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Bytedance·Data Scientist·Technical Phone Screen·Intermediate

Intermediate
May 2026

Summary

Bytedance data scientist interview with a classic probability geometry problem. Pretty clean question but the derivation takes a minute to set up properly if you haven't seen it before.

Questions Asked (1)

Q1

A stick of length 1 is broken at two points chosen independently and uniformly at random. What is the probability that the three resulting pieces can form a triangle?

Algorithms & Data Structures
Author's notes

I'd seen this problem before so I knew the answer was 1/4, but actually deriving it cleanly under pressure is another thing.

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AI HintsAI Generated

Suggested Approach

Model the two break points as independent uniform random variables on [0,1]. The three pieces form a triangle if and only if each piece is less than 1/2. Compute the probability by integrating over the region in the unit square where this condition holds.

Pro tip: Emphasize that the triangle condition is equivalent to the longest piece being less than the sum of the other two, which simplifies to each piece < 1/2. This geometric insight avoids messy algebra and shows strong probabilistic intuition.

1. Define random variables

Let X and Y be the two break points, independently and uniformly distributed on [0,1]. Assume X < Y without loss of generality (by symmetry).

2. Express piece lengths

The three pieces have lengths X, Y - X, and 1 - Y. The triangle condition is that each length is less than 1/2.

3. Translate to inequalities

The conditions X < 1/2, Y - X < 1/2, and 1 - Y < 1/2 simplify to X < 1/2, Y < X + 1/2, and Y > 1/2.

4. Compute probability via geometry

In the unit square (X,Y) with X < Y, the valid region is a triangle with vertices (0, 1/2), (1/2, 1/2), and (1/2, 1). Its area is 1/8. Since the total area of the region X < Y is 1/2, the conditional probability is (1/8) / (1/2) = 1/4.

5. State final answer

The probability is 1/4. Optionally, verify by simulation or mention that the result is a classic problem.

Key Points to Mention

  • Uniform distribution of break points and independence.
  • Triangle inequality condition: each piece must be less than the sum of the other two, equivalent to each piece < 1/2.
  • Geometric representation in the unit square and area calculation.
  • Symmetry argument to assume X < Y and adjust probability accordingly.
  • The final probability is 1/4.
  • Connection to the classic 'broken stick' problem and its relevance to data science (e.g., probabilistic reasoning, geometric probability).

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