of the problem solving experience. The assumptions about the problem are made rigorous during this discussion.

The interviewer use the questions of this type to test the "modeling" abilities of the candidates. The candidate has to take a "real world problem" and create a model that can be addressed using the rigorous language and tools of mathematics. In this particular problem, once the formulation is made precise enough, the problem becomes much easier. In a way, the entire point of this problem is its translation from the "real world" to rigorous mathematics. The book format is not a very suitable medium for discussion in which the problem solver works towards creating a precise formulation from a more vague version, so we will just present an intermediate step in which a more precise formulation is displayed. However, if we were to post this more precise formulation instead of the vague one, then we would be showing our readers an "easier version," not the version that was appearing on the quant interviews.

More precise formulation: An infinitesimally small marker is placed on a road. The marker can be assumed to be a point in Euclidean geometry. During a time

interval of length 20 the probability that at least one car passes over the marker is $90\%$ . Determine the probability that at least one car passes over the marker during a time interval of length 5 .