Compute the Probability Two Bloom Intervals Overlap
Company: Sig
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Online Assessment
# Compute the Probability Two Bloom Intervals Overlap
A purple flower starts blooming at a time chosen uniformly at random from the continuous interval `[0, 60]` days and remains in bloom for 10 days. Independently, a red flower starts at a uniformly random time in `[0, 60]` and remains in bloom for 20 days. A bloom may extend beyond day 60.
What is the probability that their bloom intervals overlap for at least one instant? Express the answer as a fraction in simplest form.
### What a Strong Answer Covers
- A two-dimensional sample space for the independent start times.
- The two disjoint nonoverlap regions and their boundary inequalities.
- Correct triangle areas relative to the `60 by 60` square.
- Simplification of the overlap probability.
### Follow-up Questions
- Why do strict versus nonstrict inequalities not change this continuous probability?
- What formula results for durations `a` and `b` within a start window of length `T`?
Overview: Compute the overlap probability for two independently timed flower blooms with different durations. The geometric solution subtracts two nonoverlap triangles from the start-time square, handles boundary events and post-window bloom time, and simplifies the result to 31/72.
Read the full Sig Data Scientist interview experience this question came from