Calculate Particle Survival Probability After Time t

Quick Overview

This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Calculate Particle Survival Probability After Time t states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Calculate Particle Survival Probability After Time t

Company: Upstart

Role: Data Scientist

Category: Statistics & Math

Difficulty: easy

Interview Round: Technical Screen

##### Scenario Radioactive-decay style process: 100 independent particles, each has a known half-life. Interviewer wants theoretical probability of how many survive after a given time. ##### Question Given 100 identical particles with half-life H (each decays independently with exponential distribution), derive the probability that exactly k (or at least one) particles remain undecayed after time t. ##### Hints Recall exponential survival function P(T>t)=e^{-λt} where λ=ln2/H; use Binomial distribution on independent survival events.

Quick Answer: This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Calculate Particle Survival Probability After Time t states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

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Aug 4, 2025, 10:55 AM
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Calculate Particle Survival Probability After Time t

Radioactive-Decay Style Probability

Context

You have 100 identical, independent particles. Each particle's lifetime is exponentially distributed with rate parameter λ corresponding to a known half-life H. For an exponential lifetime, the survival function is:

  • P(T > t) = e^{-λ t}, where λ = (ln 2) / H.

Task

  1. Derive the probability that exactly k particles (out of 100) remain undecayed after time t.
  2. Derive the probability that at least one particle remains undecayed after time t.

Assumptions

  • Particles decay independently.
  • All particles share the same half-life H (i.e., the same exponential rate λ).
  • Time t ≥ 0 and k ∈ {0, 1, ..., 100}.

Clarifying Questions to Ask Guidance

  • Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
  • Show enough derivation for the interviewer to follow the reasoning.
  • Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers Guidance

  • A correct setup with definitions, formulas, and boundary conditions.
  • A step-by-step derivation or estimation plan.
  • Interpretation of the result, including uncertainty and practical limitations.
  • Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions Guidance

  • How would the result change if the assumptions were relaxed?
  • Can you verify the answer with a simulation?
  • What is the most likely source of estimation error?
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