SIG (Susquehanna)·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jul 2026

Summary

A probability brain teaser from SIG for a Data Scientist role. Just the one question from what I can tell, classic quant-style stuff you'd expect from a trading firm.

Questions Asked (1)

Two flowers bloom independently. The purple flower starts blooming at a uniformly random time over the next 30 days and stays in bloom for 9 days. The red flower also starts at a uniformly random time over the same 30-day window and stays in bloom for 12 days. What is the probability that they overlap in bloom at some point?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

This is a geometric probability problem.

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AI HintsAI Generated

Suggested Approach

Model the start times as independent uniform random variables on [0,30] and compute the probability of overlap by integrating over the joint distribution. Alternatively, use geometric reasoning: the non-overlap condition is that one flower finishes before the other starts, which defines a region in the 2D start-time plane. Compute the area of the overlap region and divide by the total area (900).

Pro tip: Draw a diagram of the 30x30 square with axes representing start times. Shade the non-overlap regions (where red ends before purple starts or purple ends before red starts) and compute their combined area. This visual method is faster and less error-prone than integration, and it clearly communicates your reasoning to the interviewer.

1. Define random variables and sample space

Let X be the start time of the purple flower and Y be the start time of the red flower, both uniformly distributed on [0,30]. The sample space is the square [0,30] x [0,30] with area 900.

2. Express overlap condition

The flowers overlap if their bloom intervals intersect. The purple flower blooms from X to X+9, and the red from Y to Y+12. Overlap occurs if X <= Y+12 and Y <= X+9. The complement (no overlap) occurs if X+9 < Y or Y+12 < X.

3. Compute area of non-overlap region

In the (X,Y) plane, the non-overlap region consists of two right triangles: one where Y > X+9 (red starts after purple ends) and one where X > Y+12 (purple starts after red ends). Calculate the area of each triangle using the formula 0.5 * base * height.

4. Calculate probability

Sum the areas of the two triangles to get the total non-overlap area, then subtract from 900 to get the overlap area. Divide the overlap area by 900 to obtain the probability.

5. Verify and present result

Simplify the fraction and check that the probability is between 0 and 1. Present the final answer clearly, explaining each step.

Key Points to Mention

Uniform distribution of start times over a continuous interval.
Independence of the two start times.
Geometric interpretation: area in the 2D plane represents probability.
Complement rule: P(overlap) = 1 - P(no overlap).
Triangle area calculation for the non-overlap regions.
Final probability as a simplified fraction or decimal.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.