Jane Street Quantitative Trader Interview Question: Probability of Overlapping Random Intervals
Are you preparing for a Jane Street quantitative trader interview? One of the classic probability puzzles you may encounter deals with random intervals and their intersection properties. Understanding these questions not only tests your mathematical rigor but also your ability to model and analyze stochastic processes—a skill highly valued at quantitative trading firms like Jane Street. In this article, we’ll break down the famous "Probability of Overlapping Random Intervals" problem, explain key mathematical concepts involved, and guide you step by step through its solution.
Question: Five intervals are independently generated by sampling two points uniformly from [0,1] and taking them as endpoints. What is the probability that all five intervals share a common non-empty intersection?
Let’s formalize the question. For each interval \(I_i\) for \(i = 1, 2, 3, 4, 5\):
