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Compute the Probability Two Bloom Intervals Overlap

SIG (Susquehanna)Data Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

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Calculate the probability that two independent events with different known durations overlap within a fixed time window using geometric probability.

Problem Statement

In a botanical garden, two rare night-blooming flowers—Flower A and Flower B—each open once independently during a 6-hour observation window from 12:00 AM to 6:00 AM.

  • Flower A begins blooming at a random time uniformly distributed between 12:00 AM and 6:00 AM, and once open, it remains in bloom for exactly 2 hours.
  • Flower B begins blooming at a random time uniformly distributed between 12:00 AM and 6:00 AM, and once open, it remains in bloom for exactly 1 hour.
  • A flower can remain in bloom after 6:00 AM as long as its start time occurred within the 6-hour window.

Compute the exact probability that there is some point in time at which both flowers are blooming simultaneously (i.e., their blooming intervals overlap).

Questions to Answer

  1. Define the sample space and the mathematical condition for the two blooming intervals to overlap.
  2. Use a geometric probability approach to find the probability of the complement (non-overlapping intervals).
  3. State the final simplified fraction representing the probability of overlap.

Hints

Hint 1

Represent the start times of Flower A and Flower B as independent uniform random variables X and Y on the interval [0, 6], forming a square sample space of area 36.

Hint 2

Instead of computing the overlap region directly, compute the probability of the complement: the regions where Flower A finishes before Flower B starts, or Flower B finishes before Flower A starts.

Hint 3

The complement corresponds to two right triangles in the coordinate plane with right-angle side lengths (6 - 1) and (6 - 2).

Follow-up questions

  • How does the probability change if both flowers are constrained to completely finish their bloom by 6:00 AM (i.e., their start times must satisfy X <= 4 and Y <= 5)?
  • What would be the probability of overlap if the start times followed an exponential arrival process (Poisson process) instead of a uniform distribution?
  • If three flowers with durations d1, d2, and d3 choose independent uniform start times on [0, T], how would you set up the integral or geometric volume to find the probability that all three bloom concurrently?